GALOIS ACTIONS ON HOMOTOPY GROUPS OF ALGEBRAIC VARIETIES

J.P.PRIDHAM

Abstract. We study the Galois actions on the ℓ-adic schematic and Artin–Mazur homotopy groups of algebraic varieties. For proper varieties of good reduction over a local field K, we show that the ℓ-adic schematic homotopy groups are mixed representations explicitly determined by the Galois action on cohomology of Weil sheaves, whenever ℓ is not equal to the residue characteristic p of K. For quasi-projective varieties of good reduction, there is a similar characterisation involving the Gysin spectral sequence. When ℓ = p, a slightly weaker result is proved by comparing the crystalline and p-adic schematic homotopy types. Under favourable conditions, a comparison theorem transfers all these descriptions to the Artin–Mazur homotopy groups π_{n}^{ét}(XK̄) ⊗ℤ̂ ℚ_{ℓ}.

Introduction

In [AM], Artin and Mazur introduced the étale homotopy type of an algebraic variety. This gives rise to étale homotopy groups π_{n}^{ét}(X, x̄); these are pro-finite groups, abelian for n ≥ 2, and π_{1}^{ét}(X, x̄) is the usual étale fundamental group. In [Toë, §3.5.3], an approach for defining ℓ-adic schematic homotopy types was discussed, giving ℓ-adic schematic homotopy groups ϖ_{n}(X, x̄); these are (pro-finite-dimensional) ℚ_{ℓ}-vector spaces when n ≥ 2. In [Ols1], Olsson introduced a crystalline schematic homotopy type, and established a comparison theorem with the p-adic schematic homotopy type.

Thus, given a variety X defined over a number field K, there are many notions of homotopy group:

• for each embedding K ↪ ℂ, both classical and schematic homotopy groups of the topological space X_{ℂ};

• the étale homotopy groups of XK̄;

• the ℓ-adic schematic homotopy groups of XK̄;

• over localisations K_{𝔭} of K, the crystalline schematic homotopy groups of X_{K_{𝔭}}. However, despite their long heritage, very little was known even about the relation between étale and classical homotopy groups, unless the variety is simply connected.

The étale and ℓ-adic homotopy types carry natural Galois actions, and the main aim of this paper is to study their structure. In many respects, the analogous question for X_{ℂ} has already been addressed, with [KPT1] and [Pri5] describing mixed Hodge structures on the classical and real schematic homotopy types.

In [Pri3], a new approach to studying non-abelian cohomology and schematic homotopy types of topological spaces was introduced. Its primary application was to transfer cohomological data (in particular mixed Hodge structures) to give information about homotopy groups. The bulk of this paper is concerned with adapting those techniques to pro-simplicial sets. This allows us to study Artin–Mazur homotopy types of algebraic varieties, and to translate Lafforgue’s Theorem and Deligne’s Weil II theorems into statements about homotopy types. We thus establish arithmetic analogues of the results of [Pri5], with Galois actions replacing mixed Hodge structures.

This work was supported by Trinity College, Cambridge; and by the Engineering and Physical Sciences Research Council [grant number EP/F043570/1].

The main comparison results are Proposition 1.39 (showing when étale homotopy groups are pro-finite completions of classical homotopy groups), Theorem 3.40 (describing ℓ-adic schematic homotopy groups in terms of étale homotopy groups), and Proposition 7.26 (comparing p-adic and crystalline homotopy groups).

If X is smooth or proper and normal, then Corollary 6.7 shows that the Galois actions on the ℓ-adic schematic homotopy groups are mixed, with Remark 6.9 indicating when the same is true for étale homotopy groups. Corollaries 6.11 and 6.16 then show how to determine ℓ-adic schematic homotopy groups of smooth varieties over finite fields as Galois representations, by recovering them from cohomology groups of smooth Weil sheaves, thereby extending [Pri4] from fundamental groups to higher homotopy groups, and indeed to the whole homotopy type. Corollaries 7.4 and 7.36 give similar results for ℓ-adic and p-adic homotopy groups of varieties over local fields.

The structure of the paper is as follows.

In Section 1, we recall standard definitions of pro-finite homotopy types and homotopy groups, and then establish some fundamental results. Proposition 1.29 shows how Kan’s loop group can be used to construct the pro-finite completion X̂ of a space X, and Proposition 1.39 describes homotopy groups of X̂.

Section 2 reviews the pro-algebraic homotopy types of [Pri3], with the formulation of multipointed pro-algebraic homotopy types from [Pri5], together with some new material on hypercohomology.

We adapt these results in Section 3 to define non-abelian cohomology of a variety with coefficients in a simplicial algebraic group over ℚ_{ℓ}. The machinery developed in [Pri3] applies to give a pro-ℚ_{ℓ}-algebraic homotopy type, which is a non-nilpotent generalisation of the ℚ_{ℓ}-homotopy type of Weil II ([Del2]). Its homotopy groups are ℓ-adic schematic homotopy groups, and Theorem 3.40 gives conditions for relating these to étale homotopy groups. Explicitly, if π_{1}X is algebraically good (see Definition 3.35), and the higher homotopy groups have finite rank, then the higher homotopy groups of the pro-ℚ_{ℓ}-algebraic homotopy type are just π_{n}^{ét}X ⊗ℤ̂ ℚ_{ℓ}. For complex varieties, we also compare the proalgebraic homotopy types associated to the étale and analytic topologies.

Section 4 contains technical results showing how to extend the machinery of Section 3 to relative and filtered homotopy types. The former facilitate p-adic Hodge theory, while the latter are developed in order to study quasi-projective varieties. We also explore what it means for a pro-discrete group to act algebraically on a homotopy type. In Section 5, we investigate properties of homotopy types endowed with algebraic Galois actions.

In Section 6, the techniques of [Pri4] for studying Galois actions on algebraic groups then extend the finite characteristic results of [Pri1] to non-nilpotent and higher pro-ℚ_{ℓ}- algebraic homotopy groups. The results are similar to [Pri5], substituting Frobenius actions for Hodge structures. Over finite fields, Theorem 6.10 uses Lafforgue’s Theorem and Deligne’s Weil II theorems to show that the pro-ℚ_{ℓ}-algebraic homotopy type of a smooth projective variety is formal — this means that it can be recovered from cup products on cohomology of local systems. For quasi-projective varieties, Corollary 6.15 establishes a related property we call quasi-formality, which is analogous to Morgan’s description of the rational homotopy type ([Mor]) in terms of the Leray spectral sequence.

Section 7 then addresses the same question, but over local fields. In unequal characteristic, smooth specialisation suffices to adapt results from finite characteristic for varieties with good reduction. In equal characteristic, we show how pro-ℚ_{p}-algebraic homotopy types relate to the framework of p-adic Hodge theory. Proposition 7.26 is a reworking of Olsson’s non-abelian p-adic Hodge theory, and this has various consequences for Galois actions on Artin–Mazur homotopy types (Theorems 7.28–7.35). Explicitly, the homotopy type becomes formal as a Galois representation only after tensoring with the ring B_{cris}^{σ} of Frobenius-invariant periods, which means that the Hodge filtration is the only really new structure on the relative Malcev homotopy type (Remark 7.37.2).

Acknowledgements. I would like to record my sincere thanks to the team of referees for patiently reading through the manuscript. As well as identifying numerous errors, their suggestions have greatly improved the exposition.

Contents

1. Pro-finite homotopy types

Definition 1.1. Let 𝕊 be the category of simplicial sets, and take sGpd to consist of those simplicial objects in the category of groupoids whose spaces of objects are discrete (i.e. sets, rather than simplicial sets).

Let Top denote the category of compactly generated Hausdorff topological spaces.

Definition 1.2. Given G ∈ sGpd, we define π_{0}G to be the groupoid with objects Ob G, and morphisms (π_{0}G)(x, y) = π_{0}G(x, y).

Definition 1.3. A map f: X → Y in Top is said to be a weak equivalence if it gives an isomorphism π_{0}X → π_{0}Y on path components, and for all x ∈ X, the maps π_{n}(f): π_{n}(X, x) → π_{n}(Y, fx) are all isomorphisms.

We give 𝕊 the model structure of [GJ, Theorem V.7.6]; in particular, a map f: X → Y in 𝕊 is said to be a weak equivalence if the map |f|: |X| → |Y | of topological spaces is so, where | · | is the realisation functor of [GJ, §I.2]. Likewise, for x ∈ X_{0} we write π_{n}(X, x) := π_{n}(|X|, x).

A map f: G → H in sGpd is a weak equivalence if the map π_{0}G → π_{0}H is an equivalence, and for all objects x ∈ Ob G, the maps π_{n}(G(x, x)) → π_{n}(H(fx, fx)) are all isomorphisms.

For each of these categories, we define the corresponding homotopy categories Ho(𝕊), Ho(Top), Ho(sGpd) by localising at weak equivalences.

Note that there is a functor from Top to 𝕊 which sends X to the simplicial set

Sing(X)_{n} = Hom_{Top}(|Δ^{n}|, X).

This is right adjoint to realisation, and these functors are a pair of Quillen equivalences, so become quasi-inverse on the corresponding homotopy categories. From now on, we will thus restrict our attention to simplicial sets.

Definition 1.4. Given G ∈ sGpd, define the category 𝕊_{G} of G-spaces to consist of simplicial representations of G. Explicitly, X ∈ 𝕊_{G} consists of X(a) ∈ 𝕊 for each a ∈ Ob G, together with maps G(a, b) × X(b) → X(a), satisfying the obvious associativity and unit axioms.

Definition 1.5. Recall from [GJ, §V.4] that for G ∈ sGpd, the G-space W G is defined by

(W G)_{n}(x) = ∐_{y_{n},...,y_{0}∈Ob G} G_{n}(x, y_{n}) × G_{n−1}(y_{n}, y_{n−1}) × … × G(y_{1}, y_{0})

with operations

∂_{i}(g_{n}, g_{n−1}, …, g_{0}) = \begin{cases}(∂_{i}g_{n}, ∂_{i−1}g_{n−1}, …, (∂_{0}g_{n−i})g_{n−i−1}, g_{n−i−2}, …, g_{0}) & i < n, \\ (∂_{n}g_{n}, ∂_{n−1}g_{n−1}, …, ∂_{1}g_{1}) & i = n, \end{cases}
σ_{i}(g_{n}, g_{n−1}, …, g_{0}) = (σ_{i}g_{n}, σ_{i−1}g_{n−1}, …, σ_{0}g_{n−i}, id, g_{n−i−1}, …, g_{0}),

and for h ∈ G_{n}(z, x) and (g_{n}, g_{n−1}, …, g_{0}) ∈ (W G)(x),

h(g_{n}, g_{n−1}, …, g_{0}) = (hg_{n}, g_{n−1}, …, g_{0}).

Note that W G(x) is contractible for each x ∈ Ob G.

Definition 1.6. As in [GJ, Ch.V.7], there is a classifying space functor W̄: sGpd → 𝕊, given by W̄ G = G\W G, the co-invariants of the G-action. This has a left adjoint G: 𝕊 → sGpd, Dwyer and Kan’s loop groupoid functor ([DK]), and these form a pair of Quillen equivalences, so give equivalences Ho(𝕊) ∼ Ho(sGpd). The objects of G(X) are X_{0}, and for any x, y ∈ X_{0}, the geometric realisation |G(X)(x, y)| is weakly equivalent to the space of paths from x to y in |X|. These functors have the additional properties that π_{0}G(X) ≅ π_{f}|X| (the fundamental groupoid), π_{f}(|W̄ G|) ≅ π_{0}G, π_{n}(G(X)(x, x)) ≅ π_{n+1}(|X|, x) and π_{n+1}(|W̄ G|, x) ≅ π_{n}(G(x, x)). This allows us to study simplicial groupoids instead of topological spaces.

Definition 1.7. If X ∈ 𝕊, then a local system is just a representation of the groupoid π_{f}X, i.e. a functor π_{f}X → Gp from the fundamental groupoid to the category of groups. As in [GJ, §VI.5], homotopy groups form a local system π_{n}X, whose stalk at x is π_{n}(X, x).

1.1. Pro-simplicial L-groupoids.

Definition 1.8. Given a set L of primes, we say that an L-group is a finite group G for which only primes in L divide its order. We define an L-groupoid to be a groupoid H for which H(x, x) is an L-group for all x ∈ Ob H.

Definition 1.9. Given a category C, recall that the category pro(C) of pro-objects in C has objects consisting of filtered inverse systems {A_{α}} in C, with

Hom_{pro(C)}(\{A_{α}\}, \{B_{β}\}) = lim_{\overleftarrow{β}} lim_{\overrightarrow{α}} Hom_{C}(A_{α}, B_{β}).

Remark 1.10. A discrete topological space is just a set. Given a pro-set {X_{α}}, we can thus take the limit \underleftarrow{lim}_{α} X_{α} in the category of topological spaces. This functor gives a faithful embedding of pro-finite sets into topological spaces, so for X_{α} finite, \underleftarrow{lim}_{α} X_{α} is discrete if and only if {X_{α}} lies in the essential image of Set → pro(Set). Motivated by this correspondence, we will refer to the essential image of Set → pro(Set) as the discrete objects.

In fact, pro-sets endow a topological structure which cannot be detected by weak equivalences, which is why shape theory is modelled using the category pro(𝕊), as in [Isa].

Definition 1.11. Given a groupoid G and a set L of primes, define G^{∧_{L}} ∈ pro(Gpd) by requiring that G^{∧_{L}} be the completion of G with respect to all L-groupoids H. In other words, G^{∧_{L}} is an inverse system of L-groupoids, with a canonical map G → G^{∧_{L}} inducing isomorphisms

Hom(G^{∧_{L}}, H) → Hom(G, H)

for all L-groupoids H.

In particular, Ob G^{∧_{L}} = Ob G and G^{∧_{L}}(x, x) is the pro-L completion of the group G(x, x) (in the sense of [Fri, §6]). If L is the set of all primes, we write Ĝ := G^{∧_{L}}, so Ĝ(x, x) is the pro-finite completion of G(x, x) (in the sense of [Ser2, §1]).

Note that G^{∧_{L}} is a pro-L-groupoid in the sense of Definition 1.9. However, beware that a pro-groupoid can be isomorphic to a pro-L-groupoid without actually being an inverse system of L-groupoids, since {Γ_{α}}_{α∈I} ≅ {Γ_{α}}_{α≥α_{0}} for any α_{0} ∈ I.

Definition 1.12. Say that a simplicial groupoid Γ is a simplicial L-groupoid if Γ_{i} is an L-groupoid for all i. Denote the category of such groupoids by sGpd^{L}.

Definition 1.13. Given a groupoid Γ, define a disconnected normal subgroupoid K ⊲ Γ to consist of subgroups K(x) ≤ Γ(x, x) for all x ∈ Ob Γ, with aK(x)a^{−1} ∈ K(y) for all a ∈ Γ(y, x).

Note that disconnected normal subgroupoids K ⊲ Γ are in one-to-one correspondence with isomorphism classes of those surjections f: Γ → H for which Ob f: Ob Γ → Ob H is an isomorphism. The equivalence is given by setting H(x, y) = Γ(x, y)/K(y) = K(x)\Γ(x, y), and conversely by setting K(x) := ker(f: Γ(x, x) → H(fx, fx)).

Definition 1.14. Given Γ ∈ sGpd, define a simplicial disconnected normal subgroupoid K ⊲Γ to consist of disconnected normal subgroupoids K_{n}⊲Γ_{n}, closed under the operations ∂_{i}, σ_{j}.

Definition 1.15. Given Γ ∈ sGpd, define Γ^{∧_{L}} ∈ pro(sGpd^{L}) to be the inverse system {Γ/K}_{K}, where K ranges over the poset of all simplicial disconnected normal subgroupoids K ⊲ Γ for which Γ/K is a simplicial L-groupoid.

Given Γ = {Γ_{α}}_{α} ∈ pro(sGpd), define Γ^{∧_{L}} ∈ pro(sGpd^{L}) by

Γ^{∧_{L}} = lim_{\overleftarrow{α}} Γ_{α}^{∧_{L}},

where the limit is taken in pro(sGpd^{L}). This corresponds to saying that Γ^{∧_{L}} is the proobject {Γ_{α}/K_{α}}_{(α,K_{α})} indexed by pairs (α, K_{α}), for K_{α} ⊲ Γ_{α}.

Lemma 1.16. For Γ ∈ pro(sGpd) and A ∈ pro(sGpd^{L}), the canonical map

Hom_{pro(sGpd^{L})}(Γ^{∧_{L}}, A) → Hom_{pro(sGpd)}(Γ, A)

is an isomorphism.

Proof. By the definition of morphisms in pro-categories, it suffices to prove this when A ∈ sGpd^{L}. Then A is cofinite in both pro(sGpd^{L}) and pro(sGpd) (i.e. Hom(\underleftarrow{lim}_{α} Γ_{α}, A) ≅ \underrightarrow{lim}_{α} Hom(Γ_{α}, A) for filtered inverse systems \{Γ_{α}\}_{α}), so we may also assume that Γ ∈ sGpd.

Now, for any morphism f: Γ → A, the image H is a simplicial L-groupoid of the form H = Γ/K, for K ⊲ Γ a disconnected normal subgroupoid. Therefore

Hom_{sGpd}(Γ, A) = \underrightarrow{lim}_{K} Hom_{sGpd}(Γ/K, A) = Hom_{pro(sGpd)}(Γ^{∧_{L}}, A),

as required. □

Lemma 1.17. For Γ ∈ pro(sGpd), the pro-L-groupoid (Γ^{∧_{L}})_{n} is just the pro-L completion of Γ_{n}.

Proof. Given A ∈ Gpd^{L}, define A^{Δ_{n}} (not to be confused with A^{Δ^{n}}) to be the simplicial groupoid on objects Ob A with

A^{Δ_{n}}(x, y)_{i} := A(x, y)^{Δ_{n}^{i}},

with ∂_{j}: (A^{Δ_{n}})_{i} → (A^{Δ_{n}})_{i−1} coming from ∂^{j}: Δ^{i−1} → Δ^{i}, and σ_{j} coming from σ^{j}: Δ^{i+1} → Δ^{i}. Then A^{Δ_{n}} is clearly an L-groupoid, and has the key property that

Hom_{sGpd}(Γ, A^{Δ_{n}}) ≅ Hom_{Gpd}(Γ_{n}, A)

for all Γ.

Taking colimits extends this to all Γ ∈ pro(sGpd), and then

Hom_{pro(sGpd^{L})}(Γ^{∧_{L}}, A^{Δ_{n}}) ≅ Hom_{pro(Gpd^{L})}((Γ^{∧_{L}})_{n}, A),

but the left-hand side is just

Hom_{pro(sGpd)}(Γ, A^{Δ_{n}}) ≅ Hom_{pro(Gpd)}(Γ_{n}, A),

so (Γ^{∧_{L}})_{n} is the pro-L completion of Γ_{n}. □

Definition 1.18. Given X = {X_{α}} ∈ pro(𝕊), define the category of local systems on X to be the direct limit (over α) of the categories of local systems on X_{α} (in the sense of Definition 1.7).

Remark 1.19. Our motivation for working with pro(𝕊) comes from [Fri, Definition 4.4], which associates an object X_{ét} ∈ pro(𝕊) to each locally Noetherian simplicial scheme X. Finite local systems on X_{ét} then correspond to finite locally constant étale sheaves on X.

Definition 1.20. Given a pro-simplicial set X, and a map π_{f}X → Γ to a pro-groupoid with discrete objects, define the covering system X̃ by

X̃(a) := X ×_{BΓ} B(Γ↓a) ∈ pro(𝕊)

for a ∈ Ob Γ, noting that this is equipped with a natural associative action Γ(a, b)×X̃(a) → X̃(b) in pro(𝕊). Here, B is the nerve functor (equal to W̄ in this context), and Γ↓a denotes the slice category of morphisms in Γ with target a.

Definition 1.21. Given π_{f}X → Γ as above, with a continuous representation S of Γ in pro-sets (i.e. S(a) ∈ pro(Set) for a ∈ Ob Γ, equipped with an associative action Γ(a, b) × S(a) → S(b) of pro-sets), define the cosimplicial set C^{•}(X, S) by

C^{n}(X, S) := Hom_{Γ,pro(Set)}(X̃_{n}, S).

From now on, local systems will be abelian unless stated otherwise.

Definition 1.22. Given X = {X_{α}} ∈ pro(𝕊) and a local system M on X_{β} define cohomology groups by

H^{∗}(X, M) := lim_{\overrightarrow{α}} H^{∗}(X_{α}, M),

where H^{∗}(X_{α}, −) is cohomology with local coefficients, and we also write M for the pullbacks of M to X_{α} and to X. Given G ∈ pro(sGpd), set H^{∗}(G, −) := H^{∗}(W̄ G, −).

Note that the cosimplicial complex C^{•}(X, M) extends [GJ, §VI.4] to pro-spaces, and that H^{∗}(X, M) = H^{∗}(C^{•}(X, M)), the cohomology groups with local coefficients.

Definition 1.23. Given X ∈ pro(𝕊) with X_{0} discrete, and an inverse system M = {M_{i}}_{i∈ℕ} of local systems on X, define the continuous cohomology groups H^{∗}(X, M) as follows. First form the cosimplicial complex C^{•}(X, M) := \underleftarrow{lim} C^{•}(X, M_{i}), for C^{•} as in Definition 1.21, then set

H^{∗}(X, M) := H^{∗}(C^{•}(X, M)),

noting that this agrees with Definition 1.22 when M_{i} = M for all i.

Remark 1.24. Observe that there is a short exact sequence

0 → \underleftarrow{lim}^{1}H^{n−1}(X, M_{i}) → H^{n}(X, M) → \underleftarrow{lim} H^{n}(X, M_{i}) → 0,

so H^{n}(X, M) ≅ \underleftarrow{lim} H^{n}(X, M_{i}) whenever the inverse system \{H^{n−1}(X, M_{i})\}_{i∈ℕ} satisfies the Mittag–Leffler condition (for instance if the groups are finite).

When working with the étale homotopy type X_{ét}, we will usually apply this construction to ℤ_{ℓ}-local systems {M_{i} = M/ℓ^{i}}_{i}. In that case, the exact sequence above becomes the comparison between étale cohomology and Jannsen’s continuous étale cohomology (see Example 3.18 for details).

Lemma 1.25. Given X ∈ 𝕊 and an inverse system M = {M_{i}}_{i∈ℕ} of local systems on X, there is an isomorphism

H^{∗}(X, \underleftarrow{lim} M_{i}) ≅ H^{∗}(X, M).

Proof. As in Definition 1.22, H^{∗}(X, \underleftarrow{lim} M_{i}) is cohomology of the complex \underleftarrow{lim} C^{•}(X, M_{i}) = C^{•}(X, \underleftarrow{lim} M_{i}), but

C^{n}(X, \underleftarrow{lim} M_{i}) = Hom_{Set}(X_{n}, \underleftarrow{lim} M_{i}) = \underleftarrow{lim} Hom_{Set}(X_{n}, M_{i}) = C^{n}(X, M),

as required. □

We will occasionally refer to groups and groupoids as “discrete”, to distinguish them from topological (or simplicial) groups and groupoids. As in Remark 1.10, we regard a pro-groupoid as analogous to a topological groupoid, so “discrete” will indicate that both simplicial and pro structures are trivial.

Definition 1.26. Given a set L of primes, say that a pro-groupoid G with discrete object set is (L, n)-good if for all G^{∧_{L}}-representations M in abelian L-groups, the canonical map

ϕ_{M} : H^{i}(G^{∧_{L}}, M) → H^{i}(G, M)

is an isomorphism for all i ≤ n and an inclusion for i = n + 1. When L is the set of all primes, we say that G is n-good. Observe that any inverse system of (L, n)-good groupoids is (L, n)-good. Say that G is L-good if it is (L, n)-good for all n.

Lemma 1.27. Free groups are L-good for all L.

Proof. Let F = F (X) be a free group generated by a set X, and let Γ := F^{∧_{L}}. By the argument of [Ser2, I§2.6 Ex. 1(a)], it suffices to show that H^{∗}(Γ, M) → H^{∗}(F, M) is surjective for all discrete Γ-representations M in abelian L-groups. Since F is free, H^{n}(F, M) = 0 for n > 1, so it only remains to establish surjectivity for n = 1.

This amounts to showing that every derivation α: F → M factors through Γ. The derivation gives rise to a map β: F → M ⋊ G, for some finite L-torsion quotient G of F. Since M ⋊ G is an L-group, β factors through Γ. □

Examples 1.28. (1) L-groups are L-good.

(2) If 1 → F → Γ → Π → 1 is an exact sequence of groups, with F and Π L-good, F^{∧_{L}} → Γ^{∧_{L}} injective, and H^{a}(F, M) finite for all finite L-torsion Γ-modules, then Γ is L-good.

(3) All finitely generated nilpotent groups are L-good for all L.

(4) The fundamental group of a compact Riemann surface is L-good for all L.

Proof. (2) This is essentially [Ser2, I§2.6 Ex. 2(c)].

(3) Express Γ as a successive extension of finite groups and ℤ, then apply (2).

(4) Choose a smooth complex projective curve C of genus g > 0, with π_{1}(C) = Γ. It suffices to show that for all finite L-torsion Γ^{∧_{L}}-representations M, the map

H^{∗}(Γ^{∧_{L}}, M) → H_{ét}^{∗}(C, M)

is an isomorphism.

Letting C̃ be the universal étale pro-L cover of C, this is equivalent (by the Serre spectral sequence) to showing that H_{ét}^{∗}(C̃, 𝔽_{p}) = 𝔽_{p} for all p ∈ L. C̃ is the inverse limit all finite L-covers C′ → C, giving

H_{ét}^{i}(C̃, 𝔽_{p}) = lim_{\overleftarrow{C′}} H_{ét}^{i}(C′, 𝔽_{p}),

which can only be non-zero for i = 0, 1, 2.

Note that π_{1}(C̃) = ker(Γ̂ → Γ^{∧_{L}}). Thus the pro-L completion π_{1}(C̃)^{ab,L} of the abelianisation of π_{1}(C̃) must be 0, or we would have a larger pro-L quotient of Γ̂ than Γ^{∧_{L}}. Hence H_{ét}^{1}(C̃, 𝔽_{p}) = 0 for all p ∈ L.

We now adapt the proof of [Sch, Proposition 15]. Since any curve C′ has a cover C′′ of degree p, with the map H_{ét}^{2}(C′, 𝔽_{p}) → H_{ét}^{2}(C′′, 𝔽_{p}) thus being 0, we deduce that H_{ét}^{2}(C̃, 𝔽_{p}) = 0, which completes the proof.

□

Proposition 1.29. For any X ∈ 𝕊, the canonical morphism

X → W̄ (G(X)^{∧_{L}})

in pro(𝕊) induces an isomorphism (π_{f}X)^{∧_{L}} → π_{f} W̄ (G(X)^{∧_{L}}) of pro-groupoids, and has the property that for all finite abelian (π_{f}X)^{∧_{L}}-representations M in L-groups, the canonical map

H^{∗}(W̄ (G(X)^{∧_{L}}), M) → H^{∗}(X, M)

is an isomorphism.

Proof. The statement about fundamental groupoids is immediate, since completion commutes with taking quotients. Now, observe that

H^{n}(W̄ (G(X)^{∧_{L}}), M) ≅ H^{n}(G(X)^{∧_{L}}, M),

tautologically from Definition 1.22.

It thus suffices to show that the simplicial groupoid G(X) is L-good, in the sense that H^{∗}(G(X), M) ≅ H^{∗}(G(X)^{∧_{L}}, M) for all π_{0}G(X)^{∧_{L}}-representations in abelian L-groups M. This is equivalent to showing that for all x ∈ X_{0}, the simplicial groups G(X)(x, x) are L-good. This will follow if the groups G_{n}(x, x) are all L-good, because there is a spectral sequence

H^{q}(G_{p}, M) ⟹ H^{p+q}(G, M).

Since the groups G_{n}(x, x) are all free, this then follows from Lemma 1.27. □

Given a property P of groups, we will say that a groupoid Γ locally satisfies P if the groups Γ(x, x) satisfy P, for all x ∈ Ob Γ.

Definition 1.30. Define pro(𝕊)_{δ} to be the full subcategory of pro(𝕊) consisting of prospaces X for which X_{0} is discrete (as in Remark 1.10, so X_{0} is a set, not just a pro-set).

Define 𝕊^{∧_{L}} to be the full subcategory of pro(𝕊)_{δ} consisting of spaces X for which the groups π_{n}(X, x) are all pro-L-groups. If L is the set of all primes, we write 𝕊̂ := 𝕊^{∧_{L}}.

Definition 1.31. A morphism f: X → Y in pro(𝕊)_{δ} is said to be an Artin–Mazur weak equivalence if π_{0}X → π_{0}Y is an isomorphism, and the maps π_{n}(X, x) → π_{n}(Y, fx) are pro-isomorphisms for all n ≥ 1 and all x ∈ X_{0}.

Define Ho(pro(𝕊)_{δ}) and Ho(𝕊^{∧_{L}}) by formally inverting all Artin–Mazur weak equivalences.

In [Isa], Isaksen established a model structure on pro(𝕊) with the right properties for modelling pro-homotopy types. In particular, [Isa, Corollary 7.5] shows that a morphism in pro(𝕊)_{δ} is a weak equivalence in pro(𝕊) if and only if it is an Artin–Mazur weak equivalence.

Proposition 1.32. Fix N ∈ [1, ∞], and let f: X → Y be a morphism in pro(𝕊)_{δ} such that (π_{f}X)^{∧_{L}} → (π_{f}Y)^{∧_{L}} is a pro-equivalence of pro-groupoids, with the property that for all abelian (π_{f}Y)^{∧_{L}}-representations M in L-groups, the map

H^{n}(f): H^{n}(Y, M) → H^{n}(X, M)

is an isomorphism for all n ≤ N and injective for n = N + 1. Then for all Z ∈ 𝕊^{∧_{L}} with π_{i}Z = 0 for i > N (resp. i > N + 1), the map

f^{∗}: Hom_{Ho(pro(𝕊)_{δ})}(Y, Z) → Hom_{Ho(pro(𝕊)_{δ})}(X, Z)

is an isomorphism (resp. an inclusion).

Proof. First observe that if M is a π_{f}(Y)^{∧_{L}}-representation in abelian pro-L groups, we can express it as an inverse system {M_{α}} of π_{f}(Y)-representations in L-groups. Then the complex C^{•}(Y, M) of M-cochains is given by

C^{•}(Y, M) ≃ R \underleftarrow{lim}_{α} C^{•}(Y, M_{α}).

This implies that for all such M, the map H^{n}(f): H^{n}(Y, M) → H^{n}(X, M) is an isomorphism for all n ≤ N, and injective for n = N + 1.

Now consider the Moore–Postnikov tower ([GJ, Definition VI.3.4]) P_{n}Z of a fibrant replacement for Z. The pro-equivalence on π_{f} gives the required isomorphism if Z = P_{1}Z, and we can proceed by induction.

Assume that we have a homotopy class of maps X → P_{n}Z, for n < N. The obstruction to lifting this to a homotopy class of maps X → P_{n+1}Z lies in H^{n+2}(X, π_{n+1}Z), and if non-empty, the latter homotopy class is a principal H^{n+1}(X, π_{n+1}Z)-space. As π_{n+1}Z is a pro-L-group, the isomorphism H^{n+1}(Y, −) ≅ H^{n+1}(X, −) and the inclusion H^{n+2}(Y, −) ↪ H^{n+2}(X, −) (resp. the inclusion H^{n+1}(Y, −) ↪ H^{n+1}(X, −)) mean that the pro-homotopy class of lifts Y → P_{n+1}Z is similarly determined (resp. embeds into the class of lifts X → P_{n+1}Z), completing the inductive step.

Since the map Z → P_{N}Z (resp. Z → P_{N+1}Z) is an Artin–Mazur weak equivalence, this completes the proof for N < ∞. In the case N = ∞, the analysis above gives an isomorphism

f^{∗}: Hom_{Ho(pro(𝕊)_{δ})}(Y, \underleftarrow{lim}_{n} P_{n}Z) → Hom_{Ho(pro(𝕊)_{δ})}(X, \underleftarrow{lim} P_{n}Z);

since the canonical map Z → \underleftarrow{lim}_{n} P_{n}Z is an Artin–Mazur weak equivalence, this completes the proof. □

Corollary 1.33. The inclusion functor 𝕊^{∧_{L}} → pro(𝕊)_{δ} has a homotopy left adjoint, which we denote by X ⇝ X^{∧_{L}}. This has the property that for X ∈ 𝕊^{∧_{L}}, X^{∧_{L}} ≃ X.

Proof. Propositions 1.29 and 1.32 imply that for X ∈ 𝕊, the object X^{∧_{L}} := W̄ (G(X)^{∧_{L}}) ∈ 𝕊^{∧_{L}} has the required properties. Given an inverse system X = {X_{α}}, set X^{∧_{L}} := \underleftarrow{lim}(X_{α})^{∧_{L}}. □

Remarks 1.34. Comparing with [Fri, Theorem 6.4 and Corollary 6.5], we see that this gives a generalisation of Artin and Mazur’s pro-L homotopy type ([AM]) to unpointed spaces. Their context for pro-homotopy theory was formulated slightly differently, in terms of pro(Ho(𝕊)), which is not very well-behaved. See [Isa] for details of the comparison.

Since this paper was first written, an alternative pro-finite completion functor has been developed in [Qui1]. However, the category of pro-finite homotopy types in [Qui1] is larger than ours, because for its pro-spaces X, the pro-set π_{0}X is pro-finite rather than discrete. The pro-finite completion functor thus differs from ours in that it also takes the pro-finite completion of the set π_{0}X.

An important feature of [Qui1] is the existence of a model structure for pro-finite spaces, and this raises the question of whether there is a model structure on pro(sGpd^{L}), and how the respective model structures compare. The most likely solution is that there is a fibrantly cogenerated model structure on pro(sGpd_{F}^{L}), where sGpd_{F}^{L} is the full subcategory of sGpd^{L} consisting of simplicial groupoids with finite object set. For this model structure, the cogenerating fibrations should be morphisms in sGpd_{F}^{L} which are fibrations in sGpd, possibly with some additional Artinian condition analogous to [Pri7, Theorem 2.14]. The right adjoint pro(sGpd_{F}^{L}) → pro(sGpd^{L}) should then induce a fibrantly cogenerated structure on the latter, while the functor W̄ from pro(sGpd_{F}^{L}) to simplicial pro-finite sets should be a right Quillen equivalence when L is the set of all primes.

1.2. Comparing homotopy groups. We now investigate when we can describe the homotopy groups of X^{∧_{L}} in terms of the homotopy groups of X.

Lemma 1.35. If A is a finitely generated abelian group, then for n ≥ 2, completion of the Eilenberg-Maclane space is given by K(A, n)^{∧_{L}} = K(A^{∧_{L}}, n).

Proof. By Proposition 1.32, we need to show that the maps

H^{∗}(K(A^{∧_{L}}, n), M) → H^{∗}(K(A, n), M)

are isomorphisms for all abelian L-groups M. By considering the spectral sequence associated to a filtration, it suffices to consider only the cases M = 𝔽_{p}, for p ∈ L.

If A = A′ × A′′, then K(A, n) = K(A′, n) × K(A′′, n), so H^{∗}(K(A, n), 𝔽_{p}) = H^{∗}(K(A′, n), 𝔽_{p}) ⊗ H^{∗}(K(A′′, n), 𝔽_{p}). The structure theorem for finitely generated abelian groups therefore allows us to assume that A = ℤ/q, for q a prime power or 0.

Now, if q is neither zero nor a power of p, then H^{r}(K(A, n), 𝔽_{p}) = 0 for r > 0; since A^{∧_{L}} is a quotient of A, we also get H^{r}(K(A^{∧_{L}}, n), 𝔽_{p}) = 0. If q = p^{s}, then A^{∧_{L}} = A, making isomorphism automatic.

If q = 0, then A = ℤ, A^{∧_{L}} = ∏_{ℓ∈L} ℤ_{ℓ}, and H^{r}(K(ℤ_{ℓ}, n), 𝔽_{p}) = 0 for r > 0 and ℓ ≠ p. We need to show that

H^{∗}(K(ℤ_{p}, n), 𝔽_{p}) → H^{∗}(K(ℤ, n), 𝔽_{p})

is an isomorphism, or equivalently that K(ℤ, n)^{∧_{p}} = K(ℤ_{p}, n). This follows from [Qui3, Theorem 1.5]. □

Proposition 1.36. Take a morphism f: X → Y in pro(𝕊)_{δ} such that (π_{f}X)^{∧_{L}} → (π_{f}Y)^{∧_{L}} is a pro-equivalence of pro-groupoids. Then the following are equivalent:

(1) For all abelian (π_{f}Y)^{∧_{L}}-representations M in L-groups, the map

H^{n}(f): H^{n}(Y, M) → H^{n}(X, M)

is an isomorphism for all n ≤ N and injective for n = N + 1.

(2) The map

π_{n}(f) : π_{n}(X^{∧_{L}}, x) → π_{n}(Y^{∧_{L}}, fy)

is a pro-isomorphism for n ≤ N and a pro-surjection for n = N + 1. In particular, a pro-groupoid G with discrete object set is (L, N)-good if and only if

π_{n}((BG)^{∧_{L}}) = 0

for all 2 ≤ n ≤ N.

Proof. The key observation is that Hom_{Ho(pro(𝕊)_{δ})}(Y, P_{n}Z) ≅ Hom_{Ho(pro(𝕊)_{δ})}(P_{n}Y, P_{n}Z), which is deduced from the corresponding result for 𝕊. Thus Proposition 1.32 implies that

P_{N}(X^{∧_{L}}) → P_{N}(Y^{∧_{L}})

becomes an isomorphism in Ho(pro(𝕊)_{δ}), while

P_{N+1}(X^{∧_{L}}) → P_{N+1}(Y^{∧_{L}})

is an epimorphism. Since isomorphisms in Ho(pro(𝕊)_{δ}) are just Artin–Mazur weak equivalences, this completes the “only if” part.

For the converse, note that the hypothesis is equivalent to saying that the homotopy fibre F of f^{∧_{L}}: X^{∧_{L}} → Y^{∧_{L}} is N-connected, by looking at the long exact sequence of homotopy groups. Thus H^{j}(F, A) = 0 for all 0 < j ≤ N and all abelian L-groups A. For any π_{f}Y^{∧_{L}}-representation M in abelian L-groups, the Leray spectral sequence

H^{i}(Y^{∧_{L}}, H^{j}(F, M)) ⟹ H^{i+j}(X^{∧_{L}}, f^{−1}M)

forces the maps H^{i}(Y^{∧_{L}}, M) → H^{i+j}(X^{∧_{L}}, M) to be isomorphisms for i ≤ N and injective for i = N + 1, as required.

The final statement is given by taking X = BG and Y = B(G^{∧_{L}}). □

Lemma 1.37. If f: X → Y is a morphism in pro(𝕊)_{δ} for which the map

π_{n}(f) : π_{n}(X, x) → π_{n}(Y, fy)

is a pro-isomorphism for n ≤ N and a pro-surjection for n = N + 1, then the map

π_{n}(f) : π_{n}(X^{∧_{L}}, x) → π_{n}(Y^{∧_{L}}, fy)

is a pro-isomorphism for n ≤ N and a pro-surjection for n = N + 1.

Proof. The proof of Proposition 1.36 adapts to show that for any π_{f}Y -representation M, the maps H^{i}(Y, M) → H^{i}(X, M) are isomorphisms for i ≤ N and injective for i = N + 1. Thus the hypotheses of Proposition 1.36 are satisfied, giving the required results. □

Definition 1.38. Given a group-valued representation H of a groupoid Γ (i.e. a functor from Γ to the category of groups), recall from [Pri3, Definition 2.15] that the semi-direct product H ⋊ Γ is a groupoid with objects Ob (H ⋊ Γ) = Ob (Γ) and has (H ⋊ Γ)(x, y) = H_{x} ⋊ Γ(x, y).

Proposition 1.39. Fix X ∈ 𝕊. If π_{n}(X, x) is finitely generated for all n ≤ N, and if the image of π_{1}(X, x) → Aut(π_{n}(X, x) ⊗ 𝔽_{p}) is L-torsion for all n ≤ N, all p ∈ L, and all x ∈ X, then there is an exact sequence

\xymatrix{ &  & π_{N+1}(X^{∧_{L}}, x) \ar[r] & π_{N+1}((Bπ_{1}(X, x))^{∧_{L}}) \ar@{-->}[dll] \\  & π_{N}(X, x)^{∧_{L}} \ar[r] & π_{N}(X^{∧_{L}}, x) \ar[r] & π_{N}((Bπ_{1}(X, x))^{∧_{L}}) \ar[r] & … \\ … \ar[r] & π_{2}(X, x)^{∧_{L}} \ar[r] & π_{2}(X^{∧_{L}}, x) \ar[r] & π_{2}((Bπ_{1}(X, x))^{∧_{L}}) \ar[r] & 0.}

Hence if in addition π_{f}X is (L, N + 1)-good (resp. (L, N)-good), then the natural map

π_{n}(X)^{∧_{L}} → π_{n}(X^{∧_{L}})

is a pro-isomorphism for all n ≤ N (resp. a pro-isomorphism for all n < N and a pro-surjection for n = N).

Proof. We adapt the argument of [Pri3, Theorem 1.58]. Let {X(n)}_{n} be the Postnikov tower for X. We will prove the proposition inductively for the groups X(n). Thanks to Lemma 1.37, we may replace X with X(N), so may assume that the groups π_{n}(X, x) are finitely generated for all n. Write Γ := π_{f}X.

For n = 1, X(1) is weakly equivalent to Bπ_{f}X, so (Bπ_{f}X)^{∧_{L}} ≃ X(1)^{∧_{L}} and π_{n}(X(1), x) = 0 for all n ≥ 2, making the exact sequence above immediate.

Now assume that X(n − 1) satisfies the inductive hypothesis, and consider the fibration X(n) → X(n − 1). This is determined up to homotopy by a k-invariant ([GJ, §VI.5]) κ ∈ H^{n+1}(X(n − 1), π_{n}(X)). Since π_{n}(X) ⊗ 𝔽_{p} is a finite-dimensional Γ^{∧_{L}}-representation for all p ∈ L, the group A := π_{n}(X)^{∧_{L}} is an inverse limit of finite Γ^{∧_{L}}-representations. Now, the element

κ ∈ H^{n+1}(X(n − 1), A) ≅ H^{n+1}(X(n − 1)^{∧_{L}}, A)

comes from a map

G(X(n − 1))^{∧_{L}} → (N^{−1}A[−n]) ⋊ Γ,

where N^{−1} denotes the denormalisation functor ([Wei, 8.4.4]) from chain complexes to simplicial complexes (the Dold-Kan correspondence).

Let LA be the chain complex with A concentrated in degrees n, n − 1, and d: (LA)_{n} → (LA)_{n−1} the identity, and define G to be the pullback of this map along the surjection N^{−1}LA ⋊ Γ → (N^{−1}A[−n]) ⋊ Γ of simplicial locally pro-finite L-torsion groupoids. This gives an extension

N^{−1}A[1 − n] → G → G(X(n − 1))^{∧_{L}}.

Applying W̄ gives the fibration

W̄ N^{−1}A[1 − n] → W̄ G → X(n − 1)^{∧_{L}}

in pro(𝕊), corresponding to the k-invariant f^{∗}κ ∈ H^{n}(X(n − 1)^{∧_{L}}, A) for f: X(n − 1) → X(n − 1)^{∧_{L}}. This in turn gives a map X(n) → W̄ G, compatible with the fibrations.

The long exact sequence of homotopy applied to the map W̄ G → X(n − 1)^{∧_{L}} shows that π_{m}(W̄ G, x) = π_{m}(X(n − 1)^{∧_{L}}) for all m ≠ n, n + 1, and gives an exact sequence

0 → π_{n+1}(W̄ G, x) → π_{n+1}(X(n − 1)^{∧_{L}}) → A(x) → π_{n}(W̄ G, x) → π_{n}(X(n − 1)^{∧_{L}}) → 0.

The inductive hypothesis shows that π_{m}(X(n−1)^{∧_{L}}) = π_{m}((Bπ_{1}(X, x))^{∧_{L}}) for m ≥ n+1, so we deduce that there is a long exact sequence

\xymatrix{… \ar[r] & π_{m}(X(n), x)^{∧_{L}} \ar[r] & π_{m}(W̄ G, x) \ar[r] & π_{m}((Bπ_{1}(X, x))^{∧_{L}}) \ar[r] & … \\ … \ar[r] & π_{2}(X(n), x)^{∧_{L}} \ar[r] & π_{2}(W̄ G, x) \ar[r] & π_{2}((Bπ_{1}(X, x))^{∧_{L}}) \ar[r] & 0.}

As W̄ G ∈ 𝕊^{∧_{L}}, it will therefore suffice to show that F: G(X(n))^{∧_{L}} → G is a weak equivalence. We now apply the Hochschild–Serre spectral sequence, giving

H^{p}(X(n−1), H^{q}(N^{−1}A[1−n], M)) = H^{p}(G(X(n−1))^{∧_{L}}, H^{q}(N^{−1}A[1−n], M)) ⟹ H^{p+q}(G, M).

Similarly

H^{p}(X(n − 1), H^{q}(E(n), V)) ⟹ H^{p+q}(X(n), V),

for all Γ^{∧_{L}}-representations M in abelian L-groups, where E(n) is the fibre of X(n) → X(n − 1).

Now, E(n) is a K(π_{n}(X), n)-space, and W̄ N^{−1}A[1 − n] is a K(A, n)-space. By Lemma 1.35, it follows that E(n) → W̄ N^{−1}A[1 − n] is pro-L completion, giving an isomorphism of cohomology with coefficients in M. Thus F induces isomorphisms on homology groups, hence must be a weak equivalence by Proposition 1.32.

Finally, if Γ is (L, m)-good, Corollary 1.36 shows that π_{n}((BΓ)^{∧_{L}}, x) = 0 for all 1 < n ≤ m. □

2. Review of pro-algebraic homotopy types

Here we give a summary of the results from [Pri3] and [Pri5]. The motivation for these is that they provide a framework to transfer information about local systems and their cohomology to statements about homotopy types. Fix a field k of characteristic zero.

2.1. Pro-algebraic groupoids. Given a local system 𝕍 of finite-dimensional k-vector spaces on a topological space X, we can form the affine k-scheme Iso(𝕍_{x}, 𝕍_{y}) of isomorphisms of stalks, for each pair of points x, y ∈ X. These combine to form a kind of groupoid G whose objects are the points of X. This is the motivating example of a pro-algebraic groupoid; in this case it comes equipped with a canonical groupoid homomorphism π_{f}X → G(k).

For the general case, we now recall some definitions from [Pri3, §§2.1–2.3].

Definition 2.1. Define a pro-algebraic groupoid G over a field k to consist of the following data:

(1) A discrete set Ob (G).

(2) For all x, y ∈ Ob (G), an affine scheme G(x, y) (possibly empty) over k.

(3) A groupoid structure on G, consisting of a multiplication morphism m: G(x, y) × G(y, z) → G(x, z), identities Spec k → G(x, x) and inverses G(x, y) → G(y, x), satisfying associativity, identity and inverse axioms.

Note that a pro-algebraic group is just a pro-algebraic groupoid on one object. We say that a pro-algebraic groupoid is reductive (resp. pro-unipotent) if the pro-algebraic groups G(x, x) are so for all x ∈ Ob (G). An algebraic groupoid is a pro-algebraic groupoid for which the G(x, y) are all of finite type.

If G is a pro-algebraic groupoid, let O(G(x, y)) denote the global sections of the structure sheaf of G(x, y).

Remark 2.2. The terminology “pro-algebraic groupoid” follows the characterisation of proalgebraic groups in [DMOS, Ch. II]. A linear algebraic group is an affine group scheme of finite type, and there is an equivalence of categories between affine group schemes and pro-objects in linear algebraic groups. A more accurate term for pro-algebraic groupoids would thus be “linear pro-algebraically enriched groupoids”.

Definition 2.3. Given morphisms f, g: G → H of pro-algebraic groupoids, define a natural isomorphism η between f and g to consist of morphisms

η_{x}: Spec k → H(f(x), g(x))

for all x ∈ Ob (G), such that the following diagram commutes, for all x, y ∈ Ob (G):

G(x, y) ⟶^{f(x,y)} H(f(x), f(y))
g(x,y)↓  ↓^{·η_{y}}
H(g(x), g(y)) ⟶^{η_{x}·} H(f(x), g(y)).

[If we reversed our order of composition in Definition 2.1, this would be the same as a natural transformation of functors of categories enriched in affine k-schemes.]

A morphism f: G → H of pro-algebraic groupoids is said to be an equivalence if there exists a morphism g: H → G such that fg and gf are both naturally isomorphic to identity morphisms. This is the same as saying that for all y ∈ Ob (H), there exists x ∈ Ob (G) such that H(f(x), y)(k) is non-empty (essential surjectivity), and that for all x_{1}, x_{2} ∈ Ob (G), G(x_{1}, x_{2}) → H(f(x_{1}), f(x_{2})) is an isomorphism.

Definition 2.4. Given a pro-algebraic groupoid G, define a finite-dimensional linear Grepresentation to be a functor ρ from G to the category of finite-dimensional k-vector spaces, respecting the algebraic structure. Explicitly, this consists of a set {V_{x}}_{x∈Ob (G)} of finite-dimensional k-vector spaces, together with morphisms ρ_{xy}: G(x, y) → Hom(V_{y}, V_{x}) of affine schemes, respecting the multiplication and identities.

A morphism f: (V, ρ) → (W, ϱ) of G-representations consists of f_{x} ∈ Hom(V_{x}, W_{x}) such that

f_{x} ∘ ϱ_{xy} = ρ_{xy} ∘ f_{y}: G(x, y) → Hom(V_{x}, W_{y}).

Definition 2.5. Given a pro-algebraic groupoid G, define the reductive quotient G^{red} of G by setting Ob (G^{red}) = Ob (G), and

G^{red}(x, y) = G(x, y)/R_{u}(G(y, y)) = R_{u}(G(x, x))\G(x, y),

where R_{u}(G(x, x)) is the pro-unipotent radical of the pro-algebraic group G(x, x). The equality arises since if f ∈ G(x, y), g ∈ R_{u}(G(y, y)), then fgf^{−1} ∈ R_{u}(G(x, x)), so both equivalence relations are the same. Multiplication and inversion descend similarly. Observe that G^{red} is then a reductive pro-algebraic groupoid. Representations of G^{red} correspond to semisimple representations of G, since k is of characteristic 0.

Definition 2.6. Recall from [DMOS, Definition II.1.7] that a tensor category C is said to be rigid if it has an internal Hom-functor Hom, satisfying

• Hom(X, Y) ⊗ Hom(X′, Y′) ≅ Hom(X ⊗ X′, Y ⊗ Y′) and

• (X^{∨})^{∨} ≅ X for all X ∈ C,

where X^{∨} = Hom(X, 1), with 1 the unit for ⊗.

Definition 2.7. Recall from [DMOS, §II.2] that a neutral Tannakian category over k is a k-linear rigid abelian tensor category C, equipped with a faithful exact tensor functor ω (the fibre functor) from C to the category of finite-dimensional k-vector spaces.

In [Pri3, §2.1], this was extended to multifibred Tannakian categories, which have several exact tensor functors {ω_{x}}_{x∈S}, jointly faithful in the sense that Hom(U, V) ↪ ∏_{x∈S} Hom(ω_{x}U, ω_{x}V).

A Tannakian subcategory D ⊂ C is a full subcategory closed under the formation of subquotients, direct sums, tensor products, and duals.

Tannakian duality ([DMOS, Theorem II.2.11]) then states that for any neutral Tannakian category (C, ω) over a field, there is a canonical equivalence between C and the category of finite-dimensional representations of a unique affine group scheme G. Explicitly, G is the scheme of tensor automorphisms of ω.

If C is multifibred, with a set S of fibre functors, we form a pro-algebraic groupoid G on objects S by setting G(x, y) to be the affine scheme of tensor isomorphisms from ω_{x} to ω_{y}. This gives a canonical equivalence between C and the category of finite-dimensional G-representations, with ω_{x} being pullback along the inclusion {x} ↪ G.

Definition 2.8. Let AGpd denote the category of pro-algebraic groupoids over k, and observe that this category contains all limits.

Lemma 2.9. Consider the functor G ↦ G(k) from AGpd to Gpd, the category of abstract groupoids. This has a left adjoint, the algebraisation functor, denoted Γ ↦ Γ^{alg}, which is determined by the finite-dimensional linear representations of Γ.

Proof. The algebraisation functor can be given explicitly by setting Ob (Γ)^{alg} = Ob (Γ), and

Γ^{alg}(x, y) = Γ(x, x)^{alg} ×^{Γ(x,x)} Γ(x, y),

where Γ(x, x)^{alg} is the pro-algebraic (or Hochschild–Mostow) completion of the group Γ(x, x) ([HM2]), and X ×^{G} Y is the quotient of X × Y by the relation (gx, y) ∼ (x, gy) for g ∈ G.

Alternatively, the finite-dimensional linear representations of Γ (as in Definition 2.4) correspond to those of Γ^{alg} (if the latter exists). These form a multifibred Tannakian category (with one fibre functor for each object of Γ), so Tannakian duality provides unique pro-algebraic groupoid G with the same finite-dimensional representations as Γ. For any pro-algebraic groupoid H and any groupoid homomorphism Γ → H(k), we then have a functor from H-representations to Γ representations, and thus a unique compatible morphism G → H, so Γ^{alg} ≅ G. □

Example 2.10. The motivating example for this setup is when Γ = π_{f}X, the fundamental groupoid of a topological space. Then (π_{f}X)^{alg} is the pro-algebraic groupoid corresponding to the multifibred Tannakian category of local systems of finite-dimensional k-vector spaces on X. The fibre functors are given by 𝕍 ↦ 𝕍_{x}. Likewise, (π_{f}X)^{red} is the object corresponding to the Tannakian category of semisimple local systems.

Definition 2.11. Given a pro-algebraic groupoid G, and U = {U_{x}}_{x∈Ob (G)} a collection of pro-algebraic groups parametrised by Ob (G), we say that G acts on U if there are morphisms U_{x} × G(x, y) ⟶^{∗} U_{y} of affine schemes, satisfying the following conditions:

(1) (uv) ∗ g = (u ∗ g)(v ∗ g), 1 ∗ g = 1 and (u^{−1}) ∗ g = (u ∗ g)^{−1}, for g ∈ G(x, y) and

u, v ∈ U_{x}.

(2) u ∗ (gh) = (u ∗ g) ∗ h and u ∗ 1 = u, for g ∈ G(x, y), h ∈ G(y, z) and u ∈ U_{x}.

If G acts on U, we construct G ⋉ U as in Definition 1.38.

Definition 2.12. Given a pro-algebraic groupoid G, define the pro-unipotent radical R_{u}(G) to be the collection R_{u}(G)_{x} = R_{u}(G(x, x)) of pro-unipotent pro-algebraic groups, for x ∈ Ob (G). G then acts on R_{u}(G) by conjugation, i.e.

u ∗ g := g^{−1}ug,

for u ∈ R_{u}(G)_{x}, g ∈ G(x, y).

Now assume that the field k is of characteristic 0.

Proposition 2.13. For any pro-algebraic groupoid G, there is a Levi decomposition G = G^{red} ⋉ R_{u}(G), unique up to conjugation by R_{u}(G).

Proof. [Pri3, Proposition 2.17]. □

2.2. The pro-algebraic homotopy type of a topological space. We now recall the results from [Pri3, §2.4]. The motivation here is that we wish to study the whole homotopy type, not just fundamental groupoids. This will involve working with the loop groupoid, which is a simplicial groupoid, so we need a simplicial framework.

Definition 2.14. Given a simplicial object G_{•} in the category of pro-algebraic groupoids, with Ob (G_{•}) constant, define the fundamental groupoid π_{0}(G_{•}) of G_{•} to have objects Ob (G), and for x, y ∈ Ob (G), set π_{0}(G)(x, y) to be the coequaliser

\xymatrix{G_{1}(x, y) \ar[r]^{∂_{1}} \ar[r]_{∂_{0}} & G_{0}(x, y) \ar[r] & π_{0}(G)(x, y)}

in the category of affine schemes. Thus π_{0}(G) is a pro-algebraic groupoid on objects Ob (G), with multiplication inherited from G_{0}.

Definition 2.15. Define a pro-algebraic simplicial groupoid to consist of a simplicial complex G_{•} of pro-algebraic groupoids, such that

(1) Ob (G_{•}) is constant, and

(2) for all x ∈ Ob (G), G(x, x)_{•} ∈ sAGp, i.e. the maps G_{n}(x, x) → π_{0}(G)(x, x) are

pro-unipotent extensions of pro-algebraic groups. We denote the category of pro-algebraic simplicial groupoids by sAGpd.

For any G_{•} ∈ sAGpd and x ∈ Ob (G_{•}), observe that G_{•}(x, x) is a simplicial affine group scheme, so has homotopy groups π_{n}(G_{•}(x, x)). That these are also affine group schemes follows from the standard characterisation

π_{n}(G_{•}(x, x)) = H_{n}(NG_{•}(x, x), ∂_{0})

of homotopy groups of simplicial groups.

Lemma 2.16. There is a model structure on sAGpd in which a morphism f: G_{•} → H_{•} is:

(1) a weak equivalence if the map π_{0}(f): π_{0}(G_{•}) → π_{0}(H_{•}) is an equivalence of proalgebraic groupoids, and the maps π_{n}(f, x): π_{n}(G_{•}(x, x)) → π_{n}(H_{•}(fx, fx)) are isomorphisms for all n and for all x ∈ Ob (G);

(2) a fibration if the morphism N_{n}(f): N(G(x, x))_{n} → N(H(x, x))_{n} of normalised groups is surjective for all n > 0 and all x ∈ Ob (G), and f satisfies the pathlifting condition that for all x ∈ Ob (G), y ∈ Ob (H), and h ∈ H_{0}(fx, y)(k), there exists z ∈ Ob (G), g ∈ G_{0}(x, z)(k) with fg = h. Equivalently, this says that G(k) → H(k) is a fibration in the category of simplicial groupoids.

Proof. This is [Pri3, Theorem 2.25]. □

We define Ho(sAGpd) to be the localisation of sAGpd at weak equivalences.

There is a forgetful functor (k): sAGpd → sGpd, given by sending G_{•} to G_{•}(k). This functor has a left adjoint G_{•} ↦ (G_{•})^{alg}. We can describe (G_{•})^{alg} explicitly. First let (π_{0}(G))^{alg} be the pro-algebraic completion of the abstract groupoid π_{0}(G), then let (G^{alg})_{n} be the relative Malcev completion (defined in [Hai] for pro-algebraic groups) of the morphism

G_{n} → (π_{0}(G))^{alg}.

In other words, G_{n} → (G^{alg})_{n} ⟶^{f} (π_{0}(G))^{alg} is the universal diagram with f a pro-unipotent extension.

Proposition 2.17. The functors (k) and (−)^{alg} give rise to a pair of adjoint functors

\xymatrix{Ho(sGpd) \ar[r]^{𝕃^{alg}} & Ho(sAGpd) \ar[l]^{(k)}|{⊥}},

with 𝕃^{alg}G(X) = G(X)^{alg}, for any X ∈ 𝕊 and G as in Definition 1.6.

Proof. [Pri3, Proposition 2.26] shows that the functors are a Quillen pair, so the statement follows from the observation that all objects in sAGpd are fibrant, making (k) its own derived right Quillen functor. Since G(X) is cofibrant, 𝕃^{alg}G(X) = G(X)^{alg}. □

The reason that we need to take 𝕃^{alg} in the Proposition is that (−)^{alg} is not an exact functor, so only preserves weak equivalences between cofibrant objects (which roughly correspond to free simplicial groupoids). In Examples 2.24, we will see examples of discrete groups Γ for which the map 𝕃^{alg}Γ → Γ^{alg} is not a weak equivalence.

Definition 2.18. Given a simplicial set (or equivalently a topological space), define the pro-algebraic homotopy type of X over k to be the object

G(X)^{alg}

in Ho(sAGpd), where G(X) is the loop groupoid of Definition 1.6. Define the pro-algebraic fundamental groupoid by ϖ_{f}(X) := π_{0}(G(X)^{alg}). Note that π_{0}(G^{alg}) is the pro-algebraic completion of the fundamental groupoid π_{0}(G).

We then define the higher pro-algebraic homotopy groups ϖ_{n}(X) (as ϖ_{f}Xrepresentations) by

ϖ_{n}(X) := π_{n−1}(G(X)^{alg}),

where π_{n}(G) is the representation x ↦ π_{n}(G(x, x)), for x ∈ Ob (G).

Remark 2.19. We can interpret G(X)^{alg} as the classifying object for non-abelian cohomology. Given G ∈ sAGpd, we can define H^{1}(X, G) to be the homotopy class of maps G(X)^{alg} → G, which is just [X, W̄ G(k)]. When G is just a linear algebraic group, this recovers the usual definition of the set H^{1}(X, G) of classes of G-torsors on X. When A is a simplicial finite-dimensional vector space (regarded as a simplicial algebraic group), this definition gives

H^{1}(X, A) = ℍ^{1}(X, NA),

hypercohomology of the normalised complex associated to A.

2.3. Relative Malcev homotopy types.

Definition 2.20. Assume we have an abstract groupoid G, a reductive pro-algebraic groupoid R, and a representation ρ: G → R(k) which is an isomorphism on objects and Zariski-dense on morphisms (i.e. ρ: G(x, y) → R(k)(ρx, ρy) is Zariski-dense for all x, y ∈ Ob G). Define the Malcev completion (G, ρ)^{Mal} (or G^{ρ,Mal}, or G^{R,Mal}) of G relative to ρ to be the universal diagram

G → (G, ρ)^{Mal} ⟶^{p} R,

with p a pro-unipotent extension, and the composition equal to ρ. Explicitly, Ob (G, ρ)^{Mal} = Ob G and

(G, ρ)^{Mal}(x, y) = (G(x, x), ρ)^{Mal} ×^{G(x,x)} G(x, y).

If G and R are groups, observe that this agrees with the usual definition (of [Hai]).

If ϱ: G → R(k) is any any Zariski-dense representation (i.e. essentially surjective on objects and Zariski-dense on morphisms) to a reductive pro-algebraic groupoid (in most examples, we take R to be a group), we can define another reductive groupoid R̃ by setting Ob R̃ = Ob G, and R̃(x, y) = R(ϱx, ϱy). This gives a representation ρ: π_{f}X ⟶^{ρ} R̃ satisfying the above hypotheses, and we define the Malcev completion of G relative to ϱ to be the Malcev completion of G relative to ρ. Note that R̃ → R is an equivalence of pro-algebraic groupoids.

Definition 2.21. Given a Zariski-dense morphism ρ: π_{f}X → R(k), let the Malcev completion G(X, ρ)^{Mal} of X relative to ρ be the pro-algebraic simplicial group (G(X), ρ)^{Mal}. Observe that the Malcev completion of X relative to (π_{f}X)^{red} is just G(X)^{alg}. Let ϖ_{f}(X, ρ)^{Mal} = π_{0}G(X, ρ)^{Mal} and ϖ_{n}(X, ρ)^{Mal} = π_{n−1}G(X, ρ)^{Mal}. Note that π_{f}((X, ρ)^{Mal}) is the relative Malcev completion of ρ: π_{f}X → R(k).

Beware that the relative Malcev completion of X is defined by completing a loop space for X, rather than X itself. However, Theorem 2.74 will give other equivalent formulations of the homotopy type, effectively by completing a covering space for X.

Lemma 2.22. Let f: X → Y be a morphism in 𝕊 for which the map

π_{n}(f) : π_{n}(X) → π_{n}(Y)

is an isomorphism for n ≤ N and a surjection for n = N + 1, and take a Zariski-dense morphism ρ: π_{f}Y → R(k). Then the map

ϖ_{n}(f) : ϖ_{n}(X, ρ ∘ f)^{Mal} → ϖ_{n}(Y, ρ)^{Mal}

is an isomorphism for n ≤ N and a surjection for n = N + 1.

Proof. As in the proof of Lemma 1.37, for any π_{f}Y -representation M, the maps H^{i}(Y, M) → H^{i}(X, M) are isomorphisms for i ≤ N and injective for i = N + 1.

Now, [Pri3, Proposition 4.37] gives a convergent Adams spectral sequence

E_{pq}^{1}(X) = (Lie_{−p}(H̃^{∗+1}(X, 𝕆(R))^{∨}))_{p+q} ⟹ ϖ_{p+q+1}(X, ρ ∘ f)^{Mal},

in the category of pro-finite-dimensional vector spaces, where H̃ denotes reduced cohomology, Lie_{∗} is the free graded Lie algebra functor, and 𝕆(R) is the local system of Definition 2.75. Since E_{pq}^{1}(X) → E_{pq}^{1}(Y) is an isomorphism for p+q < N and surjective for p+q = N, the result follows. □

Definition 2.23. Say that a groupoid Γ is n-good with respect to a Zariski-dense representation ρ: Γ → R(k) to a reductive pro-algebraic groupoid if for all finite-dimensional Γ^{ρ,Mal}-representations V, the map

H^{i}(Γ^{ρ,Mal}, V) → H^{i}(Γ, V)

is an isomorphism for all i ≤ n and an inclusion for i = n + 1. Say that Γ is good with respect to ρ if it is n-good for all n.

See Lemma 3.36 for alternative criteria to determine when a groupoid is n-good.

Examples 2.24. By [Pri3, Examples 2.24], finite groups, free groups, finitely generated nilpotent groups and fundamental groups of compact Riemann surfaces are all good with respect to all Zariski-dense representations. Super rigid groups (such as SL_{3}(ℤ)) give examples of groups which are not good with respect to any real (or complex) representations. This is because Γ^{R,Mal} = R in these cases, but H^{∗}(Γ, ℝ) ≠ ℝ.

Theorem 2.25. If X is a topological space with fundamental groupoid Γ, equipped with a Zariski-dense representation ρ: Γ → R(k) to a reductive pro-algebraic groupoid for which:

(1) Γ is (N + 1)-good with respect to ρ,

(2) π_{n}(X, −) is of finite rank for all 1 < n ≤ N, and

(3) the Γ-representation π_{n}(X, −) ⊗_{ℤ} k is an extension of R-representations (i.e. a

Γ^{ρ,Mal}-representation) for all 1 < n ≤ N, then the canonical map

π_{n}(X, −) ⊗_{ℤ} k → ϖ_{n}(X^{ρ,Mal}, −)

is an isomorphism for all 1 < n ≤ N.

Proof. When N = ∞, this is [Pri3, Theorem 3.21], but the same proof gives the conclusion above if we only assume that Γ is (N + 1)-good (while still requiring the other conditions to hold for all n). For arbitrary N, and X as above, this means that the Nth stage X(N) in the Postnikov tower for X gives isomorphisms

π_{n}(X, −) ⊗_{ℤ} k → ϖ_{n}(X(N)^{ρ,Mal}, −)

for all 1 < n ≤ N, since π_{i}X(N) = 0 for i > N, while π_{i}X(N) = π_{i}X for i ≤ N. Applying Lemma 2.22 to the morphism X → X(N) now completes the proof. □

2.4. Cohomology and hypercohomology.

2.4.1. Simplicial groupoids.

Definition 2.26. For a simplicial groupoid Γ_{•}, a cosimplicial Γ_{•}-representation consists of the following

(1) a Γ_{n}-representation V^{n} for all n, with g · ∂^{i}v = ∂^{i}((∂_{i}g) · v), for g ∈ Γ_{n+1}, v ∈ V^{n};

(2) operations ∂^{i}, σ^{i} making V^{•}(x) into a cosimplicial complex for each x ∈ Ob Γ_{•}, satisfying the additional conditions that

g · (∂^{i}v) = ∂^{i}((∂_{i}g) · v) h · (σ^{i}v) = σ^{i}((σ_{i}g) · v)

for g ∈ Γ_{n+1}(x, y), h ∈ Γ_{n−1}(x, y), v ∈ V^{n}(y).

Remark 2.27. If Γ_{•} = G(X), then we can think of a cosimplicial Γ_{•}-representation as being a kind of hyper-local system on X. As we will see below, these give a sufficiently large category to recover cohomology, but objects with constant cosimplicial structure are still just local systems.

Definition 2.28. Given a simplicial groupoid Γ_{•} and a cosimplicial Γ_{•}-representation V, define the cosimplicial complex C^{•}(Γ_{•}, V) by

C^{n}(Γ_{•}, V) = Hom_{Γ_{n}}((W Γ_{•})_{n}, V^{n}),

for the functor W from Definition 1.5, with operations (∂^{i}f)(x) = ∂_{V}^{i} (f(∂_{i}x)) for x ∈ (W Γ_{•})_{n+1}, and (σ^{i}f)(x) = σ_{V}^{i} (f(σ_{i}x)) for x ∈ (W Γ_{•})_{n−1}.

Then define hypercohomology groups ℍ^{i}(Γ_{•}, V) by ℍ^{i}(Γ_{•}, V) = H^{i}C(Γ_{•}, V). If V is a π_{0}Γ_{•}-representation, regard V as a cosimplicial Γ_{•}-representation (with constant cosimplicial structure) and write H^{i}(Γ_{•}, V) := ℍ^{i}(Γ_{•}, V).

Lemma 2.29. If Γ_{•} is a simplicial groupoid and V a π_{0}Γ_{•}-representation, then

H^{i}(Γ_{•}, V) = H^{i}(W̄ Γ_{•}, V).

Proof. Observe that π_{0}(Γ_{•}) ×^{Γ_{•}} (W Γ_{•}) is the universal covering system of W̄ Γ_{•}. Since V is a π_{0}Γ_{•}-representation,

Hom_{Γ_{n}}((W Γ_{•})_{n}, V) = Hom_{π_{0}Γ_{•}}((π_{0}Γ_{•}) ×^{Γ_{n}} (W Γ_{•})_{n}, V) = Hom_{π_{f} W̄ Γ}({ \atop (W̄ Γ_{•})}_{n}, V),

so C^{•}(Γ_{•}, V) = C^{•}(W̄ Γ_{•}, V) (as defined in Definition 1.21), giving the required result. □

Lemma 2.30. If Γ_{•} is a simplicial groupoid and V a cosimplicial Γ_{•}-representation, then there is a convergent spectral sequence

H^{i}(Γ_{•}, H^{j}(V)) ⟹ ℍ^{i+j}(Γ_{•}, V),

where H^{j}(V) is the π_{0}Γ_{•}-representation given by setting H^{∗}(V)(x) to be cohomology of the cosimplicial complex V (x), for all x ∈ Ob Γ_{•}.

Proof. Form the filtration {F_{n}V }_{n} of V by setting F_{n}V to be the image of the n-skeleton sk^{n}V → V; F_{n}V is the subcomplex of V generated under the operations ∂^{i} by V^{≤n}, and its Dold-Kan normalisation is given by

N(F_{n}V)^{i} = \begin{cases}N^{i}V & i ≤ n \\ dN^{n}V & i = n + 1 \\ 0 & i ≥ n + 2. \end{cases}

Note that the condition g∂^{i}v = ∂^{i}((∂_{i}g)v implies that F_{n}V is Γ_{•}-equivariant. Also note that F_{n}V/F_{n−1}V is quasi-isomorphic to the denormalisation DH^{n}(V)[−n]. The spectral sequence associated to this filtration is thus

ℍ^{i+j}(Γ_{•}, DH^{j}(V)[−j]) ⟹ ℍ^{i+j}(Γ_{•}, V).

Let K_{•} := ker(Γ_{•} → π_{0}Γ_{•}); since H^{j}V is a π_{0}Γ_{•}-representation, there is a bicosimplicial complex C^{a,b} := Hom_{π_{0}(Γ_{•})}(K_{a}\(W Γ_{•})_{a}, D^{b}H^{j}(V)[−j]), with ℍ^{n}(Γ_{•}, DH^{j}(V)[−j]) = H^{n}(diag C). By the Eilenberg-Zilber Theorem ([Wei, Theorem 8.5.1]), Ndiag C is quasiisomorphic to the total complex of Hom_{π_{0}Γ_{•}}(Nℤ(K_{•}\W Γ_{•}), H^{j}(V)[−j]), so H^{n}(C) = H^{n−j}(G, H^{j}(V)), and the spectral sequence becomes

ℍ^{i}(Γ_{•}, H^{j}(V)) ⟹ ℍ^{i+j}(Γ_{•}, V).
□

Lemma 2.31. Given a weak equivalence f : Γ_{•} → Δ_{•} of simplicial groupoids, and a cosimplicial Γ_{•}-representation V, the map

f^{∗}: ℍ^{∗}(Δ_{•}, V) → ℍ^{∗}(Γ_{•}, f^{−1}V)

is an isomorphism.

Proof. Lemma 2.30 gives a morphism of convergent spectral sequences, so we may assume that V is a π_{0}Δ_{•}-representation. Since W̄ f: W̄ Γ_{•} → W̄ Δ_{•} is a weak equivalence of simplicial sets, Lemma 2.29 completes the proof. □

Lemma 2.32. Given a simplicial group Γ_{•}, a cosimplicial Γ_{•}-representation V and a simplicial abelian group A, the simplicial abelian group

Tot (V ⊗ A)

has a canonical Γ_{•}-action, where Tot : 𝕊^{Δ} → 𝕊 is the total space functor of [GJ, Ch. VIII], originally defined in [BK, Ch. X].

Proof. Given X ∈ 𝕊^{Δ} and K ∈ 𝕊, define e(X, K) ∈ 𝕊^{Δ} by e(X, K)^{n} := (X^{n})^{K_{n}}, with obvious cosimplicial operations. Note that Tot (e(X, K)) = Tot (X)^{K}.

The Γ_{•}-action on V is the same as a cosimplicial map f: V → e(V, Γ_{•}), so we have maps

V ⊗ A ⟶^{f} e(V, Γ_{•}) ⊗ A → e(V ⊗ A, Γ_{•}),

and hence a map Tot (V ⊗ A) → Tot (V ⊗ A)^{Γ_{•}}. this is equivalent to a map Γ_{•} × Tot (V ⊗ A) → Tot (V ⊗ A) of simplicial sets, and the argument above adapts to show that this action is associative. □

In order to simplify the definitions and exposition, we will now take Γ_{•} to be simplicial group, although everything can be extended to simplicial groupoids.

Definition 2.33. For a simplicial group Γ_{•}, a simplicial Γ_{•}-representation consists of a simplicial abelian group A, together with a Γ_{n}-action on A_{n} for all n, compatible with the simplicial operations. Let sRep(Γ_{•}) be the category of simplicial Γ_{•}-representations.

Note that Lemma 2.32 provides us with examples of simplicial Γ_{•}-representations constructed from cosimplicial Γ_{•}-representations. Also note that for any simplicial Γ_{•}- representation V, taking duals levelwise gives a cosimplicial Γ_{•}-representation V^{∨} given by (V^{∨})^{n} = (V_{n})^{∨}.

Lemma 2.34. Given a simplicial group Γ_{•}, there is a cofibrantly generated model structure on sRep(Γ_{•}), in which a morphism f: A → B is:

(1) a weak equivalence if the maps π_{i}(f): π_{i}(A) → π_{i}(B) are isomorphisms for all i;

(2) a fibration if the underlying map in 𝕊 is a fibration, or equivalently if the maps N_{i}(f): N_{i}(A) → N_{i}(B) on the Dold-Kan normalisation are surjective for all i > 0.

Proof. The forgetful functor from sRep(Γ_{•}) to simplicial sets preserves filtered direct limits and has a left adjoint F (S) = ℤ(Γ_{•} ×S). Thus for any finite object I ∈ 𝕊, the object F I is finite in sRep(Γ_{•}), so a fortiori permits the small object argument. The model structure on 𝕊 is cofibrantly generated by finite objects, so [Hir, Theorem 11.3.2] gives the required model structure on sRep(Γ_{•}). □

Lemma 2.35. We may characterise hypercohomology groups by

ℍ^{i}(Γ_{•}, V) = Hom_{Ho(sAb(Γ_{•}))}(ℤ, Tot (V ⊗_{ℤ} N^{−1}ℤ[−i])).

Proof. We first note that ℤ[W Γ_{•}] is a cofibrant replacement for ℤ, so for a simplicial abelian group A,

R\underline{Hom}_{sAb(Γ_{•})}(ℤ, Tot (V ⊗_{ℤ} A)) ≃ \underline{Hom}_{sAb(Γ_{•})}(ℤ[W Γ_{•}], Tot (V ⊗_{ℤ} A));

as observed in the proof of Lemma 2.32,

\underline{Hom}_{sAb}(ℤ[W Γ_{•}], Tot (V ⊗_{ℤ} A)) ≅ Tot (e(V ⊗_{ℤ} A, W Γ_{•})),

so

\underline{Hom}_{sAb(Γ_{•})}(ℤ[W Γ_{•}], Tot (V ⊗_{ℤ} A)) ≅ Tot (e(V ⊗_{ℤ} A, W Γ_{•})^{Γ_{•}}).

Now, e(V ⊗_{ℤ} A, W Γ_{•})^{Γ_{•}} is given in simplicial level n by C^{•}(Γ_{•}, V ⊗_{ℤ} A_{n}). When A_{n} is free and finitely generated, this becomes C^{•}(Γ_{•}, V) ⊗_{ℤ} A_{n}. Taking A = N^{−1}ℤ[−i] thus gives

Hom_{Ho(sAb(Γ_{•}))}(ℤ, Tot (V ⊗_{ℤ} N^{−1}ℤ[−i])) ≅ π_{0}R\underline{Hom}_{sAb(Γ_{•})}(ℤ, Tot (V ⊗_{ℤ} N^{−1}ℤ[−i]))
≅ Tot (C^{•}(Γ_{•}, V) ⊗_{ℤ} N^{−1}ℤ[−i]).

Given a cosimplicial simplicial abelian group B, the normalisation NTot B is equivalent to the good truncation in non-negative chain degrees of the product total complex Tot ∏N_{c}NB of the binormalisation of B (which is a cochain chain complex). Thus

π_{0}Tot (C^{•}(Γ_{•}, V) ⊗_{ℤ} N^{−1}ℤ[−i]) ≅ H_{0}Tot ∏((N_{c}C^{•}(Γ_{•}, V)) ⊗_{ℤ} ℤ[−i]),

and Tot ∏((N_{c}C^{•}(Γ_{•}, V)) ⊗_{ℤ} ℤ[−i]) is just the complex N_{c}C^{•}(Γ_{•}, V) turned upside down and shifted i places, so

H_{0}Tot ∏((N_{c}C^{•}(Γ_{•}, V)) ⊗_{ℤ} ℤ[−i]) = H^{i}N_{c}C^{•}(Γ_{•}, V) = ℍ^{i}(Γ_{•}, V),

as required. □

The following is an analogue of the Leray spectral sequence, and will play a key rôle in Theorem 3.32.

Proposition 2.36. Given a surjection Γ_{•} → Δ_{•} of simplicial groups with kernel B_{•}, and a cosimplicial Γ_{•}-representation V, there is a canonical convergent spectral sequence

H^{i}(Δ_{•}, ℍ^{j}(B_{•}, V)) ⟹ ℍ^{i+j}(Γ_{•}, V),

which we refer to as the Hochschild–Serre spectral sequence.

Proof. Given T ∈ sAb(Δ_{•}) and U, W ∈ sAb(Γ_{•}), we have an isomorphism

\underline{Hom}_{sAb(Δ_{•})}(T, \underline{Hom}_{sAb(B_{•})}(U, W)) ≅ \underline{Hom}_{sAb(Γ_{•})}(T ⊗ U, W).

This defines a right Quillen functor sAb(Δ_{•})^{opp} × sAb(Γ_{•})^{opp} × sAb(Γ_{•}) → 𝕊; since any cofibrant Γ_{•}-representation is cofibrant as a B_{•}-representation, the isomorphism above gives an equivalence

RHom_{sAb(Δ_{•})}(T, R\underline{Hom}_{sAb(B_{•})}(U, W)) ≃ R\underline{Hom}_{sAb(Γ_{•})}(T ⊗^{L} U, W).

In particular,

R\underline{Hom}_{sAb(Γ_{•})}(ℤ, W) ≃ RHom_{sAb(Δ_{•})}(ℤ, R\underline{Hom}_{sAb(B_{•})}(ℤ, W)).

Setting W = Tot (V ⊗_{ℤ} N^{−1}ℤ[−n]), this gives an isomorphism

ℍ^{n}(Γ_{•}, V) ≅ H^{n}C^{•}(Δ_{•}, C^{•}(B_{•}, V)),

so the morphism C^{•}(Γ_{•}, V) → C^{•}(Δ_{•}, C^{•}(B_{•}, V)) is a quasi-isomorphism, and the result now follows from Lemma 2.30. □

2.4.2. Simplicial pro-algebraic groupoids.

Definition 2.37. Given G ∈ sAGpd, define a cosimplicial G-representation to be an O(G)-comodule V in cosimplicial k-vector spaces. Thus we have cosimplicial complexes V (x) for all x ∈ Ob G, together with a coassociative coaction V (x) → O(G)(x, y) ⊗ V (y).

Note that the category of cosimplicial G-representations is opposite to the category s\begin{matrix} \\ FD & Rep \end{matrix}(G) of pro-finite-dimensional simplicial G-representations from [Pri3, §1.5].

Definition 2.38. Given G ∈ sAGpd and a cosimplicial G-representation V, define the cosimplicial complex C^{•}(G, V) by

C^{n}(G, V) = O((W G)_{n}) ⊗^{G_{n}} V^{n},

for the functor W from Definition 1.5, with operations ∂^{i} ⊗ ∂^{i} and σ^{i} ⊗ σ^{i}.

Then define hypercohomology groups ℍ^{i}(G, V) by ℍ^{i}(G, V) = H^{i}C(G, V). If V is a π_{0}Grepresentation, regard V as a cosimplicial G-representation (with constant cosimplicial structure) and write H^{i}(G, V) := ℍ^{i}(G, V).

Now, [Pri3, Example 1.45] ensures that ℍ^{i}(G, V)^{∨} = H_{i}(G, V^{∨}) in the notation of [Pri3, Definition 1.48]. In particular, this means that hypercohomology groups of G are an invariant of the homotopy type of G.

Proposition 2.39. A morphism G ⟶^{f} K of pro-algebraic simplicial groupoids is a weak equivalence if and only if

(1) f(R_{u}(G)) ≤ R_{u}(K), with the quotient map

G^{red} → K^{red}

an equivalence, and

(2) for all finite-dimensional irreducible K-representations V, the maps

H^{i}(f): H^{i}(K, V) → H^{i}(G, f^{∗}V)

are isomorphisms for all i > 0.

Proof. This is [Pri3, Corollary 1.55], adapted from groups to groupoids. □

Note that the analogue of Lemma 2.32 for pro-algebraic simplicial groupoids thus ensures that weak equivalences induce isomorphisms on hypercohomology.

Lemma 2.40. For a cofibrant pro-algebraic simplicial group G (for the model structure of Lemma 2.16), and a finite-dimensional π_{0}G-representation V, the cohomology group H^{i}(G, V) is isomorphic to the homotopy class of maps G → G ⋉ (N^{−1}V [1 − i]) in the model category sAGpd↓G.

Proof. Consider the morphism k → O(G), and let the cokernel be C. As in the proof of [Pri3, Proposition 1.50], C is fibrant as a cosimplicial G-representation. Likewise, V ⊗O(G) and V ⊗ C are both fibrant, so H^{∗}(G, V) is cohomology of the cone complex of

V ⊗^{G} O(G) → V ⊗^{G} C.

Now, V ⊗^{G} O(G) is just V, so we need to describe V ⊗^{G} C.

Letting E := O(G)^{∨}, we see that C^{∨} is the kernel of E → k. Elements θ of V ⊗_{n}^{G} C^{n} are then just morphisms θ: (C^{∨})_{n} → V satisfying α(gc) = gα(c), for g ∈ G_{n}, c ∈ (C^{∨})_{n}. There is a map E → C^{∨} given by a ↦ a − 1, so θ composed with this gives a linear morphism θ′: E_{n} → V, satisfying θ′(ga) = gθ′(a) + θ′(g) for g ∈ G_{n}.

Regarding E_{n} as an affine scheme, there is a morphism G_{n} → E_{n}, so we see that θ corresponds to a derivation θ′: G_{n} → V. Since derivations G → V are just morphisms G → G ⋉ V over G, the statement now follows from the description of the path object in sAGpd from [Pri3, Lemma 2.29]. □

Lemma 2.41. If Γ_{•} is a cofibrant simplicial groupoid (e.g. G(X) for X ∈ 𝕊), and V is a finite-dimensional π_{0}Γ_{•}^{R,Mal}-representation, then the map

H^{∗}(Γ_{•}^{R,Mal}, V) → H^{∗}(Γ_{•}, V)

is an isomorphism.

Proof. This is implicit in [Pri3, §1.5.3]. Replacing Γ_{•} with a disjoint union of simplicial groups, Lemma 2.40 gives that H^{∗}(Γ_{•}^{R,Mal}, V) is the homotopy class of maps from Γ_{•}^{R,Mal} to Γ_{•}^{R,Mal} ⋉ (N^{−1}V [1 − i]) over Γ_{•}^{R,Mal}. Since any map from Γ_{•}^{alg} to a pro-unipotent extension of R factors through Γ_{•}^{R,Mal}, this is the same as the homotopy class of maps from Γ_{•}^{alg} to Γ_{•}^{R,Mal} ⋉ (N^{−1}V [1 − i]) over Γ_{•}^{R,Mal}.

The Quillen adjunction of Proposition 2.17 then shows that this is equivalent to the homotopy class of maps from Γ_{•} to Γ_{•} ⋉ (N^{−1}V [1 − i]) in the slice category sGpd ↓ Γ_{•}, which is just H^{i}(Γ_{•}, V). □

Note that if we have Γ_{•} ∈ sGpd and G ∈ sAGpd together with a morphism f: Γ_{•} → G(k) of simplicial groupoids, then every cosimplicial G-representation V naturally gives rise to a cosimplicial Γ_{•}-representation f^{∗}V. For any coalgebra C, every C-comodule is a nested union of finite-dimensional comodules. Thus every cosimplicial G-representation V is a filtered direct limit \underrightarrow{lim}_{α} V_{α} of levelwise finite-dimensional cosimplicial G-representations, and we tweak the construction of pullbacks slightly by regarding f^{∗}V as the ind-object (i.e. filtered direct system) {f^{∗}V_{α}} of levelwise finite-dimensional cosimplicial Γ_{•}-representations. We then define C^{•}(Γ_{•}, f^{∗}V) := \underrightarrow{lim}_{α} C^{•}(Γ_{•}, f^{∗}V_{α}), and ℍ^{∗}(Γ_{•}, f^{∗}V) := H^{∗}C^{•}(Γ_{•}, f^{∗}V) = \underrightarrow{lim}_{α} ℍ^{∗}(Γ_{•}, f^{∗}V_{α}).

Also note that the category of cosimplicial G-representations is opposite to the category s\begin{matrix} \\ FD & Rep \end{matrix}(G) of [Pri3, §1.5].

Lemma 2.42. Given a cofibrant simplicial groupoid Γ_{•} and a cosimplicial O(Γ_{•}^{R,Mal})- comodule V, the canonical map

ℍ^{∗}(Γ_{•}^{R,Mal}, V) → ℍ^{∗}(Γ_{•}, V)

(induced by the morphism W Γ_{•} → W (Γ_{•}^{R,Mal})) is an isomorphism.

Proof. By Lemma 2.30 and its analogue for sAGpd, we have convergent spectral sequences

ℍ^{i}(Γ, H^{j}(V)) ⟹ ℍ^{i+j}(Γ_{•}, V)
ℍ^{i}(G, H^{j}(V)) ⟹ ℍ^{i+j}(G, V).

For ind-finite-dimensional π_{0}G-representations U, the maps H^{i}(G, U) → H^{i}(Γ_{•}, U) are isomorphisms by Lemma 2.41, so the maps H^{i}(G, H^{j}(V)) → H^{i}(Γ_{•}, H^{j}(V)) are isomorphisms, making the morphism of spectral sequences an isomorphism. □

Theorem 2.43. Take a fibration f: (X, x) → (Y, y) (of pointed connected topological spaces) with connected fibres, and set F := f^{−1}(y). Take a Zariski-dense representation ρ: π_{1}(X, x) → R(k) to a reductive pro-algebraic group R, let K be the closure of ρ(π_{1}(F, x)), and set T := R/K. If the monodromy action of π_{1}(Y, y) on H^{∗}(F, V) factors through ϖ_{1}(Y, y)^{T,Mal} for all K-representations V, then G(F, x)^{K,Mal} is the homotopy fibre of G(X, x)^{R,Mal} → G(Y, y)^{T,Mal}.

Proof. This is [Pri5, Theorem 3.10], which uses Lemma 2.42 to show that H^{∗}(G(F, x)^{K,Mal}, O(K)) and cohomology H^{∗}(F, O(K)) of the homotopy fibre F are both ℍ^{∗}(G(X, x), O(R) ⊗_{O(T)} O(G(Y, y)^{T,Mal})) ≅ H^{j}(F, O(K)). □

2.5. Equivalent formulations. Fix a reductive pro-algebraic groupoid R.

2.5.1. Lie algebras.

Definition 2.44. Recall that a Lie coalgebra C is said to be conilpotent if the iterated cobracket Δ_{n}: C → C^{⊗n} is 0 for sufficiently large n. A Lie coalgebra C is ind-conilpotent if it is a filtered direct limit (or, equivalently, a nested union) of conilpotent Lie coalgebras.

Definition 2.45. Recall from [Pri3, Definition 5.8] that for any k-algebra A, we define N̂_{A}(R) to be opposite to the category of R-representations in ind-conilpotent Lie coalgebras over A, and denote the contravariant equivalence by C ↦ C^{∨}.

Note that there is a continuous functor N̂_{k}(R) → N̂_{A}(R) given by C^{∨} ↦ (C ⊗_{k} A)^{∨}. We denote this by 𝔤 ↦ 𝔤 ˆ⊗A.

Remark 2.46. Observe that 𝔤 ∈ N̂_{A}(R) can be regarded as an object of the category Aff_{A}(R) of R-representations in affine A-schemes, by regarding it as the functor

𝔤(B) := Hom_{A,R}(𝔤^{∨}, B),

for B ∈ Alg_{A}(R) := A ↓ Alg(R). In fact, 𝔤(B) is then a Lie algebra over B, so the Campbell–Baker–Hausdorff formula defines a group structure on 𝔤(B), and the resulting group is denoted by exp(𝔤)(B). Thus exp(𝔤) is an R-representation in affine group schemes over A (i.e. a group object of Aff_{A}(R)).

Definition 2.47. Write sN̂_{A}(R) for the category of simplicial objects in N̂_{A}(R). A weak equivalence in sN̂_{A}(R) is a map which gives isomorphisms on cohomology groups of the duals (which are just A-modules). We denote by Ho(sN̂_{A}(R)) the localisation of sN̂_{A}(R) at weak equivalences.

For k = A, we will usually drop the subscript, so N̂ (R) := N̂_{k}(R), and so on.

Definition 2.48. Define E(R) to be the full subcategory of AGpd↓R consisting of those morphisms ρ: G → R of pro-algebraic groupoids which are pro-unipotent extensions. Similarly, define sE(R) to consist of the pro-unipotent extensions in sAGpd ↓ R, and Ho(sE(R)_{∗}) to be full subcategory of Ho(Ob R↓sAGpd) on objects sE(R).

Definition 2.49. Given a pro-algebraic groupoid R, define the category sP_{A}(R) to have the same objects as sN̂_{A}(R), with morphisms given by

Hom_{sP(R)}(𝔤, 𝔥) = exp( ∏_{x∈Ob R} π_{0}𝔥(x)) ×^{exp(𝔥_{0}^{R})} Hom_{Ho(sN̂ (R))}(𝔤, 𝔥),

where 𝔥_{0}^{R} (the Lie subalgebra of R-invariants in 𝔥_{0}) acts by conjugation on the set of homomorphisms. Composition of morphisms is given by (u, f) ∘ (v, g) = (u ∘ f(v), f ∘ g).

The following is a key comparison result, which will be used in Proposition 2.76 and Theorem 3.30 as a step towards reformulating Malcev homotopy types in terms of Godement resolutions.

Proposition 2.50. For any reductive pro-algebraic groupoid R, the categories Ho(sE(R)_{∗}) and sP(R) are equivalent.

Proof. This is part of [Pri3, Theorem 2.74], adapting [Pri3, Proposition 2.50] to the unpointed case. The proof just exploits the Levi decomposition of Proposition 2.13.

Explicitly, the functor maps 𝔤 ∈ sP(R) to the simplicial pro-algebraic group given in level n by R ⋉ exp(𝔤_{n}). Given a morphism

(u, f) ∈ exp( ∏_{x∈Ob R} π_{0}𝔥(x)) ×^{exp(𝔥^{R})} Hom_{Ho(sN (R))}(𝔤, 𝔥),

lift u to ũ ∈ ∏_{x∈Ob R} exp(𝔥_{0}(x)), and construct the morphism

adũ ∘ (R ⋉ exp(f)): R ⋉ exp(𝔤) → R ⋉ exp(𝔥)

in sE(R), where for a ∈ (R ⋉ exp(𝔥))(x, y), we set adũ(a) = ũ(x) · a · ũ(y)^{−1}. □

Definition 2.51. We can now define the multipointed Malcev homotopy type of X relative to ρ to be the image of G(X, ρ)^{Mal} in Ho(sE(R̃)_{∗}), or equivalently R_{u}G(X, ρ)^{Mal} in sP(R̃). Define the unpointed Malcev homotopy type of X relative to ρ to be the image of G(X, ρ)^{Mal} in Ho(sE(R̃)).

Since R̃ → R is an equivalence of groupoids, there is an equivalence Ho(sE(R)) → Ho(sE(R̃)), so may discard some basepoints to give an object of sP(R) (or equivalently of Ho(sE(R)_{∗})) whenever ρ is surjective on objects.

2.5.2. Chain Lie algebras.

Definition 2.52. Let dgN̂_{A} be opposite to the category of non-negatively graded indconilpotent cochain Lie coalgebras over A. Define dgN̂_{A}(R) to be the category of Rrepresentations in dgN̂_{A}. For k = A, we will usually drop the subscript, so dgN̂ (R) := dgN̂_{k}(R), and so on.

The following is [Pri3, Lemma 5.9] :

Lemma 2.53. There is a closed model structure on dgN̂_{A}(R) in which a morphism f: 𝔤 → 𝔥 is (1) a fibration whenever the underlying map f^{∨}: 𝔥^{∨} → 𝔤^{∨} of cochain complexes over

A is injective in strictly positive degrees;

(2) a weak equivalence whenever the maps H^{i}(f^{∨}): H^{i}(𝔥^{∨}) → H^{i}(𝔤^{∨}) are isomorphisms for all i.

Remark 2.54. It follows from the construction in [Pri3, Lemma 5.9] that for cofibrant objects 𝔤 ∈ dgN̂ (R) (taking A to be a field), 𝔤^{∨} is freely cogenerated as a graded Lie coalgebra. Thus 𝔤^{∨}[−1] is a positively graded strong homotopy commutative algebra without unit (in the sense of [Kon, Lectures 8 and 15]), and a choice of cogenerators on 𝔤^{∨} is the same as a positively graded E_{∞} (a.k.a. C_{∞}) algebra — this is an aspect of Koszul duality.

Definition 2.55. We say that a morphism f: 𝔤 → 𝔥 in dgN̂ (R) is free if there exists a (pro-finite-dimensional) sub-R-representation V ⊂ 𝔥 such that 𝔥 is the free pro-nilpotent graded Lie algebra over 𝔤 on generators V.

Proposition 2.56 (Minimal models). For every object 𝔤 of dgN̂ (R), there exists a free chain Lie algebra 𝔪 with d = 0 on the abelianisation 𝔪/[𝔪, 𝔪], unique up to non-unique isomorphism, together with a weak equivalence 𝔪 → 𝔤.

Proof. [Pri3, Proposition 4.7]. □

The significance of this result is that, together with Proposition 2.50, it allows us to reformulate Malcev homotopy types in terms of extra structure on cohomology groups, since (𝔪/[𝔪, 𝔪])_{n} is dual to H^{n+1}(𝔤, k).

Definition 2.57. Let dgP(R) be the category with the same objects as dgN̂_{A}(R), and morphisms given by

Hom_{dgP(R)}(𝔤, 𝔥) = exp( ∏_{x∈Ob R} H_{0}𝔥(x)) ×^{exp(𝔥_{0}^{R})} Hom_{Ho(dgN̂_{A}(R))}(𝔤, 𝔥),

where 𝔥_{0}^{R} (the Lie subalgebra of R-invariants in 𝔥_{0}) acts by conjugation on the set of homomorphisms. Composition of morphisms is given by (u, f) ∘ (v, g) = (u ∘ f(v), f ∘ g).

Proposition 2.58. There is a normalisation functor N: sN̂_{A}(R) → dgN̂_{A}(R) such that

H_{i}(N𝔤) ≅ π_{i}(𝔤),

giving equivalences Ho(sN̂_{A}(R)) ≃ Ho(dgN̂_{A}(R)), and sP_{A}(R) ≃ dgP_{A}(R).

Proof. This is essentially [Pri3, Propositions 4.12 and 5.11], adapted as in [Pri5, Theorem 3.28]. □

2.5.3. Cosimplicial algebras.

Definition 2.59. Let cAlg(R) be the category of of R-representations in cosimplicial k-algebras.

Proposition 2.60. There is a simplicial model category structure on cAlg(R), in which a map f: A → B is

(1) a weak equivalence if H^{i}(f): H^{i}(A) → H^{i}(B) is an isomorphism in Rep(R) for all i;

(2) a fibration if f^{i}(x): A^{i}(x) → B^{i}(x) is a surjection for all x ∈ Ob (R) and all i.

Proof. This is [Pri3, Proposition 2.60], adapting [Toë, §2.1]. □

Definition 2.61. Let cAlg(R)_{∗} be the category of of R-representations in cosimplicial k-algebras, equipped with an augmentation to ∏_{x∈Ob R} O(R)(x, −). This inherits a model structure from cAlg(R). Denote the opposite category by sAff(R)_{∗} = ∐_{x∈Ob R} R(x, −) ↓ sAff(R), where the coproduct is taken in the category of affine schemes.

Definition 2.62. Given representations V, W ∈ Rep(R), define V ⊗^{R} W := Hom_{Rep(R)}(k, V ⊗ W).

Definition 2.63. Given A ∈ cAlg(R) and 𝔤 ∈ sN̂ (R), define the Maurer-Cartan space MC(A, G) to consist of sets {ω_{n}}_{n≥0}, with ω_{n} ∈ exp(A^{n+1} ˆ⊗^{R}𝔤_{n}), such that

∂_{i}ω_{n} = \begin{cases}∂^{i+1}ω_{n−1} & i > 0 \\ (∂^{1}ω_{n−1}) · (∂^{0}ω_{n−1})^{−1} & i = 0, \end{cases}
σ_{i}ω_{n} = σ^{i+1}ω_{n+1},
σ^{0}ω_{n} = 1,

where exp(A^{n+1} ˆ⊗^{R}𝔤_{n}) is the group whose underlying set is the Lie algebra A^{n+1} ˆ⊗𝔤_{n−1}, with multiplication given by the Campbell–Baker–Hausdorff formula.

Definition 2.64. Given A ∈ cAlg(R) and 𝔤 ∈ sN̂ (R), define the gauge group Gg(A, 𝔤) ≤ ∏_{n} exp(A^{n} ˆ⊗^{R}𝔤_{n}) to consist of those g satisfying

∂_{i}g_{n} = ∂^{i}g_{n−1} ∀i > 0,
σ_{i}g_{n} = σ^{i}g_{n+1} ∀i.

This has an action on MC(A, 𝔤) given by

(g ∗ ω)_{n} = (∂_{0}g_{n+1}) · ω_{n} · (∂^{0}g_{n}^{−1}).

Definition 2.65. Let cAlg(R)_{0∗} be the full subcategory of cAlg(R)_{∗} whose objects satisfy H^{0}(A) ≅ k. Let Ho(cAlg(R)_{0∗}) be the full subcategory of Ho(cAlg(R)_{0∗}) with objects in cAlg(R)_{0∗}. Let sAff(R)_{0∗} be the category opposite to cAlg(R)_{0∗}, and Ho(sAff(R)_{0∗}) opposite to Ho(cAlg(R)_{0∗}).

Definition 2.66. Given a topological space X, and a sheaf ℱ on X, define

C^{n}(X, ℱ) := ∏_{f: |Δ^{n}|→X} Γ(|Δ^{n}|, f^{−1}ℱ).

Together, these form a cosimplicial complex C^{•}(X, ℱ).

2.5.4. Cochain algebras.

Definition 2.67. Define DGAlg(R) to be the category of R-representations in nonnegatively graded cochain k-algebras, and let dgAff(R) be the opposite category.

Lemma 2.68. There is a closed model structure on DGAlg(R) in which a morphism f: A → B is:

(1) a weak equivalence if H^{i}(f): H^{i}(A) → H^{i}(B) is an isomorphism in Rep(R) for all i;

(2) a fibration if f^{i}: A^{i} → B^{i} is a surjection for all i;

(3) a cofibration if it has LLP with respect to all trivial fibrations.

Proof. This is standard (see e.g. [KPT2, Proposition 4.1]). □

Definition 2.69. Define DGAlg(R)_{∗} to be the category of R-representations in non-negatively graded cochain k-algebras, equipped with an augmentation to ∏_{x∈Ob R} O(R)(x, −). This inherits a model structure from DGAlg(R). Define dgAff(R)_{∗} to be the category opposite to DGAlg(R)_{∗}.

Let DGAlg(R)_{0∗} be the full subcategory of DGAlg(R)_{∗} whose objects A satisfy H^{0}(A) = k. Let Ho(DGAlg(R)_{∗})_{0} be the full subcategory of Ho(DGAlg(R)_{∗}) on the objects of DGAlg(R)_{0}. Let dgAff(R)_{0∗} and Ho(dgAff(R)_{∗})_{0} be the opposite categories to DGAlg(R)_{0∗} and Ho(DGAlg(R)_{∗})_{0}, respectively.

Proposition 2.70. There is a denormalisation functor D: DGAlg(R) → cAlg(R) such that

H^{i}(DA) ≅ H^{i}(A).

This is a right Quillen equivalence, with left adjoint D^{∗}, so gives an equivalence Ho(cAlg(R)) ≃ Ho(DGAlg(R)).

Proof. This is [Pri3, Proposition 4.27]. □

Definition 2.71. Given a cochain algebra A ∈ DGAlg(R), and a chain Lie algebra 𝔤 ∈ dgN̂ (R), define the Maurer-Cartan space by

MC(A, 𝔤) := \{ω ∈ ⨁_{n} A^{n+1} ˆ⊗^{R}𝔤_{n} | dω + \frac{1}{2}[ω, ω] = 0\}.

Definition 2.72. Given A ∈ DGAlg(R) and 𝔤 ∈ dgN̂ (R), we define the gauge group by

Gg(A, 𝔤) := exp(∏_{n} A^{n} ˆ⊗^{R}𝔤_{n}).

Define a gauge action of Gg(A, 𝔤) on MC(A, 𝔤) by

g(ω) := g · ω · g^{−1} − (dg) · g^{−1}.

Definition 2.73. Recall that the Thom-Sullivan (or Thom-Whitney) functor Th from cosimplicial algebras to DG algebras is defined as follows. Let Ω(|Δ^{n}|) be the DG algebra of rational polynomial forms on the n-simplex, so

Ω(|Δ^{n}|) = ℚ[t_{0}, …, t_{n}, dt_{0}, …, dt_{n}]/(1 − ∑_{i} t_{i}, ∑_{i} dt_{i}),

for t_{i} of degree 0. The usual face and degeneracy maps for simplices yield ∂_{i}: Ω(|Δ^{n}|) → Ω(|Δ^{n−1}|) and σ_{i}: Ω(|Δ^{n}|) → Ω(|Δ^{n−1}|), giving a simplicial complex of DGAs. Given a cosimplicial algebra A, we then set

Th (A) := {a ∈ ∏_{n} A^{n} ⊗ Ω(|Δ^{n}|) : ∂_{A}^{i}a_{n} = ∂_{i}a_{n+1}, σ_{A}^{j}a_{n} = σ_{j}a_{n−1} ∀i, j}.

The following is a major comparison result, which will be used in Theorem 3.30 as the main step towards reformulating Malcev homotopy types in terms of Godement resolutions.

Theorem 2.74. We have the following commutative diagram of equivalences of categories:

\xymatrix{Ho(dgAff(R)_{∗})_{0} \ar[r]^{Spec D} \ar[d]^{Ḡ} & Ho(sAff(R)_{∗})_{0} \ar[l]^{_{Spec Th}} \ar[d]^{Ḡ} \\ dgP(R) \ar[u]^{W̄} & sP(R), \ar[u]^{W̄} \ar[l]^{N}}

with the pair

\xymatrix{Ho(dgAff(R)_{∗})_{0} \ar[r]^{Ḡ} & dgP(R) \ar[l]^{W̄}},

characterised by the property that

Hom_{Ho(dgAff(R)_{∗})}(Spec A, W̄ 𝔤) = Hom_{dgP(R)}(Ḡ(A), 𝔤) = MC(A, 𝔤)×^{Gg(A,𝔤)} ∏_{x∈Ob R} exp(H_{0}𝔤(x)).

Proof. This is [Pri5, Theorem 3.28], which adapts [Pri3, Corollary 4.41] to the pointed case. The vertical equivalences come from [Pri3, Proposition 3.48], while the horizontal equivalences are from [Pri3, Theorems 4.39] and Theorem 4.44 or, for a shorter and more conceptual proof, [Pri6, Theorem 6.23]. The results of [HS, 4.1] imply that D and Th are homotopy inverses. □

Definition 2.75. Recall that O(R) has the natural structure of an R × R-representation. Since every R-representation has an associated semisimple local system on |BR(k)|, we will also write O(R) for the R-representation in semisimple local systems on |BR(k)| corresponding to the R × R-representation O(R). We then define the R-representation 𝕆(R) in semisimple local systems on X by 𝕆(R) := ρ^{−1}O(R).

Proposition 2.76. Under the equivalences of Theorem 2.74, the relative Malcev homotopy type G(X)^{ρ,Mal} of a topological space X corresponds to

C^{•}(X, 𝕆(R)) ∈ cAlg(R),

equipped with its augmentation to ∏_{x∈X} C^{•}(x, 𝕆(R)) ≅ ∏_{x∈X} O(R)(x, −).

Proof. This is essentially the same as [Pri3, Theorem 3.55] (which considers the unpointed case). □

Corollary 2.77. Pro-algebraic homotopy types are equivalent to the schematic homotopy types of [Toë], in the sense that the full subcategory of the homotopy category Ho(sPr) on objects X^{sch} is equivalent to the full subcategory of Ho(sAGpd) on objects G(X)^{alg}. Under this equivalence, X^{sch} is represented by the simplicial scheme W̄ G(X)^{alg}, and pro-algebraic homotopy groups are isomorphic to schematic homotopy groups.

Proof. [Pri3, Corollary 3.57]. □

Definition 2.78. Given a manifold X, denote the sheaf of real n-forms on X by 𝒜^{n}. Given a real sheaf ℱ on X, write

A^{n}(X, ℱ) := Γ(X, ℱ ⊗_{ℝ} 𝒜^{n}).

Proposition 2.79. The real Malcev homotopy type of a manifold X relative to ρ: π_{f}X → R(ℝ) is given in DGAlg(R) by the de Rham complex A^{•}(X, 𝕆(R)), equipped with its augmentation to ∏_{x∈X} A^{•}(x, 𝕆(R)) ≅ ∏_{x∈X} O(R)(x, −).

Proof. [Pri3, Proposition 4.50]. □

3. Pro-ℚ_{ℓ}-algebraic homotopy types

The purpose if this section is to transfer the framework of Section 2 to an ℓ-adic setting, replacing topological spaces with pro-finite spaces (and hence étale homotopy types of algebraic varieties).

Fix a prime ℓ. Although all results here will be stated for the local field ℚ_{ℓ}, they hold for any of its algebraic extensions.

3.1. Algebraisation of locally pro-finite groupoids.

Definition 3.1. Given a pro-groupoid Γ with Ob (Γ) a discrete set (in the sense of Remark 1.10), we define the pro-algebraic completion Γ^{alg} to be the pro-ℚ_{ℓ}-algebraic groupoid prorepresenting the functor

AGpd → Set
H ↦ Hom_{TopGpd}(Γ, H(ℚ_{ℓ})),

where TopGpd denotes the category of topological groupoids, and H(ℚ_{ℓ}) is endowed with the topology induced from ℚ_{ℓ}. Note that this exists by the Special Adjoint Functor Theorem ([Mac, Theorem V.8.2]), with the algebraic groups GL_{n} providing the data for the solution set condition (by Tannakian duality). Given a set of primes L, define the L-algebraic completion Γ^{L,alg} to be (Γ^{∧_{L}})^{alg}. If P is the set of all primes, we simply write Γ̂^{alg} := Γ^{P,alg}.

Remarks 3.2. Since representations with finite monodromy are algebraic there is a canonical retraction Γ^{L,alg} → Γ^{∧_{L}} of pro-algebraic groupoids.

The motivating example for this definition is when Γ = π_{f}^{ét}(X), the étale fundamental groupoid of an algebraic variety.

The following definition is a slight generalisation of [Pri4, Definition 2.1], and extends Definition 2.20 to pro-groupoids:

Definition 3.3. Given a pro-groupoid Γ with Ob (Γ) discrete, a reductive pro-algebraic groupoid R over ℚ_{ℓ}, and a Zariski-dense (i.e. essentially surjective on objects and Zariskidense on morphisms) continuous map

ρ: Γ^{∧_{L}} → R(ℚ_{ℓ}),

where the latter is given the ℓ-adic topology, we define the relative Malcev completion Γ^{L,ρ,Mal} (or Γ^{L,R,Mal}) to be the universal diagram

Γ^{∧_{L}} ⟶^{g} Γ^{L,ρ,Mal}(ℚ_{ℓ}) ⟶^{f} R̃(ℚ_{ℓ}),

where R̃ is the groupoid equivalent to R on objects Ob Γ (as in Definition 2.20), with f: Γ^{L,ρ,Mal} → R̃ a pro-unipotent extension of pro-ℚ_{ℓ}-algebraic groupoids, g a continuous map of topological groupoids, and their composition equal to ρ.

To see that this universal object exists, we note that this description determines the linear representations of Γ^{L,ρ,Mal} (as described in Remarks 3.4). Since these form a multifibred tensor category, Tannakian duality ([Pri3, Remark 2.6]) then gives a construction of Γ^{L,ρ,Mal}.

Remarks 3.4. By considering groupoid homomorphisms Γ^{∧_{L}} → ∐_{n} GL_{n}(ℚ_{ℓ}), observe that finite-dimensional linear representations of Γ^{L,alg} are just continuous ℚ_{ℓ}-representations of Γ^{∧_{L}}.

Finite-dimensional representations of Γ^{L,ρ,Mal} are only those continuous ℚ_{ℓ}- representations whose semisimplifications are R-representations. Moreover, if we let R be the reductive quotient Γ^{L,red} of Γ^{L,alg}, then Γ^{L,alg} = Γ^{L,R,Mal}.

Definition 3.5. Given an n-dimensional ℚ_{ℓ}-vector space V, a lattice Λ in V is a rank n ℤ_{ℓ}-submodule Λ ⊂ V.

Lemma 3.6. If Γ is a pro-finite group, V an n-dimensional ℚ_{ℓ}-vector space, and ρ: Γ → GL(V) a continuous representation (where the latter is given the ℓ-adic topology) then there exists a lattice Λ ⊂ V such that ρ factors through GL(Λ).

Proof. Since Γ is pro-finite, it is compact, and hence ρ(Γ) ≤ GL(V) must be compact. [Ser1, LG 4 Appendix 1 Theorems 1 and 2] show that every compact subgroup of GL(V) is contained in a maximal compact subgroup, and that the maximal compact subgroups are of the form GL(Λ). Explicitly, we choose a lattice Λ_{0} ⊂ V, then set Λ = ∑_{γ∈Γ} ρ(γ)Λ_{0} (with compactness ensuring the sum is finite). □

Remark 3.7. In particular, when Γ = π_{f}^{ét}(X), this means that finite-dimensional representations of Γ^{alg} are smooth ℚ_{ℓ}-sheaves on X, while finite-dimensional representations of Γ^{red} are semisimple ℚ_{ℓ}-sheaves. The Zariski-dense map ρ: Γ → R(ℚ_{ℓ}) identifies R-representations with a full tensor subcategory of semisimple ℚ_{ℓ}-sheaves, and Γ^{ρ,Mal}- representations are Artinian extensions of these semisimple sheaves.

Proposition 3.8. Given a locally pro-finite groupoid Γ with discrete objects (as in Remark 1.10), and a Zariski-dense continuous map

ρ: Γ^{∧_{L}} → G(ℚ_{ℓ})

to a pro-ℚ_{ℓ}-algebraic groupoid, there is a canonical model G_{ℤ_{ℓ}} for G over ℤ_{ℓ} for which ρ factors through a Zariski-dense map

ρ_{ℤ_{ℓ}}: Γ^{∧_{L}} → G_{ℤ_{ℓ}}(ℤ_{ℓ}).

Proof. Assume that ρ is an isomorphism on objects (replacing G by an equivalent groupoid). Let C be the category of continuous Γ-representations in finite free ℤ_{ℓ}-modules. For each x ∈ Ob Γ, this gives a fibre functor ω_{x} from C to finite free ℤ_{ℓ}-modules.

If we let D be the category of Γ-representations in finite-dimensional ℚ_{ℓ}-vector spaces, with the fibre functors also denoted by ω_{x}, then the category of G-representations is equivalent to a full subcategory D(G) of D, since ρ is Zariski-dense. By Tannakian duality (as in [Pri3, §2.1]), there are isomorphisms

G(x, y)(A) ≅ Iso^{⊗}(ω_{x}|_{D(G)}, ω_{y}|_{D(G)})(A),

where Iso^{⊗} is the set of natural isomorphisms of tensor functors.

Now, by Lemma 3.6, the functor ⊗ℚ_{ℓ}: C → D is essentially surjective. Let C(G) be the full subcategory of C whose objects are those Λ for which Λ ⊗ ℚ_{ℓ} is isomorphic to an object of D(G); these are Γ-lattices in G-representations. Define

G_{ℤ_{ℓ}}(x, y)(A) := Iso^{⊗}(ω_{x}|_{C(G)}, ω_{y}|_{C(G)})(A),

observing that this is an affine scheme (since it preserves all limits), with G_{ℤ_{ℓ}} ⊗ ℚ_{ℓ} = G. Equivalently, we could set O(G_{ℤ_{ℓ}}) ⊂ O(G) to be {f : f(ρ(γ)) ∈ ℤ_{ℓ} ∀γ ∈ Γ}. □

Definition 3.9. Given a finite-dimensional nilpotent Lie algebra 𝔲 over ℚ_{ℓ}, equipped with the continuous action of a pro-finite group Γ (respecting the Lie algebra structure), we say that a lattice Λ ⊂ 𝔲 is admissible if it satisfies the following:

(1) Λ is a Γ-subrepresentation;

(2) Λ is closed under all the monomials in the Campbell–Baker–Hausdorff formula

log(e^{a} · e^{b}) = ∑_{n>0} \frac{(−1)^{n−1}}{n} ∑_{\begin{matrix}r_{i}+s_{i}>0 \\ 1≤i≤n \end{matrix}} \frac{(∑_{i=1}^{n}(r_{i} + s_{i}))^{−1}}{r_{1}!s_{1}! ⋯ r_{n}!s_{n}!} [a^{r_{1}}b^{s_{1}}a^{r_{2}}b^{s_{2}} … a^{r_{n}}b^{s_{n}}],

where

[a^{r_{1}}b^{s_{1}} … a^{r_{n}}b^{s_{n}}] = [\overbrace{a, [a, … [a}^{r_{1}}, [\overbrace{b, [b, … [b}^{s_{1}}, … [\overbrace{a, [a, … [a}^{r_{n}}, [\overbrace{b, [b, … b}^{s_{n}}]] …]],

understood to be 0 if s_{n} > 1 or if s_{n} = 0 and r_{n} > 1

Lemma 3.10. If Λ ⊂ 𝔲 is an admissible lattice and 𝔲 ∈ N, then the image of Λ under the exponential map

exp: 𝔲 → exp(𝔲)

is a pro-finite subgroup.

Proof. We may regard exp(𝔲) as being the set 𝔲, with multiplication given by the Campbell-Baker-Hausdorff formula (which has only finitely many terms in this case, since 𝔲 is nilpotent). Since Λ is closed under all the operations in the formula, it is closed under multiplication. As exp is a homeomorphism, exp(Λ) is compact and thus pro-finite. □

3.2. Pro-ℚ_{ℓ}-algebraic homotopy types. We now proceed as in §2.2, extending to a simplicial framework in order to study the loop groupoid (and hence the whole homotopy type), rather than just the fundamental groupoid.

Definition 3.11. Given a pro-simplicial groupoid G with Ob (G) a discrete set, we define the pro-algebraic completion G^{L,alg} ∈ sAGpd to represent the functor

sAGpd → Set
H ↦ Hom_{sTopGpd}(G^{∧_{L}}, H(ℚ_{ℓ})),

where TopGpd denotes the category of topological groupoids. Note that Lemma 1.17 implies that we can compute this levelwise by (G^{L,alg})_{n} = (G_{n})^{L,alg}.

Remark 3.12. It is natural to ask whether G ↦ G^{L,alg} is left Quillen for any suitable model structure on pro-L simplicial groupoids. This cannot be the case, since the functor is not even a left adjoint, essentially because ℚ_{ℓ} is not pro-finite.

Definition 3.13. Given a pro-simplicial groupoid G with Ob (G) discrete, a reductive pro-algebraic groupoid R over ℚ_{ℓ}, and a Zariski-dense continuous map

ρ: π_{0}(G)^{∧_{L}} → R(ℚ_{ℓ}),

where the latter is given the ℓ-adic topology, we define the relative Malcev completion G^{L,ρ,Mal} ∈ sE(R) ⊂ sAGpd ↓ R by (G^{L,ρ,Mal})_{n} := (G_{n})^{L,ρ∘a_{n},Mal}, for a_{n}: G_{n} → π_{0}G the canonical map.

Note that π_{0}(G^{L,ρ,Mal}) = π_{0}(G)^{L,ρ,Mal}.

Lemma 3.14. If the continuous action of a pro-finite group Γ on 𝔲_{•} ∈ sN_{ℚ_{ℓ}} is semisimple, then 𝔲 is the union of its Γ-equivariant simplicial admissible sublattices.

Proof. Since the action of Γ is semisimple, we may take a complement V_{•} ⊂ 𝔲_{•} of [𝔲_{•}, 𝔲_{•}] as a simplicial Γ-representation. Given a lattice M ⊂ V, let g(M) ⊂ 𝔲 denote the ℤ_{ℓ}- submodule generated by M and the operations in the Campbell-Baker-Hausdorff formula. Since 𝔲 is nilpotent, it follows that g(M) is a finitely generated ℤ_{ℓ}-module, and hence a lattice in 𝔲. By semisimplicity and Lemma 3.6, there exists a Γ-equivariant lattice Λ_{•} ⊂ V_{•}. The lattices ℓ^{−n}Λ_{•} ⊂ V_{•} are also then Γ-equivariant for n ≥ 0, so the lattices g(ℓ^{−n}Λ_{•}) ⊂ 𝔲_{•} are all admissible.

It only remains to show that ⋃ g(ℓ^{−n}Λ) → 𝔲 is a surjective map of Lie algebras. This follows since ⋃ ℓ^{−n}Λ → 𝔲/[𝔲, 𝔲] is surjective. □

Lemma 3.15. Given a compact topological space K and a finite-dimensional nilpotent ℚ_{ℓ}-Lie algebra 𝔲, the map

Hom_{cts}(K, ℤ_{ℓ}) ⊗_{ℤ_{ℓ}} 𝔲 → Hom_{cts}(K, 𝔲)

is an isomorphism.

Proof. First observe that the map is clearly injective, since 𝔲 is a flat ℤ_{ℓ}-module. For surjectivity, note that the image of f: K → 𝔲 must be contained in an admissible sublattice Λ ⊂ 𝔲 (by compactness and Lemma 3.14). Now,

Hom_{cts}(K, Λ) ≅ Hom_{cts}(K, ℤ_{ℓ}) ⊗_{ℤ_{ℓ}} Λ,

since Λ is a finite free ℤ_{ℓ}-module. □

Definition 3.16. Given a continuous representation V of π̂_{f}X in ℚ_{ℓ}-vector spaces, recall the standard definition that

H^{∗}(X, V) := H^{∗}(X, Λ) ⊗_{ℤ_{ℓ}} ℚ_{ℓ},

for any π_{f}X-equivariant ℤ_{ℓ}-lattice Λ ⊂ V as in Lemma 3.6, and H^{∗}(X, Λ) as in Definition 1.23.

Remark 3.17. If X is discrete, note that this is not in general the same as cohomology H^{n}(X, V^{δ}) of the discrete π_{f}X-representation V^{δ} underlying V. However, both will coincide if H_{n}(G, Λ^{∨}) has finite rank, by the Universal Coefficient Theorem and Lemma 1.25.

Example 3.18. If X is a locally Noetherian simplicial scheme, we may consider the étale topological type X_{ét} ∈ pro(𝕊), as defined in [Fri, Definition 4.4]. Since (X_{ét})_{0} is the set of geometric points of X_{0}, we may then apply the constructions of this section. For a finite local system M on X, we have

H^{∗}(X_{ét}, M) ≅ H_{ét}^{∗}(X, M),

by [Fri, Proposition 5.9]. For an inverse system M = {M_{i}} of local systems, we have

H^{∗}(X_{ét}, M) = H^{∗}(\underleftarrow{lim}_{i} C_{ét}^{•}(X, M_{i})) = H_{ét}^{∗}(X, (M)),

where C_{ét}^{•} is a variant of the Godement resolution and H_{ét}^{∗}(X, (M)) is Jannsen’s continuous étale cohomology ([Jan]). If the inverse system is indexed by ℕ and the groups H_{ét}^{∗}(X, M_{i}) satisfy the Mittag-Leffler condition (in particular, if they are finite), then

H^{∗}(X_{ét}, M) ≅ \underleftarrow{lim}_{i} H_{ét}^{∗}(X, M_{i}).

[Fri, Theorem 7.3] shows that X_{ét} ∈ 𝕊̂ whenever the schemes X_{n} are connected and geometrically unibranched. It seems that this result can be extended to simplicial schemes (or even simplicial algebraic spaces) for which the homotopy groups π_{m}^{ét}(X_{n}) satisfy the π_{∗}- Kan condition ([GJ, §IV.4]), provided the simplicial set π(X)_{•}, given by π(X)_{n} := π(X_{n}), the set of connected components of X_{n}, has finite homotopy groups.

Proposition 3.19. Take X ∈ pro(𝕊) with X_{0} discrete, and a Zariski-dense continuous map

ρ: π_{f}(X)^{∧_{L}} → R(ℚ_{ℓ}),

for ℓ ∈ L, with Ob R = Ob π_{f}(X). Then G(X)^{L,ρ,Mal} is cofibrant (for the model structure of Lemma 2.16), the map G(X)^{L′,ρ,Mal} → G(X)^{L,ρ,Mal} is an isomorphism for all L ⊂ L′, and

H^{∗}(G(X)^{L,ρ,Mal}, V) ≅ H^{∗}(X, ρ^{∗}V).

Proof. Let Δ ≤ R(ℚ_{ℓ}) be the image of ρ. Write {X_{α}}_{α∈𝕀} for the inverse system X. For 𝔲 ∈ sN (R),

Hom_{sTopGpd}(G(X)^{∧_{L}}, exp(𝔲) ⋊ R)_{R} = Hom_{sTopGpd}(G(X)^{∧_{L}}, exp(𝔲) ⋊ Δ)_{Δ}.

Since 𝔲 ∈ sN (R), the normalisation N𝔲 is bounded in degrees ≤ n, say. This implies that 𝔲 = cosk_{n+1}𝔲, the n + 1-coskeleton, or equivalently that any simplicial morphism Y → 𝔲 is determined by the maps Y_{i} → 𝔲_{i} for i ≤ n + 1.

Thus any morphism f: G(X)^{∧_{L}} → exp(𝔲)⋊Δ is determined by the maps f_{i}: G(X)_{i}^{∧_{L}} → exp(𝔲_{i})⋊Δ for i ≤ n+1. Now, by Lemma 3.14, exp(𝔲)⋊Δ is the union over all admissible Δ-equivariant sublattices Λ ⊂ 𝔲 of exp(Λ) ⋊ Δ. Since each G(X)_{i}^{∧_{L}} is compact, its image in exp(𝔲_{i}) ⋊ Δ must be contained in exp(Λ_{i}) ⋊ Δ for some admissible Λ ⊂ 𝔲. By choosing Λ large enough that this holds for all i ≤ n + 1, we see that

Hom_{sTopGpd}(G(X)^{∧_{L}}, exp(𝔲) ⋊ R)_{R} = lim_{\overrightarrow{Λ⊂𝔲 admissible}} Hom_{sTopGpd}(G(X)^{∧_{L}}, exp(Λ) ⋊ Δ)_{Δ}
= lim_{\overrightarrow{Λ⊂𝔲 admissible}} Hom_{spro(Gpd^{L})}(G(X)^{∧_{L}}, exp(Λ) ⋊ Δ)_{Δ},

because pro(Gpd^{L}) is a full subcategory of TopGpd. Here, exp(Λ) ⋊ Δ ∈ pro(sGpd^{L}) denotes the pro-object {(exp(Λ)/ exp(ℓ^{m}Λ)) ⋊ Δ}_{m}. From now on, we will abuse notation by writing exp(Λ/ℓ^{n}Λ) or even exp(Λ/ℓ^{n}) for the finite group exp(Λ)/ exp(ℓ^{n}Λ).

Now, since Λ = cosk_{n+1}Λ, any morphism f: H → exp(Λ/ℓ^{m}Λ) ⋊ Δ is determined by the maps f_{i} for i ≤ n+1. As exp(Λ/ℓ^{m}Λ) is levelwise finite, and filtered colimits commute with finite limits, this means that

Hom_{spro(Gpd^{L})}(G(X)^{∧_{L}}, exp(Λ) ⋊ Δ)_{Δ} = Hom_{pro(sGpd^{L})}(G(X)^{∧_{L}}, exp(Λ) ⋊ Δ)_{Δ}.

Hence

Hom_{sTopGpd}(G(X)^{∧_{L}}, exp(𝔲) ⋊ R)_{R} = lim_{\overrightarrow{Λ⊂𝔲 admissible}} Hom_{pro(sGpd^{L})}(G(X), exp(Λ) ⋊ Δ)_{Δ}.

Under the adjunction G ⊣ W̄, this becomes

lim_{\overrightarrow{Λ⊂𝔲 admissible}} Hom_{pro(𝕊)}(X, W̄ (exp(Λ) ⋊ Δ))_{W̄ Δ}.

This expression is independent of L, so we have shown that G(X)^{L′,ρ,Mal} → G(X)^{L,ρ,Mal} is an isomorphism for all L ⊂ L′.

For p: 𝔲 → 𝔳 an acyclic small extension with kernel I in sN (R), and an admissible lattice Λ′ < 𝔲, consider the map Λ′ → p(Λ′). This is surjective, and H_{∗}(Λ′ ∩ I) ⊗ ℚ_{ℓ} = 0, since (Λ′∩I)⊗ℚ_{ℓ} ≅ I. As H_{∗}(I) = 0, we may choose a Δ-equivariant lattice Λ′∩I < M < I such that H_{∗}(M/ℓM) = 0. Let Λ := Λ′ + M, noting that this is an admissible lattice (p being small), with the maps Λ/ℓ^{n} → p(Λ)/ℓ^{n} all acyclic.

In order to show that G(X)^{L,ρ,Mal} is cofibrant, take an arbitrary map f: G(X)^{∧_{L}} → exp(𝔳) ⋊ Δ over ρ; this must factor through exp(p(Λ′)) for some admissible lattice Λ′ < 𝔲, and we may replace Λ by Λ′ as above. It therefore suffices to show that the corresponding map

f: X → W̄ (exp(p(Λ)) ⋊ Δ)

in pro(𝕊) lifts to W̄ (exp(Λ) ⋊ Δ). For each n ∈ ℕ, we have a map

f_{n}: X_{α(n)} → W̄ (exp(p(Λ)/ℓ^{n}) ⋊ Δ),

and these are compatible with the structural morphisms.

We now prove existence of the lift by induction on n. Assume we have g_{n}: X_{α(n)} → W̄ (exp(Λ/l^{n}) ⋊ Δ), such that p ∘ g_{n} = f_{n}. This gives a map

(f_{n+1}, g_{n}): X_{α(n)} → W̄ (exp((p(Λ)/ℓ^{n+1}) ×_{p(Λ)/ℓ^{n}} (Λ/ℓ^{n})) ⋊ Δ).

However, Λ/ℓ^{n+1} → (p(Λ)/ℓ^{n+1}) ×_{p(Λ)/ℓ^{n}} (Λ/ℓ^{n}) is an acyclic small extension, so

W̄ (exp(Λ/ℓ^{n+1}) ⋊ Δ) → W̄ (exp((p(Λ)/ℓ^{n+1}) ×_{p(Λ)/ℓ^{n}} (Λ/ℓ^{n})) ⋊ Δ)

is a trivial fibration, allowing us to construct a lift g_{n+1} : X_{α(n+1)} → W̄ (exp(Λ/ℓ^{n+1})⋊Δ). This completes the proof that G(X)^{L,ρ,Mal} is cofibrant.

Finally, if V is an R-representation then H^{n+1}(G(X)^{L,ρ,Mal}, V) is the coequaliser of the diagram

\xymatrix{Hom_{sAGpd↓R}(G(X)^{L,ρ,Mal}, (N^{−1}V [−n])^{Δ^{1}}) \ar[r] \ar[r] & Hom_{sAGpd↓R}(G(X)^{L,ρ,Mal}, N^{−1}V [−n]).}

For a Δ-equivariant lattice Λ ⊂ V, this is the direct limit over m of

\xymatrix{Hom_{pro(𝕊↓W̄ R)}(X, W̄ ((N^{−1}ℓ^{−m}Λ[−n])^{Δ^{1}} ⋊ R)) \ar[r] \ar[r] & Hom_{pro(𝕊↓W̄ R)}(X, W̄ (N^{−1}ℓ^{−m}Λ[−n] ⋊ R)).}

Hence

H^{n+1}(G(X)^{L,ρ,Mal}, V) ≅ lim_{\overrightarrow{m}} H^{n+1}(X, l^{−m}Λ) = H^{n+1}(X, Λ) ⊗ ℚ_{ℓ} = H^{n+1}(X, V),

as required. □

Definition 3.20. Given X and ρ as above, define the relative Malcev homotopy type

X^{ρ,Mal} := G(X)^{P,ρ,Mal},

where P is the set of all primes, noting that this is isomorphic to G(X)^{L,ρ,Mal} for all L ∋ ℓ, by Proposition 3.19 and Lemma 2.39.

Define

X^{L,alg} := G(X)^{L,alg}.

Remark 3.21. Note that if X ∈ 𝕊, this definition of Malcev completion differs slightly from the Malcev homotopy type X^{ρ,Mal} of Definition 2.51, which is given by G(X)^{ρ,Mal}. However, the following lemma rectifies the situation.

Lemma 3.22. For X ∈ 𝕊 and ρ: π_{f}(X)^{∧_{L}} → R(ℚ_{ℓ}) Zariski-dense and continuous, there is a canonical map

G(X)^{ρ,Mal} → G(X)^{L,ρ,Mal};

this is a quasi-isomorphism whenever the groups H^{n}(X, V) are finite-dimensional for all finite-dimensional R-representations V.

Proof. Existence of the map is immediate. To see that it gives a quasi-isomorphism, Lemma 2.39 shows that we need only look at cohomology groups. Given an Rrepresentation V corresponding to a local system 𝕍 over ℚ_{ℓ} on X, the map on cohomology groups is

H^{∗}(X^{∧_{L}}, 𝕍) → H^{∗}(X, 𝕍);

this is an isomorphism by Remark 3.17. □

Definition 3.23. Define pro-algebraic (or schematic) and relative homotopy groups by ϖ_{n}(X^{∧_{L}}) := π_{n−1}(G(X)^{L,alg}) and ϖ_{n}(X^{ρ,Mal}) := π_{n−1}(G(X)^{P,ρ,Mal}).

Define pro-algebraic (or schematic) and relative fundamental groupoids by ϖ_{f}(X^{∧_{L}}) :=

π_{f}(X)^{L,alg} and ϖ_{f}(X^{ρ,Mal}) := π̂_{f}X^{ρ,Mal}.

Define ϖ_{f}(X̂), ϖ_{n}(X̂) by the convention that X̂ = X^{∧_{P}}, for P the set of all primes.

Note that Lemma 3.6 implies that for a locally Noetherian scheme X, finite-dimensional ϖ_{f}(X_{ét}^{∧_{L}})-representations correspond to smooth ℚ_{ℓ}-sheaves on X.

The following now follow immediately from Lemma 2.39

Corollary 3.24. A map f: X → Y in pro(𝕊), with X_{0}, Y_{0} discrete, induces an isomorphism

f^{L,alg} : X^{L,alg} → Y^{L,alg}

of homotopy types if and only if the following conditions hold:

(1) f^{∗} induces an equivalence between the categories of finite-dimensional semisimple continuous ℚ_{ℓ}-representations of (π_{f}X)^{∧_{L}} and (π_{f}Y)^{∧_{L}};

(2) for all finite-dimensional semisimple continuous ℚ_{ℓ}-representations V of π_{f}Y, the

maps

f^{∗}: H^{∗}(Y, V) → H^{∗}(X, f^{∗}V)

are isomorphisms.

Corollary 3.25. Take a map f: X → Y in pro(𝕊), with X_{0}, Y_{0} discrete, and with a Zariski-dense morphism ρ: (π_{f}Y)^{∧_{L}} → R(ℚ_{ℓ}) such that ρ ∘ f: (π_{f}X)^{∧_{L}} → R(ℚ_{ℓ}) is also Zariski-dense. Then f induces an isomorphism

f^{R,Mal} : X^{R,Mal} → Y^{R,Mal}

of homotopy types if and only if for all R-representations V, the maps

f^{∗}: H^{∗}(Y, ρ^{∗}V) → H^{∗}(X, f^{∗}ρ^{∗}V)

are isomorphisms.

3.3. Equivariant cochains. Proposition 2.79 showed how the schematic homotopy type of a manifold can be recovered from the de Rham complex with local system coefficients. We will now establish an analogue for algebraic varieties, involving an étale Godement resolution with coefficients in smooth ℚ_{ℓ}-sheaves.

Lemma 3.26. If Λ is a Γ-representation in pro-simplicial groups such that Λ ⋊ Γ ∈ pro(sGpd), then

Hom_{Γ,pro(𝕊)}(X̃, W̄ Λ) ≅ Hom_{pro(𝕊)↓BΓ}(X, W̄ (Λ ⋊ Γ)),

for X̃ as in Definition 1.20.

Proof. The calculation is essentially the same as for [Pri3, Lemma 3.53]. □

Definition 3.27. Given an ind-finite rank ℤ_{ℓ}-local system (i.e. a filtered direct system in the category of finite rank ℤ_{ℓ}-local systems) 𝕍 = {𝕍_{α}}_{α}, define

C^{•}(X, 𝕍) := \underrightarrow{lim}_{α} C^{•}(X, 𝕍_{α}),

where the right-hand side is given in Definition 1.21.

Definition 3.28. Given a pro-algebraic groupoid G over ℤ_{ℓ}, define O(G) to be the G×Grepresentation given by global sections of the structure sheaf of G, equipped with its left and right G-actions.

Given a representation ρ: π_{f}X → G(ℤ_{ℓ}), let 𝕆(G) be the G-representation in (ind-finite rank) ℤ_{ℓ}-local systems on X given by pulling O(G) back along its right G-action.

Definition 3.29. Given X, L, ρ, R as in Proposition 3.19, let R_{ℤ_{ℓ}} be the ℤ_{ℓ}-model for R constructed in Proposition 3.8, and set

C^{•}(X, 𝕆(R)) := C^{•}(X, 𝕆(R_{ℤ_{ℓ}})) ⊗_{ℤ_{ℓ}} ℚ_{ℓ}.

Theorem 3.30. For X, L, ρ, R as in Proposition 3.19, the relative Malcev homotopy type

G(X)^{L,ρ,Mal} ∈ sAGpd↓R

corresponds under the equivalences of Proposition 2.50 and Theorem 2.74 to the Rrepresentation

C^{•}(X, 𝕆(R))

in cosimplicial k-algebras, equipped with its natural augmentation to ∏_{x∈X_{0}} C^{•}(x, 𝕆(R)) = ∏_{x∈Ob R} O(R)(x, −).

Proof. We need to show that, for 𝔲 ∈ sN (R),

Hom_{sAGpd↓R}(G(X)^{L,ρ,Mal}, exp(𝔲) ⋊ R) ≅ Hom_{sAff(R)}(Spec C^{•}(X, 𝕆(R)), W̄ (exp(𝔲))).

Adapting the proof of Proposition 3.19, we know that

Hom_{sAGpd↓R}(G(X)^{L,ρ,Mal}, exp(𝔲) ⋊ R) ≅ lim_{\overrightarrow{Λ}} Hom_{pro(𝕊)}(X, W̄ (exp(Λ) ⋊ R_{ℤ_{ℓ}}(ℤ_{ℓ})))_{BR_{ℤ_{ℓ}}(ℤ_{ℓ})},

where the limit is taken over Λ ⊂ 𝔲 admissible. By Lemma 3.26,

Hom_{pro(𝕊)}(X, W̄ (exp(Λ) ⋊ R_{ℤ_{ℓ}}(ℤ_{ℓ})))_{BR_{ℤ_{ℓ}}(ℤ_{ℓ})} ≅ Hom_{R_{ℤ_{ℓ}}(ℤ_{ℓ}),pro(𝕊)}(X̃, W̄ exp(Λ)).

If we regard exp(Λ) as the ℤ_{ℓ}-valued points of the group scheme exp(Λ)(A) := exp(Λ ⊗ A), then this is an affine space, so

Hom_{pro(𝕊)}(X̃, W̄ exp(Λ)) ≅ Hom_{sAff_{ℤ_{ℓ}}} (Spec C^{•}(X̃, ℤ_{ℓ}), W̄ exp(Λ)).

Since Λ ≅ Λ ⊗_{ℤ_{ℓ}}^{R} O(R_{ℤ_{ℓ}}), we then have

Hom_{R_{ℤ_{ℓ}}(ℤ_{ℓ}),pro(𝕊)}(X̃, W̄ exp(Λ)) ≅ Hom_{sAff(R_{ℤ_{ℓ}})}(Spec C^{•}(X, 𝕆(R_{ℤ_{ℓ}})), W̄ exp(Λ)).

The map

lim_{\overrightarrow{Λ}} Hom_{sAff(R_{ℤ_{ℓ}})}(Spec C^{•}(X, 𝕆(R_{ℤ_{ℓ}})), W̄ exp(Λ))
→ lim_{\overrightarrow{Λ}} Hom_{sAff(R_{ℤ_{ℓ}})}(Spec C^{•}(X, 𝕆(R_{ℤ_{ℓ}})) ⊗ ℚ_{ℓ}, W̄ exp(Λ))

is clearly injective. However, since there exists an admissible lattice Λ′ with l^{−n}Λ ⊂ Λ′, the map must also be surjective. Finally, note that

Hom_{sAff(R_{ℤ_{ℓ}})}(Spec C^{•}(X, 𝕆(R_{ℤ_{ℓ}})) ⊗ ℚ_{ℓ}, W̄ exp(Λ))
= Hom_{sAff(R)}(Spec C^{•}(X, 𝕆(R)), W̄ exp(Λ ⊗ ℚ_{ℓ})),

as required. □

Remarks 3.31. We could use Proposition 2.70 to replace C^{•}(X, 𝕆(R)) with a DG algebra, giving a more reassuring analogue of the de Rham algebra used in Proposition 2.79 to govern relative Malcev homotopy types of manifolds. This is the approach taken in [Ols1], and when R = 1, it corresponds to Deligne’s ℚ_{ℓ}-homotopy type ([Del2, §V]). However, in the sequel we will work systematically with cosimplicial rather than DG objects — both approaches being equivalent, the transfer can add unnecessary complication.

Note that if we take a scheme X, then Proposition 2.76 adapts to show that C^{•}(X_{ét}, 𝕍) is a Godement resolution for the continuous étale cohomology of 𝕍. Under the comparison

of Corollary 2.77, this shows that for an algebraic variety X, { \atop G(X_{ét})}^{alg} agrees with the ℓ-adic homotopy type discussed in [Toë, §3.5.3].

Given any morphism ρ: ϖ_{f}(X̂_{ét})^{red} → R to a reductive group, there is a forgetful functor ρ^{♯}: sN̂ (R) → sN̂ (ϖ_{f}(X̂_{ét})^{red}). If we write 𝕃ρ_{♯} for the derived left adjoint and ρ is

surjective, then R_{u}({ \atop G(X_{ét})}^{ρ,Mal}) = 𝕃ρ_{♯}R_{u}({ \atop G(X_{ét})}^{alg}). Note that for C a Tannakian subcategory (see Definition 2.6) of FDRep(ϖ_{f}(X̂_{ét})^{red}), with corresponding groupoid G, the

homotopy type X_{C_{ét}} of [Ols1, 1.5] is equivalent to 𝕃ρ_{♯}R_{u}({ \atop G(X)}^{alg}), for ρ: ϖ_{f}(X̂_{ét})^{red} → G.

3.4. Completing fibrations. Observe that the definitions and results of §2.4 extend naturally to pro-groupoids and pro-spaces; we will make use of this extension without further comment.

Theorem 3.32. Take a pro-fibration f: (X, x) → (Y, y) of connected objects in pro(𝕊) with connected fibres, and set F := f^{−1}(y). Take a Zariski-dense representation ρ: π_{1}(X, x) → R(ℚ_{ℓ}) to a reductive pro-algebraic group R, let K be the Zariski closure of ρ(π_{1}(F, x)), and set T := R/K. If the monodromy action of π_{1}(Y, y) on H^{∗}(F, V) factors through ϖ_{1}(Y, y)^{T,Mal} for all K-representations V, then G(F, x)^{K,Mal} is the homotopy fibre of G(X, x)^{R,Mal} → G(Y, y)^{T,Mal}.

In particular, there is a long exact sequence

… → ϖ_{n}(F, x)^{K,Mal} → ϖ_{n}(X, x)^{R,Mal} → ϖ_{n}(Y, y)^{T,Mal} → ϖ_{n−1}(F, x)^{K,Mal} →

… → ϖ_{1}(F, x)^{K,Mal} → ϖ_{1}(X, x)^{R,Mal} → ϖ_{1}(Y, y)^{T,Mal} → 1.

Proof. We adapt the proof of Theorem 2.43.

First observe that ρ(π_{1}(F, x)) is normal in π_{1}(X, x), so K is normal in R, and T is therefore a reductive pro-algebraic group, so (Y, y)^{T,Mal} is well-defined. Next, observe that since K is normal in R, R_{u}(K) is also normal in R, and is therefore 1, ensuring that K is reductive, so (F, x)^{K,Mal} is also well-defined.

Consider the complex O(R) ⊗_{O(T)} O(G(Y, y)^{T,Mal}) of G(X, x)^{R,Mal}-representations, regarded as a cosimplicial G(X, x)-representation. Since G(F, x) → ker(G(X, x) → G(Y, y)) is a weak equivalence, the Hochschild-Serre spectral sequence for f (Proposition 2.36) with coefficients in this complex is

E_{2}^{i,j} = ℍ^{i}(G(Y, y), H^{j}(F, O(R)) ⊗_{O(T)} O(G(Y, y)^{T,Mal}))
⟹ ℍ^{i+j}(G(X, x), O(R) ⊗_{O(T)} O(G(Y, y)^{T,Mal})).

Regarding O(R) as a K-representation, H^{∗}(F, O(R)) is a ϖ_{1}(Y, y)^{T,Mal}-representation by hypothesis. Hence H^{∗}(F, O(R)) ⊗_{O(T)} O(G(Y, y)^{T,Mal}) is a cosimplicial G(Y, y)^{T,Mal}- representation, so

ℍ^{i}(G(Y, y), H^{j}(F, O(R))⊗_{O(T)}O(G(Y, y)^{T,Mal})) ≅ ℍ^{i}(G(Y, y)^{T,Mal}, H^{j}(F, O(R))⊗_{O(T)}O(G(Y, y)^{T,Mal})),

by Lemma 2.42.

Now, H^{∗}(F, O(R)) ⊗_{O(T)} O(G(Y, y)^{T,Mal}) is a fibrant cosimplicial G(Y, y)^{T,Mal}- representation, so

ℍ^{i}(G(Y, y)^{T,Mal}, H^{j}(F, O(R)) ⊗_{O(T)} O(G(Y, y)^{T,Mal}))

≅ H^{i}Γ(G(Y, y)^{T,Mal}, H^{j}(F, O(R)) ⊗_{O(T)} O(G(Y, y)^{T,Mal}))

= \begin{cases}H^{j}(F, O(R)) ⊗_{O(T)} k = H^{j}(F, O(K)) & i = 0 \\ 0 & i ≠ 0, \end{cases}

so

ℍ^{j}(G(X, x), O(R) ⊗_{O(T)} O(G(Y, y)^{T,Mal})) ≅ H^{j}(F, O(K)).

Now, let F be the homotopy fibre of G(X, x)^{R,Mal} → G(Y, y)^{T,Mal} (which is just the kernel as this map is surjective), noting that there is a natural map G(F, x)^{K,Mal} → F. Lemma 2.42 implies that

ℍ^{j}(G(X, x), O(R)⊗_{O(T)}O(G(Y, y)^{T,Mal})) = ℍ^{j}(G(X, x)^{R,Mal}, O(R)⊗_{O(T)}O(G(Y, y)^{T,Mal})),

and [Pri3, Theorem 1.51] gives a Hochschild–Serre spectral sequence

ℍ^{i}(G(Y, y)^{T,Mal}, H^{j}(F, O(R)) ⊗_{O(T)} O(G(Y, y)^{T,Mal}))
⟹ ℍ^{i+j}(G(X, x)^{R,Mal}, O(R) ⊗_{O(T)} O(G(Y, y)^{T,Mal})).

The reasoning above adapts to show that this spectral sequence also collapses, yielding

H^{j}(F, O(K)) = ℍ^{j}(G(X, x), O(R) ⊗_{O(T)} O(G(Y, y)^{T,Mal})).

We have therefore shown that the map G(F, x)^{K,Mal} → F gives an isomorphism

H^{∗}(F, O(K)) → H^{∗}(G(F, x)^{K,Mal}, O(K)),

and hence isomorphisms H^{∗}(F, V) → H^{∗}(G(F, x)^{K,Mal}, V) for all K-representations V. Since this is a morphism of simplicial pro-unipotent extensions of K, [Pri3, Corollary 1.55] implies that G(F, x)^{K,Mal} → F is a weak equivalence. □

Examples 3.33. Note that we can apply this theorem to f_{ét} : X_{ét} → Y_{ét} whenever f: X → Y is geometric fibration in the sense of [Fri, Definition 11.4]. This includes smooth projective morphisms, as well as smooth quasi-projective morphisms where the divisor is transverse to f. The fibre of f_{ét} over y will then be equivalent to (f^{−1}{y})_{ét}.

Another source of examples comes from nerves of pro-finite groups. Any surjection g: Γ → Δ of pro-finite groups gives a pro-fibration BΓ → BΔ, with fibre B(ker g).

Of course, even if f: X → Y is not a pro-fibration, we can take a fibrant replacement. This will have connected fibres if and only if π_{1}(X, x) → π_{1}(Y, y) is surjective, and the theorem then describes the homotopy fibre of f.

3.5. Comparison with Artin–Mazur homotopy groups.

Lemma 3.34. Let f: X → Y be a morphism in pro(𝕊)_{δ} for which the map

π_{n}(f) : π_{n}(X) → π_{n}(Y)

is a pro-isomorphism for n ≤ N and a pro-surjection for n = N +1, and take a continuous Zariski-dense morphism ρ: π_{f}Y → R(ℚ_{ℓ}). Then the map

ϖ_{n}(f) : ϖ_{n}(X, ρ ∘ f)^{Mal} → ϖ_{n}(Y, ρ)^{Mal}

is an isomorphism for n ≤ N and a surjection for n = N + 1.

Proof. The proof of Lemma 2.22 carries over to this generality. □

Definition 3.35. By analogy with Definition 2.23, say that a locally pro-discrete groupoid Γ is n-good with respect to a continuous Zariski-dense representation ρ: Γ → R(ℚ_{ℓ}) to a reductive pro-algebraic groupoid if for all finite-dimensional Γ^{ρ,Mal}-representations V, the map

H^{i}(Γ^{ρ,Mal}, V) → H^{i}(Γ, V)

is an isomorphism for all i ≤ n and and an inclusion for i = n + 1. Say that Γ is good with respect to ρ if it is n-good for all n.

If Γ is (n-)good relative to Γ^{red}, then we say that Γ is algebraically (n-)good.

Lemma 3.36. A pro-groupoid Γ is N-good with respect to ρ if and only if for any finitedimensional Γ^{ρ,Mal}-representation V, and α ∈ H^{n}(Γ, V) for n ≤ N, there exists an injection f: V → W_{α} of finite-dimensional Γ^{ρ,Mal}-representations, with f(α) = 0 ∈ H^{n}(Γ, W_{α}).

Proof. This is a special case of the results of [Pri8, §1.2.3], which adapt directly from groups to groupoids. □

Lemma 3.37. Let Γ be a locally finitely presented (L, N)-good groupoid and ρ: Γ^{∧_{L}} → R(ℚ_{ℓ}) a Zariski-dense representation, with ℓ ∈ L. Then Γ is N-good relative to ρ: Γ → R(ℚ_{ℓ}) if and only if Γ^{∧_{L}} is N-good relative to ρ.

Proof. Take a finite-dimensional R-representation V. By Lemma 3.22, (BΓ)^{ρ,Mal} ≃ (BΓ)^{L,ρ,Mal}. Since Γ is L-good, Lemma 1.36 gives that π_{n}((BΓ)^{∧_{L}}) = 0 for all 1 < n ≤ N. Applying Lemma 3.34 to the morphism (BΓ)^{∧_{L}} → B(Γ^{∧_{L}}), the observations above show that

ϖ_{n}(BΓ)^{ρ,Mal} → ϖ_{n}(B(Γ^{∧_{L}}))^{L,ρ,Mal}

is an isomorphism for n ≤ N and a surjection for n = N + 1.

Now, [Pri8, §1.2.3] shows that a pro-group G is N-good relative to ρ if and only if ϖ_{n}(BG)^{L,ρ,Mal} = 0 for 1 < n ≤ N, and the same proof adapts to groupoids. Thus Γ is N-good relative to ρ if and only if Γ^{∧_{L}} is so. □

Examples 3.38. A pro-finite group Γ is good with respect to a representation ρ: Γ^{∧_{L}} → R whenever any of the following holds:

(1) Γ is finite, or Γ^{∧_{L}} ≅ Δ^{∧_{L}}, for Δ a finitely generated free discrete group.

(2) Γ^{∧_{L}} ≅ Δ^{∧_{L}}, for Δ a finitely generated nilpotent discrete group.

(3) Γ^{∧_{L}} ≅ Δ^{∧_{L}}, for Δ the fundamental group of a compact Riemann surface. In particular, this applies if Γ is the fundamental group of a smooth projective curve C/k, for k a separably closed field whose characteristic is not in L.

(4) If 1 → F → Γ → Π → 1 is an exact sequence of groups, with F finite and F^{∧_{L}} → Γ^{∧_{L}} injective, assume that Π^{∧_{L}} is good relative to R/\overline{ρ(F)}, where denotes Zariski closure. Then Γ is good relative to ρ.

Proof. Combine Lemma 3.37 with Examples 1.28 and [Pri3, Examples 3.20]. □

Remark 3.39. For an example of an important pro-finite group which is not good with respect to a representation, note that Sp_{g}(ℤ_{ℓ}) is not good with respect to the natural map ρ: Sp_{g}(ℤ_{ℓ}) → Sp_{g}(ℚ_{ℓ}) for g ≥ 2. In fact, ϖ_{2}((BSp_{g}(ℤ_{ℓ}))^{ρ,Mal}) ≅ 𝔾_{a}. This issue arises in [HM1], considering the pro-finite mapping class group Γ_{g} acting on a genus g curve. The action on cohomology gives a map ρ: Γ_{g} → Sp_{g}(ℤ_{ℓ}) with kernel T_{g}, the Torelli subgroup, and the map T_{g}^{1,Mal} → ker(Γ_{g}^{ρ,Mal} → Sp_{g}) has kernel 𝔾_{a}. Theorem 3.32 allows us to interpret this copy of 𝔾_{a} as the image of the connecting homomorphism ϖ_{2}((BSp_{g}(ℤ_{ℓ}))^{ρ,Mal}) → T_{g}^{1,Mal}.

Theorem 3.40. Let L be a set of primes containing ℓ, and take X ∈ pro(𝕊)_{δ} with fundamental groupoid π_{f}X = Γ, equipped with a continuous Zariski-dense representation ρ: Γ^{∧_{L}} → R(ℚ_{ℓ}) to a reductive pro-algebraic groupoid. If

(1) π_{n}(X^{∧_{L}}, −) ⊗ℤ̂ ℚ_{ℓ} is finite-dimensional for all 1 < n ≤ N, and

(2) the Γ^{∧_{L}}-representation π_{n}(X^{∧_{L}}, −)⊗ℤ̂ℚ_{ℓ} is an extension of R-representations (i.e.

a Γ^{L,ρ,Mal}-representation) for all 1 < n ≤ N,

then for each x ∈ X there is an exact sequence

[figure: ϖ_{N+1}(X^{L,ρ,Mal}, x); ϖ_{N+1}((BΓ)^{L,ρ,Mal}); π_{N}(X^{∧_{L}}, x) ⊗ℤ̂ ℚ_{ℓ}; ϖ_{N}(X^{L,ρ,Mal}, x); ϖ_{N}((BΓ)^{L,ρ,Mal}); …; …; π_{2}(X^{∧_{L}}, x) ⊗ℤ̂ ℚ_{ℓ}; ϖ_{2}(X^{L,ρ,Mal}, x); ϖ_{2}((BΓ)^{L,ρ,Mal}); 0.]

In particular, if in addition Γ^{∧_{L}} is (N + 1)-good (resp. N-good) with respect to ρ, then the canonical map

π_{n}(X^{∧_{L}}, −) ⊗ℤ̂ ℚ_{ℓ} → ϖ_{n}(X^{L,ρ,Mal})

is an isomorphism for all n ≤ N (resp. an isomorphism for all n < N and a surjection for n = N).

Proof. Without loss of generality, we may assume that X is connected, choose a point x ∈ X, and replace R with the group R(x, x). Let (X̃, x) be the universal cover of (X, x), and note that we have a homotopy fibration sequence (X̃, x) → (X, x) → Bπ_{1}(X, x), which means that we can apply Theorem 3.32 (after taking a fibrant replacement for (X, x) → Bπ_{1}(X, x)). This immediately gives the long exact sequence

… → ϖ_{n}(X̃, x) → ϖ_{n}(X, x)^{R,Mal} → ϖ_{n}(Bπ_{1}(X, x))^{R,Mal} → ϖ_{n−1}(X̃, x)^{K,Mal} → … → ϖ_{2}(X̃, x) → ϖ_{2}(X, x)^{R,Mal} → ϖ_{2}(Bπ_{1}(X, x))^{R,Mal} → 0.

It therefore suffices to show that

π_{n}(X̃^{∧_{L}}, x) ⊗ℤ̂ ℚ_{ℓ} → ϖ_{n}(X̃^{L,alg}, x)

is an isomorphism for n ≤ N.

We may assume that X̃ = {X̃_{α}}_{α} is an inverse system of fibrant simplicial sets, and then form the tower {X̃(n)}_{n} by setting X̃(n) = {X̃_{α}(n)}_{α}, where {X̃_{α}(n)}_{n} the Moore– Postnikov tower of X̃_{α}.

Note that if X̃(N) satisfies the theorem, then we can apply Lemma 3.34 to the morphism X̃ → X̃(N), so X̃ will also satisfy the theorem. We now prove by induction on n that X̃(n) satisfies the theorem for n ≤ N.

For n = 1, X̃(1) is contractible, making the long exact sequence automatic. Now, assume that X̃(n − 1) satisfies the inductive hypothesis, and consider the pro-fibration X̃(n) → X̃(n − 1), with fibre E(n) over x. Properties of the Postnikov tower give that π_{i}X̃(n) = π_{i}X̃ for all i ≤ n, with E(n) being a K(π_{n}X, n)-space.

The long exact sequence of Theorem 3.32 gives ϖ_{i}(E(n)^{alg}) ≅ ϖ_{i}(X̃(n)^{L,ρ,Mal}) for i ≥ n, and exact sequences

ϖ_{i}(E(n)^{alg}) → ϖ_{i}(X̃(n)^{L,alg}) → π_{i}(X̃^{∧_{L}}) ⊗ℤ̂ ℚ_{ℓ} → ϖ_{i−1}(E(n)^{alg}).

Since E(n) is a K(π_{n}X, n)-space, the problem thus reduces to establishing the theorem for the case when X is a K(π, n) space (for n ≥ 2), and R = 1. Unlike [Pri3, Theorem 1.58], we cannot now immediately appeal to the Curtis convergence theorem to show that for any pro-discrete abelian group π and n ≥ 2, the map

G(K(π, n))^{L,alg} → N^{−1}(π̂ ⊗ℤ̂ ℚ_{ℓ}[1 − n])

is a weak equivalence of simplicial unipotent groups.

Instead, observe that we may replace π by πℓ̂, so assume that π is a pro-ℓ group. Since π ⊗_{ℤ_{ℓ}} ℚ_{ℓ} is finite-dimensional, we may write π = νl̂, for ν an abelian group of finite rank. On cohomology, we have maps

H^{∗}(N^{−1}(π ⊗_{ℤ_{ℓ}} ℚ_{ℓ}[1 − n]), ℚ_{ℓ}) → H^{∗}(K(π, n), ℚ_{ℓ}) → H^{∗}(K(ν, n), ℚ_{ℓ}). (†)

By [Qui2, Theorem I.3.4], the Lie algebra ν ⊗_{ℤ} ℚ_{ℓ}[1 − n] is the ℚ_{ℓ}-homotopy type of K(ν, n). Since π ⊗_{ℤ_{ℓ}} ℚ_{ℓ} = ν ⊗_{ℤ} ℚ_{ℓ}, the composite is an isomorphism in (†), while the second map is an isomorphism by Lemma 1.35. Thus the first map is also an isomorphism, as required.

For the final part, we just note that [Pri8, §1.2.3] shows that Γ is N-good relative to ρ if and only if ϖ_{n}((BΓ)^{L,ρ,Mal}) = 0 for 1 < n ≤ N. □

3.6. Comparison of homotopy types for complex varieties. Let X_{•} be a simplicial scheme of finite type over ℂ. To this we may associate the étale homotopy type X_{ét} ∈ pro(𝕊) (as in Example 3.18). There is also an analytic homotopy type X_{an} : = diag Sing(X_{•}(ℂ)) ∈ 𝕊, where diag is the diagonal functor on bisimplicial sets. We now compare the corresponding schematic homotopy types.

Lemma 3.41. If G is a pro-algebraic group over ℚ_{ℓ}, and ρ: π_{f}(X_{an}) → G(ℚ_{ℓ}) a representation with compact image (for the ℓ-adic topology on G(ℚ_{ℓ})), then ρ factorises canonically through { \atop π_{f}(X_{ét})}, giving a continuous representation

ρ: { \atop π_{f}(X_{ét})} → G(ℚ_{ℓ}).

Proof. It follows from [Fri, Theorem 8.4] that

{ \atop π_{f}(X_{ét})} ≅ { \atop π_{f}(X_{an})}.

Since G(ℚ_{ℓ}) is totally disconnected, any compact subgroup is pro-finite, completing the proof. □

Now, given a reductive pro-algebraic groupoid R, and ρ: π_{f}(X_{ℂ}) → R(ℚ_{ℓ}) with compact Zariski-dense image, we may compare the relative Malcev homotopy type X_{an}^{ρ,Mal} of [Pri3, Definition 3.16] with the relative Malcev homotopy type X_{ét}^{ρ,Mal} of Definition 3.20, since both are objects of Ho(sE(R)).

Theorem 3.42. For X, ρ as above, there is a canonical isomorphism

X_{an}^{ρ,Mal} ≅ X_{ét}^{ρ,Mal}.

Proof. We adapt [Fri, Theorem 8.4], which constructs a new homotopy type X_{s.ét}, and gives morphisms

X_{ét} ← X_{s.ét} → X_{an}

in pro(𝕊)_{δ}, inducing weak equivalences on pro-finite completions. By Lemma 3.22, X_{an}^{ρ,Mal}

is quasi-isomorphic to X̂_{an}^{ρ,Mal}. By Lemma 1.37, the maps

X̂_{ét} ← { \atop X_{s.ét}} → X̂_{an}

are weak equivalences in 𝕊̂. Lemma 3.34 then implies that the maps

X̂_{ét}^{ρ,Mal} ← { \atop X_{s.ét}}^{ρ,Mal} → X̂_{an}^{ρ,Mal}

are quasi-isomorphisms, as required. □

Remarks 3.43. In particular, this shows that there is an action of the Galois group Gal(ℂ/K) on the relative Malcev homotopy groups ϖ_{n}(X_{an}^{ρ,Mal}) whenever X is defined over a number field K and ρ is Galois-equivariant. The question of when this action is continuous will be addressed in §5.

It seems possible that the conditions of Theorem 2.25 might be satisfied in some cases where those of Theorem 3.40 do not hold, giving ϖ_{n}(X_{an}^{ρ,Mal}) ≅ π_{n}(X_{an}) ⊗_{ℤ} ℚ_{ℓ}, but no such examples are known to the author.

4. Relative and filtered homotopy types

The aims of this section are twofold. Firstly, we adapt some of the framework of proalgebraic homotopy types to work over a base ring, rather than a base field. This is motivated by the need in §7.2 to phrase the étale-crystalline comparison over variants of Fontaine’s ring B_{cris} of p-adic periods, rather than just over ℚ_{p}. Secondly, §4.3 develops techniques for transferring filtrations systematically from cochains to homotopy types. These will be used in §§6 and 7 to determine the structure of homotopy types of quasiprojective varieties. This is possible because the Gysin filtration on homotopy groups (unlike that on cohomology) is not determined by weights of Frobenius, so imposes further restrictions.

4.1. Actions on pro-algebraic homotopy types. Fix a ℚ_{ℓ}-algebra A, and a reductive pro-algebraic groupoid R over ℚ_{ℓ}.

Definition 4.1. Define cAlg_{A}(R) (resp. DGAlg_{A}(R)) to be the comma category A ↓ cAlg(R) (resp. A ↓ DGAlg(R)), with model structure induced by Proposition 2.60 (resp. Proposition 2.68). Denote the opposite category by sAff_{A}(R) (resp. dgAff_{A}(R)). Likewise, define

cAlg_{A}(R)_{∗} := cAlg_{A}(R)↓ ∏_{x∈Ob R} A ⊗ O(R)(x, −),
DGAlg_{A}(R)_{∗} := DGAlg_{A}(R)↓ ∏_{x∈Ob R} A ⊗ O(R)(x, −),

and so on.

Observe that the Quillen equivalence of Proposition 2.70 induces Quillen equivalences between dgAff_{A}(R)_{∗} and sAff_{A}(R)_{∗}, so gives the following equivalence of categories:

\xymatrix{Ho(dgAff_{A}(R)_{∗}) \ar[r]^{Spec D} & Ho(sAff_{A}(R)_{∗}). \ar[l]^{R(Spec D^{∗})}}

Although we do not have a precise analogue of Theorem 2.74 for Ho(dgAff_{A}(R)_{∗})_{0}, we have the following:

Lemma 4.2. Given X ∈ dgAff(R)_{0∗} and 𝔤 ∈ dgN̂ (R),

Hom_{Ho(dgAff_{A}(R)_{∗})}(X ⊗ A, W̄ 𝔤 ⊗ A)
≅ Hom_{Ho(dgN̂_{A}(R))}(Ḡ(X) ˆ⊗A, 𝔤 ˆ⊗A) ×^{exp(𝔤_{0}^{R} ˆ⊗A)} ∏_{x∈Ob R} exp(H_{0}𝔤(x) ˆ⊗A).

Proof. The proof of [Pri3, Proposition 3.48] adapts to this context. □

4.2. Homotopy actions.

Definition 4.3. Given 𝔤 ∈ sP̂_{R}, define a group-valued functor Aut_{R}(𝔤) on the category of ℚ_{ℓ}-algebras by setting

Aut_{R}(𝔤)(A) := Aut_{sP_{A}(R)}(𝔤 ˆ⊗A).

Given G ∈ sE(R), define RAut(G) := Aut_{R}(R_{u}(G)), noting that

RAut(G)(ℚ_{ℓ}) ≅ Aut_{Ho(sE(R))_{∗}}(G).

For G ∈ sAGpd, set RAut(G) := Aut_{G^{red}}(R_{u}(G)).

Lemma 4.4. If G ∈ sE(R) is such that H^{i}(G, V) is finite-dimensional for all i and all finite-dimensional irreducible R-representations V, then the group-valued functor

RAut(G)

is represented by a pro-algebraic group over ℚ_{ℓ}. The map

RAut(G) → ∏_{x∈Ob R} ∏_{i} Aut_{R}(H^{i}(G, O(R)(x, −)))

of pro-algebraic groups has pro-unipotent kernel.

Proof. This is a consequence of [Pri3, Theorem 5.13], which proves the corresponding statement for the group ROut(G) := RAut(G)/ ∏_{x∈Ob G} R_{u}(G)(x). Since ∏_{x∈Ob G} R_{u}(G)(x) is a pro-unipotent pro-algebraic group, the result follows. □

Definition 4.5. Given a pro-algebraic groupoid G, we may extend the automorphism group Aut(G) to a group presheaf over ℚ_{ℓ}, by setting

Aut(G)(A) := Aut_{A}(G ×_{Spec ℚ_{ℓ}} Spec A).

Lemma 4.6. For G ∈ sE(R), there is a group presheaf Aut^{h}(G) over ℚ_{ℓ}, with the properties that Aut^{h}(G)(ℚ_{ℓ}) is the group of automorphisms of G in Ho(Ob G ↓ sAGpd), and that there is an exact sequence

1 → RAut(G) → Aut^{h}(G) ⟶^{α} Aut(R) → 1,

where Aut(R) is given the algebraic structure of Definition 4.5.

If H^{i}(G, V) is finite-dimensional for all i and all finite-dimensional irreducible Rrepresentations V, then α is fibred in affine schemes.

Proof. Let R = G^{red}, take Y ∈ Ho(dgAff(R)_{∗}) corresponding to G under the equivalence of Theorem 2.74 and define

Aut^{h}(G)(A) := {(f, θ) : f ∈ Aut(R)(A), θ ∈ Iso_{Ho(dgAff_{A}(R)_{∗})}(Y ⊗ A, f^{♯}Y ⊗ A)}.

We may now take a minimal model 𝔪 for Ḡ(Y) ∈ dgN̂ (R), and observe that Lemma 4.2 then gives

Hom_{Ho(dgAff_{A}(R)_{∗})}(Y ⊗ A, f^{♯}Y ⊗ A)
≅ Hom_{Ho(dgAff_{A}(R)_{∗})}(Y ⊗ A, f^{♯}W̄ 𝔪 ⊗ A)
≅ Hom_{Ho(dgN̂_{A}(R)_{∗})}(Ḡ(Y) ˆ⊗A, 𝔪 ˆ⊗A) ×^{exp(𝔪_{0}^{R} ˆ⊗A)} ∏_{x∈Ob R} exp(H_{0}𝔪(x) ˆ⊗A)
≅ Hom_{Ho(dgN̂_{A}(R))}(𝔪 ˆ⊗A, 𝔪 ˆ⊗A) ×^{exp(𝔪_{0}^{R} ˆ⊗A)} ∏_{x∈Ob R} exp(H_{0}𝔪(x) ˆ⊗A).

The proof that α is fibred in affine schemes is now essentially the same as Lemma 4.4, which deals with the fibre over 1 ∈ Aut(R). □

Definition 4.7. Given a pro-discrete group Γ, we say that a morphism Γ → Aut^{h}(G)(ℚ_{ℓ}) is algebraic if it factors through a morphism Γ^{alg} → Aut^{h}(G) of presheaves of groups.

Corollary 4.8. If H^{i}(G, V) is finite-dimensional for all i and all finite-dimensional irreducible R-representations V, then a morphism Γ → Aut^{h}(G)(ℚ_{ℓ}) is algebraic whenever Γ → Aut(G^{red}) is so.

Proof. We have Γ^{alg} → Aut(G^{red}), so θ: Γ → (Γ^{alg} ×_{Aut(G^{red})} Aut^{h}(G))(ℚ_{ℓ}). Since Aut^{h}(G) → Aut(G^{red}) is fibred in affine schemes, the group on the right is pro-algebraic, so θ factors through Γ^{alg}, as required. □

If R = G^{red}, observe that there is canonical action of Aut^{h}(G) on ⨁_{x∈Ob R} H^{∗}(G, O(R)(x, −)). In fact, we have a homomorphism

β: Aut^{h}(G) → Aut(R) × Aut( ⨁_{x∈Ob R} H^{∗}(G, O(R)(x, −)))

of presheaves of groups.

Lemma 4.9. If H^{i}(G, V) is finite-dimensional for all i and all finite-dimensional irreducible R-representations V, then the kernel of β is a pro-unipotent pro-algebraic group.

Proof. The kernel of β is just the kernel of

RAut(G) → ∏_{x∈Ob R} ∏_{i} Aut_{R}H^{i}(G, O(R)(x, −)),

which is pro-unipotent by Lemma 4.4. □

4.3. Filtered homotopy types.

4.3.1. Commutative algebras.

Definition 4.10. Given a ℚ_{ℓ}-algebra A and a reductive pro-algebraic groupoid R over ℚ_{ℓ}, define F DGAlg_{A}(R) (resp. F cAlg_{A}(R)) to consist of R-representations B in nonnegatively graded cochain (resp. cosimplicial) algebras over A, equipped with an increasing exhaustive filtration J_{0}B ⊂ J_{1}B ⊂ … of B as a DG (resp. cosimplicial) (R, A)-module, with the property that (J_{m}B) · (J_{n}B) ⊂ J_{m+n}B. Morphisms are required to respect the filtration, and we assume that 1 ∈ J_{0}B.

Write

F cAlg(R)_{∗} := F cAlg(R)↓ ∏_{x∈Ob R} O(R)(x, −),
F DGAlg(R)_{∗} := F DGAlg(R)↓ ∏_{x∈Ob R} O(R)(x, −),

where O(R)(x, −) = J_{0}O(R)(x, −).

Given (B, J) ∈ F DGAlg_{A}(R) or F cAlg_{A}(R), there is a spectral sequence _{J}E_{∗}^{∗,∗}(B) associated to the filtration J, with

JE_{1}^{a,b}(B) = H^{a+b}(Gr_{−a}^{J}B).

Definition 4.11. We regard _{J}E_{1}^{∗,∗}(B) as an object of F DGAlg_{A}(R), with

J_{m}(_{J}E_{1}^{∗,∗}(B))^{n} = ⨁_{r≤m}_{J}E_{1}^{−r,n+r}(B),

noting that d(J_{m}(E_{1})^{n}) ⊂ J_{m−1}(E_{1})^{n+1}.

Definition 4.12. Define a map f: B → C to be a fibration if the maps J_{n}f: J_{n}B → J_{n}C are all surjective. A map f is a weak equivalence if the maps_{J}E_{1}^{∗,∗}(f): _{J}E_{1}^{∗,∗}(B) → _{J}E_{1}^{∗,∗}(C) are all isomorphisms.

Lemma 4.13. There are cofibrantly generated model structures on the categories F cMod_{A}(R) and F DGMod_{A}(R), with the classes of fibrations and weak equivalences above.

Proof. First, note that normalisation gives an equivalence F cMod_{A}(R) → F DGMod_{A}(R) of categories, preserving and reflecting fibrations and weak equivalences. It thus suffices only to consider F DGMod_{A}(R)

Let S_{n,m} denote the cochain complex consisting of A concentrated in degree n, with J_{m}S_{n,m} = S_{n,m} and J_{m−1}S_{n,m} = 0. Let D_{n,m} denote the cochain complex consisting of A concentrated in degrees n, n − 1 with differential d^{n−1} the identity, J_{m}D_{n,m} = D_{n,m} and J_{m−1}D_{n,m} = 0. By convention, D_{0,m} = 0. Note that there are natural maps S_{n,m} → D_{n,m}.

For a set {V } of representatives of irreducible R-representations in ℚ_{ℓ}-vector spaces, define I to be the set of morphisms A ⊗ S_{n,m} ⊗ V → A ⊗ D_{n,m} ⊗ V, for n ≥ 0. Define J to be the set of morphisms 0 → A ⊗ D_{n,m} ⊗ V, for n ≥ 0.

Now,

Hom_{F DGMod_{A}(R)}(A ⊗ S_{n,m} ⊗ V, M) = Hom_{R}(V, J_{m}Z^{n}M)
Hom_{F DGMod_{A}(R)}(A ⊗ D_{n,m} ⊗ V, M) = Hom_{R}(V, J_{m}M^{n−1}),

so a map f: M → N in F DGMod_{A}(R) is then I-injective when

J_{m}M^{n−1} ⟶^{f,d} J_{m}N^{n−1} ×_{d,J_{m}Z^{n}N,f} Z^{n}M

is surjective, for all m, n, and J-injective when

J_{m}M^{n−1} ⟶^{f} J_{m}N^{n−1}

is surjective for all n. Thus I-injectives are trivial fibrations, and J-injectives are fibrations.

Since S_{n,m} = D_{n+1,m}/S_{n+1,m}, the map 0 → D_{n,m} is a composition 0 → S_{n,m} → D_{n,m} of pushouts of maps in I, so maps in J are all I-cofibrations. Since maps in J are all weak equivalences, we have satisfied the conditions of [Hov, Theorem 2.1.19,] giving the model structure claimed. □

Lemma 4.14. In the category F DGMod(R) = F DGMod_{ℚ_{ℓ}}(R), all objects V are cofibrant.

Proof. Given V ∈ F DGMod_{ℚ_{ℓ}}(R), it will suffice to show that J_{0}V is cofibrant, and that all the maps J_{m−1}V → J_{m}V are cofibrations, since V = \underrightarrow{lim} J_{m}V. To do this, we will show that these maps are transfinite compositions of pushouts of generating cofibrations.

Now, since all R-representations are semisimple, we may choose decompositions gr_{m}^{J}V^{n} = M^{n} ⊕ N^{n} ⊕ dN^{n−1}, with dM^{n} = 0. By semisimplicity, we may also lift the R-modules M^{i}, N^{i} to M̃^{i}, Ñ^{i} ⊂ J_{m}V. Now dM̃ ⊂ J_{m−1}V, so the map J_{m−1}V → J_{m}V is a pushout of ⨁_{n}(S_{n+1,m} ⊗ M̃^{n}) → ⨁_{n}(D_{n+1,m} ⊗ M̃^{n}) ⊕ ⨁_{n}(D_{n+1,m} ⊗ Ñ^{n}), and hence a cofibration. Since this argument also applies to 0 → J_{0}V, we deduce that V is cofibrant. □

Proposition 4.15. There is a cofibrantly generated model structure on F DGAlg_{A}(R) (resp. F cAlg_{A}(R)), for which a morphism is a fibration or weak equivalence whenever the underlying morphism in F DGMod_{A}(R) (resp. F cMod_{A}(R)) is so (in the model structure of Lemma 4.13).

Proof. The forgetful functor F DGAlg_{A}(R) → F DGMod_{A}(R) (resp. F cAlg_{A}(R) → F cMod_{A}(R)) preserves filtered colimits and has a left adjoint, the free algebra functor. Since the free algebra functor maps trivial generating cofibrations to weak equivalences, we may apply [Hir, Theorem 11.3.2], which gives the required cofibrantly generated model structure. The generating cofibrations and trivial cofibrations are given by the images under the free algebra functor of the generating cofibrations and trivial cofibrations in F DGMod_{A}(R) (resp. F cMod_{A}(R)). □

4.3.2. Lie algebras.

Definition 4.16. Define F N̂_{A}(R) to be opposite to the category F N̂_{A}(R)^{opp} of Rrepresentations in ind-conilpotent (see Definition 2.44) Lie coalgebras C over A, equipped with an exhaustive increasing filtration J_{0}C ⊂ J_{1}C ⊂ …, of C as an (R, A)-module, with the property that ∇(J_{r}C) ⊂ ∑_{m+n=r}(J_{m}C)⊗(J_{n}C), for ∇ the cobracket. Morphisms are required to respect the filtration.

Similarly, F dgN̂_{A}(R) is opposite to the category of R-representations in non-negatively filtered ind-conilpotent ℕ_{0}-graded cochain Lie coalgebras over A. F sN̂_{A}(R) is the category of simplicial objects in F N̂_{A}(R). When A = ℚ_{ℓ}, we will usually drop the subscript A.

Proposition 4.17. There is a closed model structure on F dgN̂_{A}(R) (resp. F sN̂_{A}(R)), in which a morphism f: 𝔤 → 𝔥 is a fibration or a weak equivalence whenever the underlying map f^{∨}: 𝔥^{∨} → 𝔤^{∨} in F DGMod_{A}(R) (resp. F cMod_{A}(R)) is a cofibration or a weak equivalence.

Proof. The proof of [Pri3, Lemma 5.9] carries over to this context. □

4.3.3. Equivalences.

Definition 4.18. Define F cAlg(R)_{00_{∗}} (resp. F DGAlg(R)_{00_{∗}}) to be the full subcategory of F cAlg_{A}(R)_{∗} (resp. F DGAlg_{A}(R)_{∗}) consisting of objects B with B^{0} = ℚ_{ℓ}. Let F cAlg(R)_{0∗} (resp. F DGAlg(R)_{0_{∗}}) be the full subcategory consisting of objects weakly equivalent to objects of F cAlg(R)_{00_{∗}} (resp. F DGAlg(R)_{00_{∗}}). Let Ho(F cAlg(R)_{∗})_{0} (resp. Ho(F DGAlg(R)_{∗})_{0}) be the full subcategory of Ho(F cAlg(R)_{∗}) (resp. Ho(F DGAlg(R)_{∗})) on objects F cAlg(R)_{0_{∗}} (resp. F DGAlg(R)_{0_{∗}}). Denote the opposite category to F cAlg(R)_{00∗} by F sAff(R)_{00∗}, etc.

Definition 4.19. Given 𝔤 ∈ F sN̂ (R), we define W̄ 𝔤 ∈ F sAff(R) by

(W̄ 𝔤)(B) := W̄ (exp(Hom_{F Mod(R)}(𝔤^{∨}, (B)))) ∈ 𝕊

for B ∈ Alg_{A}(R). Here, W̄ is the classifying space functor of Definition 1.6, and exp denotes exponentiation of a pro-nilpotent Lie algebra to give a pro-unipotent group.

Observe that this functor is continuous, and denote its left adjoint by G: F sAff(R) → F sN̂ (R).

Definition 4.20. Define functors \xymatrix{F dgAff(R) \ar[r]^{G} & F dgN̂ (R) \ar[l]^{W̄}_{⊥}} as follows. For 𝔤 ∈

F dgN̂ (R), the Lie bracket gives a linear map ⋀^{2} 𝔤 → 𝔤. Write Δ for the dual Δ: 𝔤^{∨} → ⋀^{2} 𝔤^{∨}, which respects the filtration. This is equivalent to a map Δ: 𝔤^{∨}[−1] → Symm^{2}(𝔤^{∨}[−1]), and we define

O(W̄ 𝔤) := Symm(𝔤^{∨}[−1])

to be the graded polynomial ring on generators 𝔤^{∨}[−1], with a derivation defined on generators by D := d + Δ. The Jacobi identities ensure that D^{2} = 0.

We define G by writing σB[1] for the brutal truncation (in non-negative degrees) of B[1], and setting

G(B)^{∨} = CoLie(σB[1]),

the free filtered graded Lie coalgebra over ℚ_{ℓ}, with differential similarly defined on cogenerators by D := d + μ, μ here being the product on B. Note also that G(B) is cofibrant for all B.

Definition 4.21. Define the category F sP(R) (resp. F dgP(R)) to have the fibrant objects of F sN̂ (R) (resp. F dgN̂ (R)), with morphisms given by

Hom_{F sP(R)}(𝔤, 𝔥) = Hom_{Ho(F sN̂ (R))}(𝔤, 𝔥) ×^{exp(𝔥_{0}^{R})} ∏_{x∈Ob R} exp(π_{0}𝔥(x)),
Hom_{F dgP(R)}(𝔤, 𝔥) = Hom_{Ho(F dgN̂ (R))}(𝔤, 𝔥) ×^{exp(𝔥_{0}^{R})} ∏_{x∈Ob R} exp(H_{0}𝔥(x)),

where 𝔥_{0}^{R} is the Lie algebra Hom_{Mod(R)}(𝔥_{0}^{∨}, ℚ_{ℓ}) = Hom_{F Mod(R)}(𝔥_{0}^{∨}, ℚ_{ℓ}), acting by conjugation on the set of homomorphisms.

Theorem 4.22. There is the following commutative diagram of equivalences of categories:

\xymatrix{Ho(F dgAff(R)_{∗})_{0} \ar[r]^{Spec D} \ar[d]_{Ḡ} & Ho(F sAff(R)_{∗})_{0} \ar[d]_{Ḡ} \ar[l]^{_{Spec Th}} \\ F dgP(R) \ar[u]_{W̄} & F sP(R), \ar[u]_{W̄} \ar[l]^{N}}

where N denotes normalisation, D is denormalisation, and Th is the functor of Thom-Sullivan cochains.

Proof. The proof of Theorem 2.74 carries over to this context, making use of Lemma 4.14, which implies that everything in the image of W̄ is fibrant, as are all objects of F dgN̂ (R) and F sN̂ (R). On objects, the functor Ḡ is defined by choosing, for any X ∈ Ho(F sAff(R)_{∗})_{0} (resp. X ∈ Ho(F dgAff(R)_{∗})_{0}) a weakly equivalent object X′ ∈ F sAff(R)_{00} (resp. X′ ∈ F dgAff(R)_{00}), and setting

Ḡ(X) := G(X′),

for the functor G from Definition 4.20. □

Although we do not have a precise analogue of this result for Ho(F dgAff_{A}(R)) for general A, we do have the following:

Lemma 4.23. Given X ∈ Ho(F dgAff(R)_{∗})_{0} and 𝔤 ∈ F dgN̂ (R),

Hom_{Ho(F dgAff_{A}(R)_{∗})}(X ⊗ A, W̄ 𝔤 ⊗ A)
≅ Hom_{Ho(F dgN̂_{A}(R))}(Ḡ(X) ˆ⊗A, 𝔤 ˆ⊗A) ×^{exp(𝔤_{0}^{R} ˆ⊗A)} ∏_{x∈Ob R} exp(H_{0}𝔤(x)).

Proof. The proof of [Pri3, Proposition 3.48] adapts to this context. □

Definition 4.24. We say that a filtered cochain algebra (B, J) ∈ F DGAlg_{A}(R) is quasiformal if it is weakly equivalent in F DGAlg_{A}(R) to _{J}E_{1}^{∗,∗}(B) (as in Definition 4.11). We say that a filtered homotopy type is quasi-formal if its associated cochain algebra is so.

4.3.4. Minimal models. Let F DGRep(R) = F DGMod_{ℚ_{ℓ}}(R) be the category of nonnegatively graded filtered complexes of R-representations.

Definition 4.25. We say that M ∈ F DGRep(R) is minimal if d(J_{m}M) ⊂ J_{m−1}M for all m.

Lemma 4.26. For any V ∈ F DGRep(R), there exists a quasi-isomorphic filtered subobject M ↪ V, with M minimal.

Proof. We prove this by induction on the filtration. Assume that we have constructed a filtered quasi-isomorphism J_{m}f: J_{m}M ↪ J_{m}V (for m = −1, this is trivial). Pick a basis v_{α} for H^{∗}(gr_{m+1}^{J}V), and lift v_{α} to v′_{α} ∈ J_{m+1}V. Thus dv′_{α} ∈ J_{m}V, and [dv′_{α}] = 0 ∈ H^{∗}(J_{m}V/J_{m}M) = 0. This means that dv′_{α} ∈ J_{m}M + dJ_{m}V. Choose u_{α} ∈ J_{m}V such that dv′_{α} − du_{α} ∈ J_{m}M, and set ṽ_{α} := v′_{α} − u_{α}.

Now, [ṽ_{α}] = v_{α} ∈ H^{∗}(gr_{m+1}^{J}V), so define

J_{m+1}M := J_{m}M ⊕ ⟨ṽ_{α}⟩_{α};

this has the properties that dJ_{m+1}M ⊂ J_{m}M and H^{∗}(gr_{m+1}^{J}M) ≅ H^{∗}(gr_{m+1}^{J}V), as required. □

Definition 4.27. We say that a cofibrant object 𝔪 ∈ F dgN̂ (R) (resp. F sN̂ (R)) is minimal if (𝔪/[𝔪, 𝔪])^{∨} (resp. N(𝔪/[𝔪, 𝔪])^{∨}) is minimal in the sense of Definition 4.25.

Proposition 4.28 (Minimal models). Every weak equivalence class in F dgN̂ (R) (resp. sN̂ (R)) has a minimal element 𝔪, unique up to non-unique isomorphism.

Proof. The proof of [Pri3, Proposition 1.16] adapts to this context, using Lemma 4.26 instead of the corresponding result for DGRep(R). □

4.3.5. Homotopy automorphisms.

Definition 4.29. Given 𝔲 ∈ F sN̂ (R), let G = exp(𝔲) ⋊ R, and define the group presheaf of filtered automorphisms by

Aut_{J}^{h}(G)(A) := {(f, θ) : f ∈ Aut(R)(A), θ ∈ Iso_{F sP_{A}(R)}(𝔲 ˆ⊗A, f^{♯}𝔲 ˆ⊗A)}.

Define RAut_{J}(G) := ker(Aut_{J}^{h}(G) → Aut(R)).

Definition 4.30. Given V ∈ Rep(R) and 𝔤 ∈ F sN̂ (R), define the spectral sequence JE_{∗}^{∗,∗}(R ⋉ exp(𝔤), V) to be the cohomology spectral sequence of the filtered complex

O(W̄ 𝔤) ⊗^{R} V,

for J_{0}V = V. Thus _{J}E_{1}^{a,b}(R ⋉ exp(𝔤), V) = H^{a+b}(Gr_{−a}^{J}O(W̄ 𝔤) ⊗^{R} V).

Lemma 4.31. Assume that G is as above, and let 𝔪 ∈ F sN̂ (R) be a minimal model for R_{u}(G). If H^{i}(G, V) is finite-dimensional for all i and all finite-dimensional irreducible R-representations V, then the group presheaves

Aut_{F sN̂ (R)}(𝔪) × ∏_{x∈Ob R} exp(π_{0}𝔪(x)) ⟶^{α} RAut_{J}(G) ⟶^{β} ∏_{a,b} Aut_{R}(_{J}E_{1}^{a,b}(G, O(R)))

are all pro-algebraic groups, the maps α and β both have pro-unipotent kernels, and β is surjective.

Proof. The proof of [Pri3, Theorem 5.13] carries over. □

4.3.6. Examples.

Definition 4.32. Given B^{•} ∈ DGAlg_{A}(R), we define the good truncation τ_{∗} on B by

(τ_{m}B)^{n} := \begin{cases}B^{n} & n < m \\ Z^{m}(B) & n = m \\ 0 & n > m. \end{cases}

Observe that (B^{•}, τ) ∈ F DGAlg_{A}(R).

Definition 4.33. Given a bicosimplicial algebra B^{•,•} ∈ ccAlg_{A}(R), we define the associated filtered cosimplicial algebra (τ′′_{0}B ≤ τ′′_{1}B ≤ …) ∈ F cAlg_{A}(R) by

(τ′′_{m}B)^{n} = (Dτ_{m}Th B^{n,•})^{n},

for D, Th as in Theorem 2.70. Observe that there is a canonical quasi-isomorphism diag B^{•,•} → τ′′_{∞}B^{•}, where diag denotes the diagonal of a bicosimplicial complex.

In practice, the only filtered homotopy types which we will encounter come from morphisms of spaces:

Definition 4.34. Given an algebraic variety X and an ind-constructible ℓ-adic sheaf 𝕍 on X, recall (e.g. from [Pri2, Definition 2.3]) that there is a natural cosimplicial complex

𝒞_{ét}^{•}(𝕍)

of ℓ-adic sheaves on 𝕍, with the property that Γ(X, 𝒞_{ét}^{•}(𝕍)) = C_{ét}^{•}(X, 𝕍), the Godement resolution (as in Remark 3.18). This construction respects tensor products.

Lemma 4.35. To any morphism j: Y → X of algebraic varieties, and any ℚ_{ℓ}-sheaf 𝒮 of algebras on Y as in Definition 4.34, there is associated a canonical filtered homotopy type C_{ét}^{•}(j, 𝒮) ∈ Ho(F cAlg_{ℚ_{ℓ}}), with the property that _{J}E_{∗}^{∗,∗}C_{ét}^{•}(j, 𝒮) is the Leray spectral sequence

JE_{1}^{a,b}C_{ét}^{•}(j, 𝒮) = H^{2a+b}(X, R^{−a}j_{∗}𝒮) ⟹ H^{a+b}(Y, 𝒮).

The associated unfiltered homotopy type is canonically weakly equivalent to C_{ét}^{•}(Y, 𝒮).

Proof. We have a ℚ_{ℓ}-sheaf j_{∗}𝒞_{ét}^{•}(𝒮) of cosimplicial algebras on X, and hence a bicosimplicial algebra

C_{ét}^{•}(X, j_{∗}𝒞_{ét}^{•}(𝒮)).

Now, set

J_{n}C_{ét}^{•}(j, 𝒮) = τ′′_{n}C_{ét}^{•}(X, j_{∗}𝒞_{ét}^{•}(𝒮)) = diag C_{ét}^{•}(X, Dτ_{n}Th j_{∗}𝒞_{ét}^{•}(𝒮)),

as in Definition 4.33, with C_{ét}^{•}(X, j_{∗}𝒞_{ét}^{•}(𝒮)) → J_{∞}C_{ét}^{•}(j, 𝒮) a quasi-isomorphism. Finally, observe that there is a quasi-isomorphism

C_{ét}^{•}(Y, 𝒮) = Γ(X, j_{∗}𝒞_{ét}^{•}(𝒮)) → diag C_{ét}^{•}(X, j_{∗}𝒞_{ét}^{•}(𝒮)),

and that gr_{n}^{τ}j_{∗}𝒞_{ét}^{•}(𝒮) is quasi-isomorphic to R^{n}j_{∗}𝒮. □

Remark 4.36. There is a similar statement for filtrations on homotopy types coming from morphisms of topological spaces, using Čech resolutions instead of Godement resolutions.

Since the construction above is functorial, for any point y ∈ Y, we have a morphism C_{ét}^{•}(j, 𝒮) → C_{ét}^{•}(id_{y}, 𝒮_{y}), where id_{y} is the identity map id_{y}: y → y. Now,

J_{n}C_{ét}^{•}(id_{y}, 𝒮_{y}) = diag C_{ét}^{•}(y, Dτ_{n}Th 𝒮_{y}).

Since 𝒮_{y} has constant simplicial structure, Th 𝒮_{y} = 𝒮_{y}, so J_{n}C_{ét}^{•}(id_{y}, 𝒮_{y}) = 𝒮_{y} for all n ≥ 0.

Definition 4.37. Given a morphism j: Y → X of algebraic varieties and a Zariski-dense continuous map

ρ: { \atop π_{f}^{ét}(Y)} → R(ℚ_{ℓ})

define the filtered homotopy type (Y^{ρ,Mal}, j) to correspond to C_{ét}^{•}(j, 𝕆(R)) ∈ F cAlg(R)_{∗}, where the augmentation map is the canonical morphism

C_{ét}^{•}(j, 𝕆(R)) → ∏_{y∈Y} C_{ét}^{•}(id_{y}, 𝕆(R)) = ∏_{y∈Y} O(R)(y, −).

5. Algebraic Galois actions

5.1. Weight decompositions. By a weight decomposition, we will mean an algebraic action of the group 𝔾_{m}. A weight decomposition on a vector space V is equivalent to a decomposition V = ⨁_{n∈ℤ} W_{n}V, given by λ ∈ 𝔾_{m} acting as λ^{n} on W_{n}V.

Fix a prime p, which need not differ from ℓ. Let ℤ^{alg} be the pro-algebraic group over ℚ_{ℓ} parametrising ℤ-representations. Since ℤ is commutative, ℤ^{alg} is commutative, so ℤ^{alg} = ℤ^{red} × R_{u}(ℤ^{alg}), where ℤ^{red} is its reductive quotient. For any unipotent algebraic group U, this means that Hom(R_{u}(ℤ^{alg}), U) ≅ Hom(ℤ, U(ℚ_{ℓ})) = U(ℚ_{ℓ}), so R_{u}(ℤ^{alg}) = 𝔾_{a}. Combining these observations gives ℤ^{alg} = 𝔾_{a} × ℤ^{red}.

Likewise, let ℤ̂^{alg} be the pro-algebraic group over ℚ_{ℓ} parametrising continuous ℤ̂- representations. Since continuous ℤ̂-representations form a full subcategory of ℤrepresentations, ℤ̂^{alg} is a quotient of ℤ^{alg}. The reasoning above adapts to show that ℤ̂^{alg} = 𝔾_{a} × ℤ̂^{red}.

Definition 5.1. Given n ∈ ℤ and a power q of p, recall that an element α ∈ ℚ̄_{ℓ} is said to be pure of weight n if it is algebraic and for every embedding ι: ℚ̄_{ℓ} ↪ ℂ the element ι(α) has complex absolute value q^{n/2}.

Let M_{q} be the quotient of ℤ̂^{red} whose representations ρ correspond to semisimple ℤ̂- representations for which the eigenvalues of ρ(1) are all of integer weight with respect to q. Such representations are called mixed.

Observe that every M_{q}-representation decomposes into “pure” representations, in which all eigenvalues have the same weight. There is thus a canonical map 𝔾_{m} → M_{q} given by λ ∈ 𝔾_{m} acting as λ^{n} on a pure representation of weight n.

Definition 5.2. Define P_{q} to be the quotient of M_{q} whose representations are pure of weight 0, so P_{q} = M_{q}/𝔾_{m}.

Definition 5.3. Given n ∈ ℤ, an embedding ι: ℚ̄_{ℓ} → ℂ and a power q of p, recall that an element α ∈ ℚ̄_{ℓ} is said to be ι-pure of weight n if |ι(α)| = q^{n/2}.

Let M_{ι,q} be the quotient of ℤ^{red} whose representations ρ correspond to semisimple ℤrepresentations for which the eigenvalues of ρ(1) are all of integer ι-weight. Note that M_{q} is a quotient of M_{ι,q}.

Observe that there is a canonical map 𝔾_{m} → M_{ι,q} given by λ ∈ 𝔾_{m} acting as λ^{n} on an ι-pure representation of weight n, and that this induces the map 𝔾_{m} → M_{q} above.

Definition 5.4. Define P_{ι,q} to be the quotient of M_{ι,q} whose representations are pure of ι-weight 0, so P_{ι,q} = M_{ι,q}/𝔾_{m}.

Definition 5.5. Given a pro-algebraic group G, let G^{0} be the connected component of the identity; if Ĝ is the maximal pro-finite quotient of G (parametrising representations with finite monodromy), then G^{0} = ker(G → Ĝ).

Lemma 5.6. If Γ is a pro-discrete group, then we may make the identification

Γ^{alg,0} = lim_{\overleftarrow{Δ}} Δ^{alg},

where Δ runs over Δ ⊲ Γ open of finite index.

Thus the category of finite-dimensional Γ^{alg,0}-representations is the direct limit \underrightarrow{lim}_{Δ} FDRep(Δ) (over Δ as above) of the categories of finite-dimensional Δrepresentations.

Proof. This is essentially [Mag, Proposition 2], which deals with the case when Γ is discrete, and refers to \underrightarrow{lim}_{Δ} FDRep(Δ) as the category of virtual Γ-representations.

First note that Γ̂^{alg} = Γ̂, where the pro-finite completion Γ̂ of Γ is characterised by the property that Hom_{pro(Gp)}(Γ, F) ≅ Hom_{pro(Gp)}(Γ̂, F) for all finite groups F. Thus Γ̂ = Γ whenever Γ is pro-finite.

The exact sequence Δ → Γ → Γ/Δ → 1 gives an exact sequence (Δ)^{alg} ⟶^{α} Γ^{alg} → Γ/Δ → 1. It suffices to show that α is injective. This follows from the observation that every finite-dimensional Δ-representation V embeds into a finite-dimensional Γ-representation Ind_{Δ}^{Γ}V. □

Thus if F is a generator for ℤ, then representations of ℤ^{alg,0} are sums of F^{r}- representations, with morphisms commuting locally with sufficiently high powers of F.

Observe that we have commutative diagrams

ℤ̂ ⟶^{r} ℤ̂
↓  ↓
M_{q^{r}} ⟶ M_{q}.

Any ℤ̂-representation with finite monodromy is pure of weight 0, giving a map P_{p} → ℤ̂. Also note that M_{q^{r}} = ker(M_{q} → ℤ/rℤ). Combining these observations gives:

Lemma 5.7.

M_{p}^{0} = \underleftarrow{lim} M_{p^{r}}, P_{p}^{0} = \underleftarrow{lim} P_{p^{r}};

writing M^{0} := M_{p}^{0} and P^{0} := P_{p}^{0}, there are quotient maps ℤ̂^{red,0} ↠ M^{0} ↠ P^{0}. There are similar results for M_{ι}^{0} := M_{ι,p}^{0}, P_{ι}^{0} := P_{ι,p}^{0}.

Definition 5.8. We say that a representation of ℤ^{alg,0} is mixed (resp. pure of weight 0, resp. ι-mixed with integral weights, resp. ι-pure) if the action of ℤ^{red,0} ⊲ ℤ^{alg,0} factors through M^{0} (resp. P^{0}, resp. M_{ι}^{0}, resp. P_{ι}^{0}).

Lemma 5.9. Observe that the canonical maps 𝔾_{m} → M_{q} are compatible, giving 𝔾_{m} → M^{0}, with trivial image in P^{0}. Similarly, we have 𝔾_{m} → M_{ι}^{0}, with trivial image in P_{ι}^{0}.

5.1.1. Slope decompositions.

Definition 5.10. Define the pro-algebraic group 𝔾̃_{m} to be the inverse limit of the étale universal covering system of 𝔾_{m}. This is the inverse system {G_{r}}_{r∈ℕ} with G_{r} = 𝔾_{m} and

morphisms G_{sr} ⟶^{[s]} G_{r}, for s ∈ ℕ.

Lemma 5.11. The category of 𝔾̃_{m}-representations is canonically equivalent to the category of ℚ-graded vector spaces.

Proof. A representation of 𝔾_{m} is equivalent to a ℤ-grading. Given a finite-dimensional vector space V with a ℚ-grading V = ⨁ V_{λ}, let d be the lowest common multiple of the denominators of the set {λ ∈ ℚ : V_{λ} ≠ 0}. Then V = ⨁_{n∈ℤ} V_{n/d}, giving a 𝔾_{m}-action on V. If we regard this copy of 𝔾_{m} as G_{d}, this defines a 𝔾̃_{m}-action.

Now, for any pro-algebraic group G, arbitrary G-representations are nested unions of finite-dimensional G-subrepresentations. Likewise, every ℚ-graded vector space can be expressed as a nested union of finite-dimensional ℚ-graded vector subspaces, so the two categories are equivalent. □

Now assume that p = ℓ.

Definition 5.12. Given a power q of p, normalise the p-adic valuation v on ℚ̄_{p} by v(q) = 1. Define the slope of α ∈ ℚ̄_{p} to be v(α) ∈ ℚ.

Lemma 5.13. There is a canonical morphism 𝔾̃_{m} → ℤ^{red}, corresponding to the functor sending a ℤ-representation V to a slope decomposition ⨁ V_{λ}.

Proof. Let F be the canonical generator for ℤ. Given a finite-dimensional semisimple ℤ-representation V, we may decompose V ⊗_{ℚ_{p}} ℚ̄_{p} into F -eigenspaces, and hence take a decomposition by slopes of the eigenvalues. Since conjugates in ℚ̄_{p} have the same slope, this descends to a slope decomposition V = ⨁_{λ∈ℚ} V_{λ}, as required. □

5.2. Potentially unramified actions. Fix a prime p ≠ ℓ, and take a local field K, with finite residue field k of characteristic p. Let G := Gal(K̄/K)^{alg}, the pro-algebraic completion of Gal(K̄/K) over ℚ_{ℓ}.

Definition 5.14. Say that a finite-dimensional continuous ℚ_{ℓ}-representation of Gal(K̄/K) is potentially unramified if there exists a finite extension K′/K for which the action of Gal(K̄/K′) is unramified. Say that an arbitrary ℚ_{ℓ}-representation of Gal(K̄/K) is potentially unramified if it is a sum of finite-dimensional potentially unramified representations.

These form a neutral Tannakian category (see Definition 2.7); let G^{pnr} be the corresponding pro-algebraic group. Since Rep(G^{pnr}) is a Tannakian subcategory of Rep(G), G^{pnr} is a quotient of G.

Lemma 5.15. We can write G^{pnr} = Gal(k̄/k)^{alg} ×_{Gal(k̄/k)} Gal(K̄/K), so G^{pnr,0} = Gal(k̄/k)^{alg,0} ≅ ℤ̂^{alg,0}.

Proof. A representation G → GL(V) (for V finite-dimensional) is potentially unramified if it annihilates ker(Gal(K̄/K′) → Gal(k̄/k′)) for some finite Galois extension K′/K. In other words, it annihilates ker(Gal(K̄/K) → Gal(k̄/k) ×_{Gal(k′/k)} Gal(K′/K)), so is an algebraic representation of Gal(k̄/k)^{alg} ×_{Gal(k′/k)} Gal(K′/K). Thus the category of finitedimensional G^{pnr}-representations is given by

FDRep(G^{pnr}) = \underrightarrow{lim}_{K′} FDRep(Gal(k̄/k)^{alg} ×_{Gal(k′/k)} Gal(K′/K))
= FDRep(\underleftarrow{lim}_{K′} Gal(k̄/k)^{alg} ×_{Gal(k′/k)} Gal(K′/K))
= FDRep(Gal(k̄/k)^{alg} ×_{Gal(k̄/k)} Gal(K̄/K)),

as required.

The final statement is an immediate consequence of Lemma 5.6. □

Definition 5.16. We say that a representation of G^{pnr} is mixed (resp. pure of weight 0) if the resulting action of ℤ^{alg,0} ↠ ℤ̂^{alg,0} is so.

5.3. Potentially crystalline actions. Now let ℓ = p, and take a local field K, with finite residue field k of order q = p^{f}. Let G := Gal(K̄/K)^{alg}, the pro-algebraic completion of Gal(K̄/K) over ℚ_{p}. Let W := W (k), with fraction field K_{0}, and let σ denote the unique lift of arithmetic Frobenius Φ ∈ Gal(k̄/𝔽_{p}) to σ ∈ Gal(K_{0}^{nr}/ℚ_{p}), for K_{0}^{nr} the maximal unramified extension of K_{0}. Note that the geometric Frobenius of the previous section is F = Φ^{−f}.

Definition 5.17. Say that a finite-dimensional continuous Gal(K̄/K)-representation over ℚ_{p} is potentially crystalline if there exists a finite extension K′/K for which the action of Gal(K̄/K′) is crystalline. Say that an arbitrary ℚ_{p}-representation of Gal(K̄/K) is potentially crystalline if it is a sum of finite-dimensional potentially crystalline representations. Note that since unramified representations are automatically crystalline, all potentially unramified representations are potentially crystalline.

These form a neutral Tannakian category (see Definition 2.7); let G^{pcris} be the corresponding pro-algebraic group. Since Rep(G^{pcris}) is a full subcategory of Rep(G) closed under subobjects, G^{pcris} is a quotient of G.

Definition 5.18. In [Fon, §4], Fontaine defined a ring B_{cris} := B_{cris}(V) of periods over ℚ_{p}, equipped with a Hodge filtration and actions of Gal(K̄/K) and Frobenius, and used it to characterise crystalline representations (adapted in Proposition 5.20 below).

In [Ols1, 6.8], Olsson defined a localisation B̃_{cris}(V) of B_{cris}(V) as follows. Fix a sequence τ_{m} of elements of V̄ with τ_{0} = p and τ_{m+1}^{p} = τ_{m} for all m ≥ 0. Define λ_{p^{−n}} to be the sequence (τ_{n+m})_{m≥0}, and let δ_{p^{−n}} be the associated Teichmüller lifting. Set

B̃_{cris}(V) := B_{cris}(V)[δ_{p^{−n}}^{−1}]_{n≥0},

noting that (δ_{p^{−n−1}})^{p} = δ_{p^{−n}}.

Definition 5.19. Given a finite-dimensional Gal(K̄/K)-representation U, set

D_{cris,K}(U) := (U ⊗_{ℚ_{p}} B_{cris})^{Gal(K̄/K)},
D_{pcris}(U) := \underrightarrow{lim} D_{cris,K′}(U),
D̃_{cris,K}(U) := (U ⊗_{ℚ_{p}} B̃_{cris})^{Gal(K̄/K)},
D̃_{pcris}(U) := \underrightarrow{lim} D̃_{cris,K′}(U),

for K′ ranging over all finite extensions of K. For an arbitrary algebraic Gal(K̄/K)- representation U, set

D_{pcris}(U) := \underrightarrow{lim} D_{pcris}(U_{α}),

for U_{α} running over all finite-dimensional subrepresentations, and similarly for D̃_{pcris}.

Observe that Spec B_{cris} is an affine G-scheme over Spec ℚ_{p}, and that the coarse quotient (Spec B_{cris})/G^{0} is Spec K_{0}^{nr}.

Proposition 5.20. An action of G on an affine ℚ_{p}-scheme Y factors through G^{pcris} if and only if there exists an affine K_{0}^{nr}-scheme Z, with

Y ×_{ℚ_{p}} Spec B̃_{cris} ≅ Z ×_{K_{0}^{nr}} Spec B̃_{cris}

a G^{0}-equivariant map (for trivial G^{0}-action on Z).

In that case, we necessarily have 𝒪_{Z} = D_{pcris}(𝒪_{Y}) = D̃_{pcris}(𝒪_{Y}).

Proof. If we replace potentially crystalline with crystalline, and K_{0}^{nr} with K_{0}, then this is just [Ols1, Theorem D.3]. Taking the direct limit over finite extensions of K gives the first expression.

Taking G^{0}-invariants gives 𝒪_{Z} = D̃_{pcris}(𝒪_{Y}), but then [Ols1, Remark D.10] shows that for potentially crystalline representations U, D̃_{pcris}(U) = D_{pcris}(U). □

5.3.1. Frobenius actions. Although we do not have a canonical map ℤ^{alg,0} → G^{pcris}, there is something nearly as strong:

Lemma 5.21. There is a canonical morphism

ℤ^{alg,0} ⊗_{ℚ_{p}} B_{cris}^{σ} → G^{pcris} ⊗_{ℚ_{p}} B_{cris}^{σ}

of affine group schemes over the σ-invariant subring B_{cris}^{σ} of B_{cris}.

Proof. Given U ∈ FDRep(G^{pcris}), U is crystalline over K′ for some finite extension K′/K with residue field k′. If |k′/k| = r and q = p^{f}, then ϕ^{fr} is a K′_{0}-linear endomorphism of D_{cris,K′}(U). This extends uniquely to give a K_{0}^{nr}-linear automorphism F_{r} of D_{pcris}(U) (note that F_{r} ≠ ϕ^{fr}, the latter being σ-semilinear).

Now, observe that D_{pcris}(U) is a sum of finite-dimensional F_{r}-representations over ℚ_{p}, since D_{cris,K′}(U) is finite-dimensional over K′, and hence over ℚ_{p}. This gives us a σequivariant ℚ_{p}-linear action of ℤ^{0,alg} on D_{pcris}(U), and hence a σ-equivariant B_{cris}^{σ}-linear action on D_{pcris}(U)⊗_{K_{0}^{nr}} B_{cris} = U ⊗_{ℚ_{p}} B_{cris}. We now take the ϕ-invariant subspace, giving a ℤ^{0,alg} ⊗_{ℚ_{p}} B_{cris}^{σ}-action on U ⊗_{ℚ_{p}} B_{cris}^{σ}.

If we took a larger extension K′′/K with residue field k′′, the we would have |k′′/k| = s with r|s. The corresponding K_{0}^{nr}-linear automorphism F_{s} of D_{pcris}(U) is given by F_{s} = F_{r}^{s/r}, so gives rise to the same ℤ^{0,alg}-action on D_{pcris}(U). This ensures that the action is functorial in U.

Given U, V ∈ FDRep(G^{pcris}), we have D_{pcris}(U ⊗_{ℚ_{p}} V) = D_{pcris}(U) ⊗_{K_{0}^{nr}} D_{pcris}(V), compatible with ϕ. Choosing K′ so that U, V are both crystalline over K′, we see that D_{pcris}(U ⊗_{ℚ_{p}} V) is isomorphic to D_{pcris}(U) ⊗_{K_{0}^{nr}} D_{pcris}(V) as an F_{r}-representation.

Hence the ℤ^{0,alg} ⊗_{ℚ_{p}} B_{cris}^{σ}-representation (U ⊗_{ℚ_{p}} V) ⊗_{ℚ_{p}} B_{cris}^{σ} is isomorphic to (U ⊗_{ℚ_{p}} B_{cris}^{σ}) ⊗_{B_{cris}^{σ}} (V ⊗_{ℚ_{p}} B_{cris}^{σ}).

For a ℚ_{p}-algebra A, Tannakian duality says that giving an element g ∈ G^{pcris}(A) is equivalent to giving A-linear automorphisms g_{U} of U ⊗ A for all G^{pcris}-representations U, functorial and compatible with tensor products and duals. Therefore the ℤ^{0,alg} ⊗_{ℚ_{p}} B_{cris}^{σ}-actions on the representations U ⊗_{ℚ_{p}} B_{cris}^{σ} give group homomorphisms ℤ^{0,alg}(C) → G^{pcris}(C), functorial in B_{cris}^{σ}-algebras C, as required. □

Definition 5.22. We say that a potentially crystalline representation U is mixed (resp. pure, resp. ι-mixed with integral weights, resp. ι-pure) if the action of ℤ^{alg,0} ⊗ B_{cris}^{σ} on U ⊗B_{cris}^{σ} factors through M_{q} (resp. P_{q}, resp. M_{ι,q}, resp. P_{ι,q}). This is equivalent to saying that the action of ℤ on D_{pcris}(U) is mixed (resp. pure, resp. ι-mixed with integral weights, resp. ι-pure).

We have the following analogue of a slope decomposition:

Lemma 5.23. There is a canonical morphism 𝔾̃_{m} → G^{pcris} ⊗_{ℚ_{p}} B_{cris}^{σ} of affine group schemes over B_{cris}^{σ}, for 𝔾̃_{m} as in definition 5.10.

Proof. Combine Lemma 5.13 with Lemma 5.21. □

6. Varieties over finite fields

Fix a variety X_{k} over a finite field k, of order q prime to ℓ. Let X := X_{k} ⊗_{k} k̄, for k̄ the algebraic closure of k. There is a Galois action on X, and hence on the pro-simplicial set X_{ét}, and on its algebraisation G(X_{ét})^{alg}. The purpose of this section is to describe this action as far as possible.

6.1. Algebraising the Weil groupoid. The morphism X → X_{k} gives a map of groupoids α: π_{f}^{ét}X → π_{f}^{ét}(X_{k}). Similarly, there is a map π_{f}^{ét}X_{k} → π_{f}^{ét}(Spec k) = Gal(k̄/k) ≅ ℤ̂. Denote the canonical generator of Gal(k̄/k) by F, the geometric Frobenius automorphism.

In constructing fundamental groupoids and étale homotopy types, we may use the same set of geometric points for both X_{k} and X, so assume that α is an isomorphism on objects. We then have

π_{f}^{ét}(X) = π_{f}^{ét}(X_{k}) ×ℤ̂ 0.

Definition 6.1. Define the Weil groupoid W_{f}(X_{k}) by

W_{f}(X_{k}) := π_{f}^{ét}(X_{k}) ×ℤ̂ ℤ,

noting that this is a pro-groupoid with discrete objects.

For any scheme Y, note that finite-dimensional representations of ϖ_{f}^{ét}(Y) := ϖ_{f}(Ŷ_{ét}) correspond to smooth ℚ_{ℓ}-sheaves on Y. We now introduce natural quotients of this groupoid.

Definition 6.2. Define^{W}ϖ_{f}^{ét}(X) to be the image of ϖ_{f}^{ét}(X) → W_{f}(X_{k})^{alg}, so W_{f}(X_{k})^{alg} = Wϖ_{f}^{ét}(X) ⋉ ℤ^{alg}.

Define ^{Gal}ϖ_{f}^{ét}(X) to be the image of ϖ_{f}^{ét}(X) → ϖ_{f}^{ét}(X_{k}), so ϖ_{f}^{ét}(X_{k}) = ^{Gal}ϖ_{f}^{ét}(X)⋉ℤ̂^{alg}. Note that ^{Gal}ϖ_{f}^{ét}(X) is a Frobenius-equivariant quotient of ^{W}ϖ_{f}^{ét}(X) (it is in fact the quotient on which ℤ̂ acts continuously).

In [Pri4], ^{W}ϖ_{1}(X, x̄) was defined to be the universal object classifying continuous W (X_{k}, x)-equivariant homomorphisms π_{1}(X, x̄) → G(ℚ_{ℓ}) to algebraic groups. In the terminology of [Pri4, Definition 1.3], ^{W}ϖ_{1}(X, x̄) is the maximal quotient of ϖ_{1}(X, x̄) on which Frobenius acts algebraically.

Note that these definitions are consistent by [Pri4, Lemma 1.11], which proceeds by establishing an action of ℤ^{alg} on ^{W}ϖ_{1}(X, x̄) generated by Frobenius, then showing that the map ℤ^{alg} ⋉^{W}ϖ_{1}(X, x̄) → W (X_{k}, x)^{alg} is an isomorphism.

It also implies that linear representations of ^{W}ϖ_{f}^{ét}(X) correspond to smooth ℚ_{ℓ}-sheaves on X arising as subsheaves of Weil sheaves, while linear representations of ^{Gal}ϖ_{f}^{ét}(X) correspond to smooth ℚ_{ℓ}-sheaves on X arising as subsheaves of pullbacks of smooth ℚ_{ℓ}- sheaves on X_{k}.

Lemma 6.3. The canonical action of F on ^{W}ϖ_{f}^{ét}(X) factors through a morphism

ℤ^{alg} → Aut(^{W}ϖ_{f}^{ét}(X))

of group presheaves, for ℤ^{alg} as in §5.1.

Proof. Write G = ^{W}ϖ_{f}^{ét}(X), H = W_{f}(X_{k})^{alg}, and observe that the orbits of F in Ob G = Ob H are finite, giving a map

ℤ̂ → Aut(Ob H).

Since ℤ̂ is pro-finite, we may regard it as the pro-algebraic group ℤ^{alg}/ℤ^{alg,0}. Now, consider the group scheme

N := ∐_{f∈Aut(Ob (H))} ∏_{x∈Ob (H)} H(x, fx),

with multiplication given by

(f, {h_{x}}) · (f′, {h′_{x}}) = (f · f′, {h_{f′x} · h_{x}}).

There is a morphism N → Aut(Ob (H)) fibred in affine schemes. Thus

ℤ̂ ×_{Aut(Ob (H))} N

is an affine scheme.

Now, F gives a collection of paths F (x) ∈ W_{f}(X_{k})(x, F x), and thus a map

ℤ → (ℤ̂ ×_{Aut(Ob (H))} N)(ℚ_{ℓ}).

Since the latter is an affine group scheme, this extends to a map ℤ^{alg} → ℤ̂ ×_{Aut(Ob (H))} N. Finally, observe that the conjugation action of H on G gives a map

N → Aut(G).

□

Theorem 6.4. If X_{k}/k is normal, then the action of ℤ^{red} on ^{W}ϖ_{f}^{ét}(X)^{red} factors through P_{q} (see Definition 5.2); in other words, the Frobenius representation O(^{W}ϖ_{f}^{ét}(X)^{red}) is a sum of finite-dimensional Galois representations, pure of weight 0.

Moreover ^{W}ϖ_{f}^{ét}(X)^{red} = ^{Gal}ϖ_{f}^{ét}(X)^{red}, so the ℤ^{red} action factors through its quotient ℤ̂^{red}.

Proof. Since ℤ^{alg} = ℤ^{red} × 𝔾_{a} (§5.1), this amounts to showing that the Frobenius action factors through P_{q} × 𝔾_{a}. We adapt the proof of [Pri4, Theorem 1.14] (to which we refer the reader for details).

Let T be the set of all isomorphism classes of irreducible representations V of^{W}ϖ_{f}^{ét}(X)^{red} over ℚ̄_{ℓ}. Since ^{W}ϖ_{f}^{ét}(X)^{red} is reductive, there is an isomorphism of ^{W}ϖ_{f}^{ét}(X)^{red} × Wϖ_{f}^{ét}(X)^{red}-representations given on objects (x, y) by

O(^{W}ϖ_{f}^{ét}(X)^{red}(x, y)) ⊗_{ℚ_{ℓ}} ℚ̄_{ℓ} ≅ ⨁_{V ∈T} Hom(V_{x}, V_{y}).

If 𝕍 is the smooth sheaf on X corresponding to the representation V, then ⨁_{V ∈T} Hom(V_{x}, V_{y}) corresponds to the smooth sheaf

⨁_{V ∈T} pr_{1}^{−1}𝕍^{∨} ⊗ pr_{2}^{−1}𝕍

on X × X.

Now, V ∈ T is an irreducible representation of ϖ_{f}^{ét}(X)^{red} which is a subrepresentation of some W_{f}(X_{k})-representation. This is the same as underlying a W_{f}(X_{k′})-representation for some finite extension k′/k, so 𝕍 underlies a smooth Weil sheaf on X_{k′}.

From Lafforgue’s Theorem ([Del2, Conjecture 1.2.10], proved in [Laf, Theorem VII.6 and Corollary VII.8]), every irreducible smooth Weil sheaf over ℚ̄_{ℓ} is of the form

𝕍 ≅ P ⊗ ℚ̄_{ℓ}^{(b)},

for some mixed sheaf P on X_{k′}. By [Del2, Theorem 3.4.1 (ii)], every irreducible smooth ι-mixed Weil sheaf is ι-pure. Thus the mixed sheaf P is ι-pure for all ι, and hence pure.

Thus

pr_{1}^{−1}𝕍^{∨} ⊗ pr_{2}^{−1}𝕍 ≅ pr_{1}^{−1}P^{∨} ⊗ pr_{2}^{−1}P,

which is a smooth sheaf on X_{k′} × X_{k′}, pure of weight 0.

Therefore O(^{W}ϖ_{f}^{ét}(X)^{red}) ⊗_{ℚ_{ℓ}} ℚ̄_{ℓ}, and hence O(^{W}ϖ_{f}^{ét}(X)^{red}), is a pure Galois representation of weight 0. Thus the action of ℤ^{alg} factors through P_{q} ×𝔾_{a}, and the discrete Galois action on ϖ_{f}^{ét}(X)^{red} descends to a continuous action on ^{W}ϖ_{f}^{ét}(X)^{red}, so

Wϖ_{f}^{ét}(X)^{red} = ^{Gal}ϖ_{f}^{ét}(X)^{red}.
□

6.2. Weight decompositions. Now assume that X is either smooth or proper and normal.

Definition 6.5. Define a weight decomposition on a multipointed homotopy type G ∈ Ho(sE(R)_{∗}) to be a morphism

𝔾_{m} → RAut(G)

of pro-algebraic groups.

Compare this with [Pri3, Definition 5.15], which considers weight decompositions on unpointed homotopy types, corresponding to outer automorphisms.

Proposition 6.6. If we let R be any Frobenius-equivariant quotient of ^{W}ϖ_{f}^{ét}(X)^{red}, then the Galois action on

X_{ét}^{R,Mal}

is mixed, giving a canonical weight decomposition. Furthermore, the Frobenius action extends canonically to a continuous algebraic Gal(k̄/k)-action.

Proof. By Theorem 6.4, the Galois action on R factors through the quotient P_{q} × 𝔾_{a} of ℤ^{alg}. By Corollary 4.8, the Gal(k̄/k) action on X_{ét}^{R,Mal} is thus algebraic. Since R is a P_{q} × 𝔾_{a}-representation, the Weil sheaf ⨁_{x∈Ob R} 𝕆(R)(x) is an arithmetic sheaf of weight 0. Deligne’s Weil II theorems ([Del2, Corollaries 3.3.4 – 3.3.6]) then imply that ⨁_{x∈X} H^{∗}(X, 𝕆(R)(x)) is a mixed Gal(k̄/k) representation (i.e. a representation of M_{q} × 𝔾_{a}). By Lemma 4.9, we may therefore conclude that the action of ℤ^{red} on X_{ét}^{R,Mal} factors through M_{q}, giving

M_{q} → Aut^{h}(X_{ét}^{R,Mal}).

Finally, use the map 𝔾_{m} → M_{q} (given after Definition 5.1) to define the weight decomposition. Since R is pure of weight zero, the 𝔾_{m}-action on R is trivial, giving

𝔾_{m} → Aut^{h}(X_{ét}^{R,Mal}),

as required. □

Corollary 6.7. The Galois actions are mixed on the duals ϖ_{n}(X_{ét}^{R,Mal}, x)^{∨} of the homotopy groups for n ≥ 2, and on the structure sheaves O(ϖ_{f}(X_{ét}^{R,Mal}))(x, y). In particular, these objects have canonical weight decompositions.

Proof. This is just the observation that there are canonical maps Aut^{h}(X_{ét}^{R,Mal}) → Aut(ϖ_{n}(X_{ét}^{R,Mal}, x)) and Aut^{h}(X_{ét}^{R,Mal}) → Aut(O(ϖ_{f}(X_{ét}^{R,Mal}))(x, y)) of group-valued presheaves, so Proposition 6.6 gives algebraic actions of M_{q} × 𝔾_{a} (and hence 𝔾_{m}) on the homotopy groups and fundamental groupoid. □

Remark 6.8. We have shown that ϖ_{n}(X_{ét}^{R,Mal}) is a mixed π_{f}^{ét}(X_{k})-representation. In par-

ticular, this means that ϖ_{n}(X_{ét}^{R,Mal}, x̄) is a mixed ˆ⟨F_{x}⟩-representation, so has a canonical weight decomposition.

Remark 6.9. If the hypotheses of Theorem 3.40 hold and π_{1}^{ét}(X, x) is N-good relative to R, then Corollary 6.7 implies that the Galois actions on the π_{n}^{ét}(X, x) ⊗ℤ̂ ℚ_{ℓ} are mixed for n ≤ N.

Alternatively, if it should happen that the Galois action on H^{n}(π_{1}^{ét}(X, x), V) is mixed for all R-representations V underlying pure π_{1}^{ét}(X_{k}, x)-representations and all n ≤ N, then Lemma 4.9 (combined with the Adams spectral sequence of [Pri3, Proposition 4.37]) implies that the Galois actions on ϖ_{n}(Bπ_{1}(X, x))^{R,Mal} is mixed for n ≤ N. Provided the first two hypotheses of Theorem 3.40 hold, the exact sequence of that theorem would then imply that the Galois actions on π_{n}^{ét}(X, x) ⊗ℤ̂ ℚ_{ℓ} are also mixed.

6.3. Formality. Now assume that X is smooth and proper. Deligne’s Weil II theorems then imply that ⨁_{x∈X} H^{n}(X, 𝕆(R)(x)) is pure of weight n.

Theorem 6.10. For R as in Proposition 6.6, the Malcev homotopy type X_{ét}^{R,Mal} ∈ sE(R) is formal, in the sense that it corresponds (under the equivalences of Proposition 2.50 and Theorem 2.74) to the R-representation

H_{ét}^{∗}(X, 𝕆(R))

in cochain algebras, equipped with the unique augmentation map ℚ_{ℓ} = H^{0}(X, 𝕆(R)) → ∏_{x∈Ob R} O(R)(x, −). This isomorphism is Galois equivariant.

Proof. We need to construct an isomorphism θ: NR_{u}(X_{ét}^{ρ,Mal}) ≅ GH_{ét}^{∗}(X, 𝕆(R)) in dgP(R) (for G as in Definition 4.20), such that ad_{θ}: Aut^{h}(X_{ét}^{ρ,Mal}) → Aut^{h}(GSpec DH_{ét}^{∗}(X, 𝕆(R)) ⋊ R) satisfies ad_{θ}F = F.

As in §4.3.4, take a minimal model 𝔪 for NR_{u}(X_{ét}^{ρ,Mal}) ∈ dgN̂ (R). This has the property that 𝔪_{n}/[𝔪, 𝔪]_{n} ≅ H^{n+1}(X, 𝕆(R))^{∨}.

From the proof of Lemma 4.4, we know that

∏_{x∈Ob R} exp(H_{0}𝔪(x)) × Aut_{dgN̂_{A}(R)}(𝔪 ˆ⊗A) → RAut(X_{ét}^{ρ,Mal})(A)

is a pro-unipotent extension of pro-algebraic groups.

Likewise, the maps

Aut(R ⋉ exp(𝔪)) ↠ Aut^{h}(X_{ét}^{ρ,Mal}) →
{(f, α) : f ∈ Aut(R), α ∈ Iso_{DGAlg(R)}(H_{ét}^{∗}(X, 𝕆(R)), f^{♯}H_{ét}^{∗}(X, 𝕆(R)))}

both have pro-unipotent kernels.

We may therefore lift the map ℤ̂^{alg} → Aut^{h}(X_{ét}^{ρ,Mal}) to give ℤ̂^{alg} → Aut(R ⋉ exp(𝔪)). This gives a lift of the weight decomposition 𝔾_{m} → RAut(X_{ét}^{ρ,Mal}) to 𝔾_{m} → Aut(R ⋉ exp(𝔪)). Since 𝔪 is of strictly negative weights, we may adapt [Pri4, Corollary 1.21] by observing that O(R ⋉ exp(𝔪))/O(R) is of strictly positive weights, and that the weight 0 part W_{0}O(R ⋉ exp(𝔪)) is just O(R), so we have a ℤ̂^{alg}-equivariant decomposition

O(R ⋉ exp(𝔪)) = O(R) ⊕ W_{+}O(R ⋉ exp(𝔪)).

This amounts to giving a ℤ̂^{alg}-equivariant section of R ⋉ exp(𝔪) → R, or equivalently a ℤ̂^{alg}-equivariant Levi decomposition, so we may assume that the ℤ̂^{alg} action on R⋉exp(𝔪) consists of actions on R and on 𝔪.

Let V_{n} := W_{−n−1}𝔪_{n}, for W as in §5.1; since cohomology is pure, we deduce that V_{n} → H^{n+1}(X, 𝕆(R))^{∨} is an isomorphism, and that 𝔪 is freely generated as a Lie algebra by the spaces V_{n}. The differential d on 𝔪 is then determined by d: V_{n} → 𝔪_{n−1}, and weight considerations show that the only non-zero contribution is V_{n} → ∏_{a+b=n−1}[V_{a}, V_{b}]. This is isomorphic to d: 𝔪/[𝔪, 𝔪] → [𝔪, 𝔪]/[𝔪, [𝔪, 𝔪]], so must be dual to the cup product.

Therefore, the choice of lift ℤ̂^{alg} → Aut(R ⋉ exp(𝔪)) has determined an isomorphism R ⋉ exp(𝔪) ≅ R ⋉ exp(GH_{ét}^{∗}(X, 𝕆(R))), and this is automatically compatible with the Galois action ℤ̂^{alg} → Aut^{h}(R ⋉ exp(𝔪)). □

Corollary 6.11. If we let R be any Frobenius-equivariant quotient of ^{W}ϖ_{f}^{ét}(X)^{red}, then the relative Malcev homotopy groups ϖ_{n}^{ét}(X^{R,Mal}, x) can be described in terms of cohomology as

ϖ_{n}^{ét}(X^{R,Mal}, x) ≅ H_{n−1}(GH^{∗}(X, 𝕆(R))),

for G as in Definition 4.20. This description is Galois-equivariant. If the conditions of Theorem 3.40 hold (including goodness), then this also calculates π_{n}^{ét}(X, x) ⊗ℤ̂ ℚ_{ℓ} as a Galois representation.

6.4. Quasi-formality. Let j: X ↪ X̄ be an open immersion of varieties over k̄, such that locally for the étale topology, the pair (X, X̄) is isomorphic to (𝔸^{m} × ∏_{i}(𝔸^{c_{i}} − {0}), 𝔸^{d}), for some d = m + ∑ c_{i}. Note that this is satisfied when X̄ − X is a normal crossings divisor (corresponding to the case c_{i} = 1 for all i). It also includes all geometric fibrations over k̄ in the sense of [Fri, Definition 11.4].

Definition 6.12. For X, X̄ as above, let T = X̄ − X, and let D be the closed subscheme of T of codimension 1 in X̄. Note that π_{f}^{ét}(X) → π_{f}^{ét}(X̄ −D) is an isomorphism, and define π_{f}^{t}(X) := π_{f}^{t}(X̄ − D) to be the tame fundamental groupoid (as in [SGA, XIII.2.1.3]).

Define π_{f}^{t}(X_{k}) similarly, with the tame Weil groupoid W_{f}^{t}(X_{k}) given by

W_{f}^{t}(X_{k}) := π_{f}^{t}(X_{k}) ×ℤ̂ ℤ.

Let ϖ_{f}^{t}(X) := π_{f}^{t}(X)^{alg}, and define ^{W}ϖ_{f}^{t}(X) to be the image of ϖ_{f}^{t}(X) → W_{f}^{t}(X_{k})^{alg}. Given a local system 𝕍 on X, observe that the direct image i_{∗}𝕍 of 𝕍 under the inclusion

i: X ↪ X̄ − D is also a local system. We say that 𝕍 is tamely ramified along the divisor if i_{∗}𝕍 is tamely ramified along D in the sense of [SGA, Definition XIII.2.1.1].

Lemma 6.13. Take j as above. If 𝕍 is a pure smooth Weil sheaf on Y of weight zero, tamely ramified along the divisor, then R^{ν}j_{∗}𝕍 is pure of weight 2ν (in the sense of [KW, Lemma-Definition II.12.7]).

Proof. This is a consequence of the following statements:

(1) R^{ν}j_{∗}𝕍 is pointwise pure of weight 2ν;

(2) the canonical map (R^{ν}j_{∗}𝕍)^{∨} → Rℋ omX̄(R^{ν}j_{∗}𝕍, ℚ_{ℓ}) is an isomorphism. If 0 → 𝕍′ → 𝕍 → 𝕍′′ → 0 is an exact sequence, with the statements holding for 𝕍 and 𝕍′′, then observe that they also hold for 𝕍, since the long exact sequence must degenerate.

The statements are local on X̄. Étale-locally, the pair (X, X̄) is isomorphic to (U, U′) = (𝔸^{m} × ∏_{i}(𝔸^{c_{i}} − {0}), 𝔸^{d}), for d = m + ∑ c_{i}. We may then reduce to the case when 𝕍 is irreducible on U, and so 𝕍 = 𝕍_{m} ⊠⨂_{i} 𝕍_{i}, for 𝕍_{i} irreducible on 𝔸^{c_{i}} −{0}. By the Künneth formula, we now need only consider the pair (𝔸^{c} − {0}, 𝔸^{c}).

If 𝕍 is constant, then the statements follow from the cohomological purity theorem ([Mil, VI.5.1]). Since the scheme 𝔸^{c} − {0} is simply connected for c > 1, this leaves only the case c = 1. [KW, Lemma I.9.1] shows that j_{∗}𝕍 is pure, and local calculations give R^{i}j_{∗}𝕍 = 0 for i > 0 (since 𝕍 is tamely ramified, and is non-constant irreducible). □

Proposition 6.14. Assume that j: X_{k} ↪ X̄_{k} is a morphism over k, with j ⊗ k̄ as in Lemma 6.13, for X̄_{k} proper. If 𝕍 is a pure smooth Weil sheaf on X of weight zero, tamely ramified along the divisor, then H^{i}(X̄, R^{ν}j_{∗}𝕍) is pure of weight i + 2ν, for j: X → X̄ the compactification map.

Proof. By [Del2, Corollary 3.3.4], we know that H^{i}(X̄, R^{ν}j_{∗}𝕍) is mixed of weights ≤ i+2ν, since R^{ν}j_{∗}𝕍 is pure of weight 2ν. Now, Poincaré duality ([KW, Corollary II.7.3]) implies that

H^{i}(X̄, R^{ν}j_{∗}𝕍)^{∨} ≅ H^{2d−i}(X̄, (R^{ν}j_{∗}𝕍)^{∨})(2d),

which is mixed of weight ≤ −i − 2ν, using the isomorphism (R^{ν}j_{∗}𝕍)^{∨} ≅ Rℋ omX̄(R^{ν}j_{∗}𝕍, ℚ_{ℓ}) of Lemma 6.13. □

Corollary 6.15. For X as above, and ρ: ϖ_{f}^{ét}X → R any Frobenius-equivariant quotient of ^{W}ϖ_{f}^{t}(X)^{red}, the filtered homotopy type (X^{ρ,Mal}, j) of Definition 4.37 is quasi-formal (in the sense of Definition 4.24). The formality quasi-isomorphism is equivariant with respect to the Galois action.

Proof. This is largely the same as Theorem 6.10. Use the equivalences of Theorem 4.22 to take a filtered minimal model (𝔪, J) ∈ F sN̂ (R) for (X^{ρ,Mal}, j). The increasing filtration J_{∗} on 𝔪^{∨} gives a decreasing filtration J^{∗} on 𝔪, with J^{r}𝔪_{n} the annihilator of J_{r−1}(𝔪^{∨}). Note that [J^{a}𝔪, J^{b}𝔪] ⊂ J^{a+b}𝔪 and J^{0}𝔪 = 𝔪.

If we write Aut_{J}(R ⋉ exp(𝔪)) for the group of filtered automorphisms of R ⋉ exp(𝔪), then similarly to Lemma 4.31, the maps

Aut_{J}(R ⋉ exp(𝔪)) ↠ Aut_{J}^{h}(X_{ét}^{ρ,Mal}) →
{(f, α) : f ∈ Aut(R), α ∈ Iso_{F DGAlg(R)}(H_{ét}^{∗}(X̄, R^{∗}j_{∗}𝕆(R)), f^{♯}H_{ét}^{∗}(X̄, R^{∗}j_{∗}𝕆(R)))}

both have pro-unipotent kernels.

We may therefore lift the Galois action ℤ̂^{alg} → Aut_{J}^{h}(X_{ét}^{ρ,Mal}) to a filtered automorphism of R ⋉ exp(𝔪). This gives a lift of the weight decomposition 𝔾_{m} → RAut_{J}(X_{ét}^{ρ,Mal}), a unique Galois-equivariant Levi decomposition of R ⋉ exp(𝔪), and a weight decomposition 𝔾_{m} → Aut_{J}(𝔪).

Now, (𝔪_{n}^{ab})^{∨} ≅ ⨁_{a+b=n+1} H^{a}(X̄, R^{b}j_{∗}𝕆(R)) =: E^{n+1}, on which J_{r} is the subspace of weights ≤ n + r + 1. Thus J^{r}(𝔪_{n}^{ab}) is the subspace of weights ≤ −(n + r + 1).

Let Γ^{r}𝔪 be the lower central series on 𝔪, so Γ^{1}𝔪 = 𝔪 and Γ^{r+1}𝔪 = [𝔪, Γ^{r}𝔪]. The weight restrictions on 𝔪^{ab} show that J^{r}(gr_{Γ}^{s}𝔪)_{n} = J^{r}(Lie_{s}(𝔪^{ab}))_{n}, which is of weights ≤ −(n + r + s). This implies that J^{r}(Γ_{s}𝔪)_{n} is of weights ≤ −(n + r + s).

We now make a canonical choice of generators by setting

W_{−(n+r+1)}V_{n} := W_{−(n+r+1)}J^{r}𝔪_{n}.

Set V := ∏_{i} W_{i}V; the weight conditions above show that this has no intersection with Γ_{s}𝔪 for s > 1, so the composition V → 𝔪 → 𝔪^{ab} is injective. Since W_{−(n+r+1)}(𝔪^{ab})_{n} = W_{−(n+r+1)}J^{r}(𝔪^{ab})_{n}, the composition is also surjective, so V is a space of generators for 𝔪.

The structure of 𝔪 is now determined by the differentials d: V_{n} → 𝔪_{n−1}. As 𝔪 = Lie(V) = V ×⋀^{2} V ×Γ_{3}𝔪, weight and filtration considerations show that we must have the projection d: V_{n} → (Γ_{3}𝔪)_{n−1} being 0. The non-zero contributions to d are V_{n} → V_{n−1}, which is dual to d_{1} on E, and V_{n} → ∏_{a+b=n−1}[V_{a}, V_{b}], which must be dual to the cup product. Thus 𝔪 = G(E), and so R ⋉ exp(𝔪) = R ⋉ exp(G(E)), as required. □

Corollary 6.16. For X and R as above, we can describe the relative Malcev homotopy groups ϖ_{n}^{ét}(X^{R,Mal}, x) explicitly in terms of the Leray spectral sequence as

H_{n−1}(G(_{J}E_{1}^{∗,∗})),

for

JE_{1}^{a,b} = H^{2a+b}(X̄, R^{−a}j_{∗}𝕆(^{W}ϖ_{f}^{ét}(X)^{L,red}))

as in Definition 4.11, and G as in Definition 4.20. If the conditions of Theorem 3.40 hold (including goodness), then this also calculates π_{n}^{ét}(X, x) ⊗ℤ̂ ℚ_{ℓ} as a Galois representation.

7. Varieties over local fields

7.1. Potentially good reduction, ℓ ≠ p. Let V′ be a complete discrete valuation ring, with residue field k′ (finite, of characteristic p ≠ ℓ), and fraction field K′ (of characteristic 0). Let k̄, K̄ be the algebraic closures of k, K′ respectively, and V̄ the algebraic closure of V′ in K̄.

Let X_{V′} = X̄_{V′} − T_{V′} be a geometric fibration over V′ (in the sense of [Fri, Definition 11.4]). Assume that we have a subfield K ⊂ K′ and a scheme X_{K}/K such that X_{K}⊗_{K}K′ ≅ X_{V′} ⊗_{V′} K′. We wish to study the Gal(K̄/K)-action on the homotopy type X_{K̄,ét}.

Recall from [SGA, Theorem X.2.1] that the map π_{f}^{ét}(X̄_{k′}) → π_{f}^{ét}(X̄_{V′}) is an equivalence. By ibid. §XIII.2.10, this generalises to an equivalence π_{f}^{t}(X_{k′}) → π_{f}^{t}(X_{V′}). Meanwhile, ibid. Corollary XIII.2.8 implies that π_{f}^{t}(XK̄) → π_{f}^{t}(XV̄) is an epimorphism, and ibid. Corollary XIII.2.9 shows that π_{f}^{ét}(XK̄)^{∧_{L}} → π_{f}^{ét}(XV̄)^{∧_{L}} is an equivalence, where L is any set of prime numbers excluding p.

Proposition 7.1. If 𝕍 is an ℓ-adic local system on XV̄, tamely ramified along the divisor (i.e. coming from a representation of π_{f}^{t}(XV̄)), then the maps

i_{η}^{∗}: H^{∗}(XV̄ , 𝕍) → H^{∗}(XK̄, i_{η}^{∗}𝕍)
i_{s}^{∗}: H^{∗}(XV̄ , 𝕍) → H^{∗}(Xk̄, i_{s}^{∗}𝕍)

are isomorphisms.

Proof. In [Fri, Theorem 11.5], this is proved for π_{f}^{ét}(XV̄)^{∧_{L}}-representations, for p ∉ L. The same proof carries over to π_{f}^{t}(XV̄)-representations, since the pro-L hypothesis is only used to restrict the monodromy around the divisor. □

Definition 7.2. Since π_{1}^{ét}(Spec V′) ≅ Gal(k̄/k′), we may define ^{W}ϖ_{f}^{t}(XV̄) analogously to Definition 6.2 as the maximal quotient of ϖ_{f}^{t}(XV̄) := π_{f}^{t}(XV̄)^{alg} on which the Frobenius action is algebraic. Define ^{pnr}ϖ_{f}^{t}(XK̄) to be the image of ϖ_{f}^{t}(XK̄) → ^{W}ϖ_{f}^{t}(XV̄), noting that this is a quotient of ϖ_{f}^{t}(XK̄) on which the Gal(K̄/K)-action is potentially unramified.

Note that these definitions are independent of the choice of extension V′/V, in the sense that a finite extension V′′/V′ would give the same construction.

Theorem 7.3. Let R be any Frobenius-equivariant reductive quotient of ^{pnr}ϖ_{f}^{t}(XK̄). Then the Gal(K̄/K)-action on the homotopy type

X_{K̄,ét}^{R,Mal}

is algebraic, potentially unramified (in the sense of §5.2) and mixed (Definition 5.16), giving a canonical Galois-equivariant weight decomposition. It is also quasi-formal, corresponding to the E_{2}-term

⨁_{a,b} H^{a}(X̄K̄, R^{b}j_{∗}𝕆(R)) ∈ F DGAlg(R),

of the Leray spectral sequence for the immersion j: X → X̄. The formality quasiisomorphism is equivariant with respect to the Gal(K̄/K)-action.

Proof. We know that the homotopy type is given by

C_{ét}^{•}(XK̄, 𝕆(R)) ∈ cAlg(R).

From the definition of ^{pnr}ϖ_{f}^{t}(XK̄), we know that 𝕆(R) is the pullback of a local system on XV̄, so i_{η∗}𝕆(R) is a local system and i_{η}^{∗}i_{η∗}𝕆(R) = 𝕆(R).

The equivalences of Proposition 7.1 now give quasi-isomorphisms

C_{ét}^{•}(XK̄, 𝕆(R)) = C_{ét}^{•}(XK̄, i_{η}^{∗}i_{η∗}𝕆(R)) ← C_{ét}^{•}(XV̄ , i_{η∗}𝕆(R)) → C_{ét}^{•}(Xk̄, i_{s}^{∗}i_{η∗}𝕆(R)),

compatible with the basepoint augmentation maps.

We may assume that K ⊂ K′ is a Galois extension, then observe that the equivalences above imply that action of Gal(K̄/K′) is unramified, so the Gal(K̄/K)^{alg} action factors through Gal(K̄/K) ×_{Gal(k̄/k′)} Gal(k̄/k′)^{alg}. In fact, Proposition 6.6 implies that the action factors through Gal(K̄/K) ×_{Gal(k̄/k′)} M_{q′}, where q′ = |k′|, so the morphism 𝔾_{m} → M_{q′}^{0} = ker(M_{q′} → Gal(k̄/k′)) provides the weight decomposition. This is compatible with the Galois action since M_{q′} is commutative (being a quotient of ℤ^{alg}), so 𝔾_{m} lies in the centre of Gal(K̄/K) ×_{Gal(k̄/k′)} M_{q′}.

We may now adapt Corollary 6.15 to see that this is quasi-formal, noting that all of the quasi-isomorphisms above extend naturally to the filtered algebras of Corollary 6.15. □

Corollary 7.4. Let X and R be as above. Then the homotopy groups ϖ_{n}^{ét}(XK̄) are potentially unramified and mixed as Galois representations, giving them a canonical weight decomposition. They may also be recovered from the Leray spectral sequence, as in Corollary 6.16.

Corollary 7.5. If L is a set of primes including ℓ, and:

(1) π_{f}^{ét}(X)^{∧_{L}} is (N + 1)-good relative to ^{pnr}ϖ_{f}^{t}(X∧_{L} ; K̄),

(2) π_{n}^{ét}(X^{∧_{L}}) ⊗ℤ̂ ℚ_{ℓ} is finite-dimensional for all 1 < n ≤ N, and (3) the action of ker(π_{f}^{ét}(XK̄)^{∧_{L}} → π_{f}^{t}(XV̄)^{∧_{L}}) on π_{n}^{ét}(X∧_{L} ; K̄) ⊗ℤ̂ ℚ_{ℓ} is unipotent for all

1 < n ≤ N,

then the Galois action on π_{n}^{ét}(X∧_{L} ; K̄) ⊗ℤ̂ ℚ_{ℓ} is potentially unramified and mixed, giving it a canonical weight decomposition. It may also be recovered from the Leray spectral sequence.

Proof. Substitute R = π_{f}^{t}(XV̄)^{L,red} into Corollary 7.4, Corollary 6.16 and Theorem 3.40. □

Note that if L does not contain p, then the third condition of the Corollary is vacuous.

7.2. Potentially good reduction, ℓ = p.

7.2.1. Convergent isocrystals. Let X, X̄, V′, K, K′, k′ etc. be as in the previous section, but with ℓ = p. Let W′ = W (k′), the ring of Witt vectors over k′, and K′_{0} the fraction field of W′; let W^{nr} := W (k̄), with K_{0}^{nr} its fraction field. Choose a homomorphism σ: K′ → K′ extending the natural action of the Frobenius operator ϕ on W (k′) ⊂ K′. Assume moreover that T_{V′} = D_{V′}, a normal crossings divisor, or more generally that D_{V′} corresponds to a log structure.

Definition 7.6. Let MF_{(X̄_{V′},D_{V′})/K′}^{∇} be the category of filtered convergent F -isocrystals on (X̄_{V′}, D_{V′}), as in [Tsu, §1] (or [Ols1, 6.9] when K′ is unramified, noting that the construction extends to ramified rings, as mentioned at the end of [Ols1, 1.14]).

Roughly speaking, an object of MF_{(X̄_{V′},D_{V′})/K′}^{∇} consists of an F -isocrystal (E, ϕ_{E}) on (X̄_{k}, D_{k})/W, together with a filtration Fil^{i}ℰ of ℰ satisfying Griffiths transversality with respect to ∇_{ℰ}, where (ℰ , ∇_{ℰ}) is the module with logarithmic connection on (X̄_{K′}, D_{K′}) obtained by base change from the evaluation of E on the p-adic completion of (X̄_{V′}, D_{V′}).

7.2.2. Crystalline étale sheaves. We now introduce crystalline étale sheaves, as in [Fal, V(f)] or [AI].

Definition 7.7. We define the category of associations on (X̄_{V′}, D_{V′}) to consist of triples (𝕍, ι, E), where

(1) 𝕍 is a smooth ℚ_{p}-sheaf on X_{K′},

(2) E ∈ MF_{(X̄_{V′},D_{V′})}^{∇}(Φ),

(3) ι is an association isomorphism ([Ols1, §6.13]), i.e. a collection of isomorphisms

ι_{U}: 𝕍 ⊗_{ℚ_{p}} B_{cris}(Û) → E(B_{cris}(Û))

for U → X_{V′} étale, compatible with the filtrations and semi-linear Frobenius automorphisms, and with morphisms over X, so that ι becomes an isomorphism of étale presheaves. Here, B_{cris}(Û) is formed by applying Fontaine’s construction to the p-adic completion Û of U.

A morphism f: (𝕍, ι, E) → (𝕍′, ι′, E′) in the category of associations consists of a morphism f^{ét} : 𝕍 → 𝕍′ and a morphism f^{cris} : E → E′ such that f^{cris} ∘ ι = ι′ ∘ f^{ét} : 𝕍 ⊗_{ℚ_{p}} B_{cris}(Û) → E′(B_{cris}(Û)) for all U.

The following lemma is a counterpart to [Fal, Lemma 5.5], which gives the corresponding statements for the forgetful functor from associations to MF_{(X_{V′},D_{V′})}^{∇}.

Proposition 7.8. The forgetful functor (𝕍, ι, E) ↦ 𝕍 from the category of associations to the category of smooth ℚ_{p}-sheaves on X_{K′} is full and faithful. Its essential image is stable under extensions and subquotients.

Proof. Given associations (𝕍, ι, E) and (𝕍′, ι′, E′), note that (𝕍^{∨} ⊗ 𝕍, (ι^{∨})^{−1} ⊗ ι′, E^{∨} ⊗ E′) is another association. Giving a morphism f^{ét} : 𝕍 → 𝕍′ amounts to giving an element of H^{0}(X_{K}, 𝕍^{∨} ⊗ 𝕍′), or equivalently a Galois-invariant element of H^{0}(XK̄, 𝕍^{∨} ⊗ 𝕍′). By [Fal, 5.6], the map

(ι^{∨})^{−1} ⊗ ι′: H^{∗}(XK̄, 𝕍^{∨} ⊗ 𝕍′) ⊗_{ℚ_{p}} B_{cris} → H_{cris}^{∗}(X_{k}/W, E^{∨} ⊗ E′) ⊗_{K′}_{0} B_{cris}

is an isomorphism. Taking Galois-invariant and Frobenius-invariant elements in Fil^{0}, this gives an isomorphism

(ι^{∨})^{−1} ⊗ ι′: H^{0}(XK̄, 𝕍^{∨} ⊗ 𝕍′)^{Gal(K̄/K′)} → Fil^{0}H_{cris}^{0}(X_{k}/W, E^{∨} ⊗ E′)^{ϕ},

so there is a unique Frobenius-equivariant morphism f^{cris} : E → E′ preserving the Hodge filtration such that the diagrams

𝕍 ⊗_{ℚ_{p}} B_{cris}(Û) ⟶^{ι} E(B_{cris}(Û))
f^{ét}⊗_{ℚp}B_{cris}↓  ↓_{f^{cris}(B_{cris}(Û))}
𝕍′ ⊗_{ℚ_{p}} B_{cris}(Û) ⟶^{ι′} E′(B_{cris}(Û))

commute. This shows that the forgetful functor is full and faithful.

To see that the essential image is stable under extensions, observe that extensions of 𝕍 by 𝕍′ are parametrised by elements a of H^{1}(X_{K′}, 𝕍^{∨} ⊗ 𝕍′). The isomorphisms above then show that ((ι^{∨})^{−1} ⊗ ι′)(a) is a Frobenius-equivariant element of Fil^{0}H_{cris}^{1}(X_{k}/W, E^{∨} ⊗ E′), so gives a unique extension of (𝕍, ι, E) by (𝕍′, ι′, E′) in the category of associations.

Finally, note that the subquotient of an extension is an extension of subquotients, so it suffices to show that the essential image contains subquotients of semisimple objects. Since such a subquotient 𝕍′ of 𝕍 is isomorphic to a direct summand, we have an idempotent endomorphism π of 𝕍 with ker π ≅ 𝕍′. Since the forgetful functor is full, π lifts to an idempotent endomorphism π̃ of (𝕍, ι, E), so 𝕍′ underlies ker π̃. □

Definition 7.9. Say that a smooth ℚ_{p}-sheaf 𝕍 on X_{K′} is crystalline if it lies in the essential image of the forgetful functor from the category of associations.

Proposition 7.10. The fibre functors (𝕍, ι, E) ↦ 𝕍x̄ make the category of associations into a multifibred Tannakian category. The corresponding pro-algebraic groupoid ϖ_{f}^{ét}(X_{K′})^{cris} is a quotient of ϖ_{f}^{ét}(X_{K′}). Moreover, ϖ_{f}^{ét}(X_{K′})^{cris} is the Malcev completion of π_{f}^{ét}(X_{K′}) with respect to the reductive quotient ϖ_{f}^{ét}(X_{K′})^{cris,red}.

Proof. Associations form a ℚ_{p}-linear rigid abelian tensor category, with (𝕍, ι, E) ⊗ (𝕍′, ι′, E′) = (𝕍 ⊗_{ℚ_{p}} 𝕍′, ι ⊗ ι′, E ⊗_{𝒪_{X_{k},cris}} E′) and (𝕍, ι, E)^{∨} = (𝕍^{∨}, (ι^{−1})^{∨}, E^{∨}).

By Proposition 7.8, associations are equivalent to the Tannakian subcategory of crystalline étale sheaves in Rep(ϖ_{f}^{ét}(X_{K′})). Thus the forgetful functor from associations to smooth ℚ_{p}-sheaves corresponds to a surjection ϖ_{f}^{ét}(X_{K′}) → ϖ_{f}^{ét}(X_{K′})^{cris} of pro-algebraic groupoids (with the same object set).

For ρ: π_{f}^{ét}(X_{K′}) → ϖ_{f}^{ét}(X_{K′})^{cris,red}, representations of ϖ_{f}^{ét}(X_{K′})^{ρ,Mal} are smooth ℚ_{p}- sheaves on X_{K′} which are Artinian extensions of semisimple crystalline étale sheaves. By Proposition 7.8, this is equivalent to the category Rep(ϖ_{f}^{ét}(X_{K′})^{cris}) of associations. □

Definition 7.11. Say that a smooth ℚ_{p}-sheaf 𝕍 on X_{K} is potentially crystalline if 𝕍|_{X_{K′′}} is crystalline for some finite extension K′ ⊂ K′′.

7.2.3. Equivariant pro-algebraic fundamental groups.

Definition 7.12. Define ^{cris,K′}ϖ_{f}^{ét}(XK̄) to be the image of ϖ_{f}^{ét}(XK̄) → ϖ_{f}^{ét}(X_{K′})^{cris}.

Note that we can also characterise ^{cris,K′}ϖ_{f}^{ét}(XK̄) as

ker(ϖ_{f}^{ét}(X_{K′})^{cris} → Gal(K̄/K′)^{cris} = ϖ_{f}^{ét}(Spec K′)^{cris}),

using the right-exactness of pro-algebraic completion. Thus

ϖ_{f}^{ét}(X_{K′})^{cris} = ^{cris,K′}ϖ_{f}^{ét}(XK̄) ⋊ Gal(K̄/K′)^{cris},

so representations of ^{cris,K′}ϖ_{f}^{ét}(XK̄) correspond to smooth ℚ_{p}-sheaves on XK̄ arising as subsheaves of pullbacks of crystalline étale ℚ_{p}-sheaves on X_{K′}.

Definition 7.13. Define

pcrisϖ_{f}^{ét}(XK̄) := \underleftarrow{lim}_{K′′} ^{cris,K′′}ϖ_{f}^{ét}(XK̄),

where the limit is taken over all finite Galois extensions K′ ⊂ K′′.

Finite-dimensional representations of ^{pcris}ϖ_{f}^{ét}(XK̄) thus correspond to smooth ℚ_{p}- sheaves on XK̄ arising as subsheaves of pullbacks of potentially crystalline smooth ℚ_{p}- sheaves on X_{K}.

Since G = \underleftarrow{lim}_{K′′}(Gal(K̄/K′′)^{cris} ×_{Gal(K̄/K′′)} Gal(K̄/K)), this gives an isomorphism

\underleftarrow{lim}_{K′′}(Gal(K̄/K) ×_{Gal(K̄/K′′)} ϖ_{f}^{ét}(X_{K′′})^{cris}) ≅ ^{pcris}ϖ_{f}^{ét}(XK̄) ⋊ G^{pcris},

so the Galois action on ^{pcris}ϖ_{f}^{ét}(XK̄) is algebraic and potentially crystalline.

Lemma 7.14. The map ϖ_{f}^{ét}(XK̄) ↠ ^{pnr}ϖ_{f}(X̄K̄) factors through ^{pcris}ϖ_{f}^{ét}(XK̄).

Proof. Since

Gal(K̄/K)^{pnr} ⋉^{pnr}ϖ_{f}(X̄K̄) = \underleftarrow{lim}_{K′′} Gal(K̄/K) ×_{Gal(k̄/k′′)} ϖ_{f}(X̄_{k′′}),

it suffices to show that the map ϖ_{f}(X_{K′′}) → ϖ_{f}(X̄_{k′′}) factors through ϖ_{f}^{ét}(X_{K′′})^{cris}. By looking at representations, this is equivalent to saying that every smooth ℚ_{p}-sheaf on X̄_{k′′} pulls back to give a crystalline étale sheaf on X_{V′′}. This now follows from [Kat, 4.1.1], which shows that smooth ℚ_{p}-sheaves on X̄_{k′′} correspond to unit-root F -lattices on X_{V′′}. □

Definition 7.15. Any field extension K′ → K′′ gives a pullback functor MF_{(X̄_{V′},D_{V′})/K′}^{∇} →

MF_{(X̄_{V′′},D_{V′′})/K′′}^{∇}, and we set

MF_{(X̄V̄ ,DV̄)/K̄}^{∇} := \underrightarrow{lim}_{K′′} MF_{(X̄_{V′′},D_{V′′})/K′′}^{∇},

where K′′ ranges over all finite field extensions K′ ⊂ K′′.

Representations of Gal(K̄/K)^{pcris,0} ⋉ ^{pcris}ϖ_{f}^{ét}(XK̄) are just representations of \underleftarrow{lim}_{K′′} ϖ_{f}^{ét}(X_{K′′})^{cris}, so the category of finite-dimensional representations is \underrightarrow{lim}_{K′′} FDRep(ϖ_{f}^{ét}(X_{K′′})^{cris}).

Definition 7.16. Making use of the forgetful functor from associations to filtered convergent F -isocrystals, the observation above gives us a ℚ_{p}-linear functor

D_{pcris}^{X}: FDRep(Gal(K̄/K)^{pcris,0} ⋉^{pcris}ϖ_{f}^{ét}(XK̄)) → MF_{(X̄V̄ ,DV̄)/K̄}^{∇}.

Say that an object of MF_{(X̄V̄ ,DV̄)/K̄}^{∇} is potentially admissible if it lies in the essential image of D_{pcris}^{X}.

Note that D_{pcris}^{Spec K} = D_{pcris}.

Definition 7.17. Given a G^{0}-equivariant affine scheme Y over ℚ_{p}, define the affine scheme D_{pcris}(Y) over K_{0}^{nr} by

D_{pcris}(Y) = Spec D_{pcris}O(Y).

Observe that O(Y) is therefore an ind-object of (i.e. a sum of objects in) the category MF_{(Spec V̄ ,∅)/K̄}^{∇}.

Proposition 7.18. The category of finite-dimensional D_{pcris}(^{pcris}ϖ_{f}^{ét}(XK̄))- representations in potentially admissible objects of MF_{(Spec V̄ ,∅)/K̄}^{∇} is equivalent to the category of finite-dimensional G^{pcris,0} ⋉^{pcris}ϖ_{f}^{ét}(XK̄)-representations, which in turn is equivalent to the category of potentially admissible objects of MF_{(X̄V̄ ,DV̄)/K̄}^{∇}.

For any point x ∈ XV̄ (K̄), the associated fibre functor from D_{pcris}(^{pcris}ϖ_{f}^{ét}(XK̄))- representations to MF_{(Spec V̄ ,∅)/K̄}^{∇} corresponds under this equivalence to the pullback

x^{∗}: MF_{(X̄V̄ ,DV̄)/K̄}^{∇} → MF_{(Spec V̄ ,∅)/K̄}^{∇}.

Proof. A D_{pcris}(^{pcris}ϖ_{f}^{ét}(XK̄))-representation V in potentially admissible objects of MF_{(Spec V̄ ,∅)/K̄}^{∇} consists of potentially admissible objects V (x) ∈ MF_{(Spec V̄ ,∅)/K̄}^{∇} for all x ∈ Ob (^{pcris}ϖ_{f}^{ét}(XK̄)), together with coassociative morphisms

V (y) → V (x) ⊗ D_{pcris}O(^{pcris}ϖ_{f}^{ét}(XK̄)(x, y))

in MF_{(Spec V̄ ,∅)/K̄}^{∇}.

Since D_{pcris} gives an equivalence between G^{pcris,0}-representations and potentially admissible objects of MF_{(Spec V̄ ,∅)/K̄}^{∇}, the description above shows that it defines the required equivalence from G^{pcris,0} ⋉^{pcris}ϖ_{f}^{ét}(XK̄)-representations.

Now, G^{pcris,0} ⋉^{pcris}ϖ_{f}^{ét}(XK̄) ≅ \underleftarrow{lim}_{K′′} ϖ_{f}^{ét}(X̄_{K′′})^{cris}, so we may apply the functor D_{pcris}^{X} from Definition 7.16, mapping to potentially admissible objects in MF_{(X̄V̄ ,DV̄)/K̄}^{∇}. By [Fal, Lemma 5.5], this functor is full and faithful, so gives us the second equivalence required. □

Definition 7.19. Define

Isoc((X̄k̄, Dk̄)/K_{0}^{nr}) := \underrightarrow{lim}_{K′′} Isoc((X̄_{k′′}, D_{k′′})/K′′)

to be the category of isocrystals on \underleftarrow{lim}_{K′′}(X̄_{k′′}, D_{k′′})/K′′, where the limit is taken over finite extensions K′ ⊂ K′′.

Proposition 7.20. The category of finite-dimensional D_{pcris}(^{pcris}ϖ_{f}^{ét}(XK̄))- representations over K_{0}^{nr} is equivalent to a full subcategory of Isoc((X̄k̄, Dk̄)/K_{0}^{nr}). This subcategory is the smallest full abelian subcategory containing the potentially admissible objects of MF_{(X̄V̄ ,DV̄)/K̄}^{∇}.

Proof. Write G := ^{pcris}ϖ_{f}^{ét}(XK̄), and let 𝕆(G) be the universal G-representation in smooth ℚ_{p}-sheaves on XK̄, as defined in Definition 2.75. Following through the proof of Proposition 7.18, the functor from D_{pcris}(G)-representations in potentially admissible objects of MF_{(Spec V̄ ,∅)/K̄}^{∇} to MF_{(X̄V̄ ,DV̄)/K̄}^{∇} is given by

F (A) := A ⊗^{D_{pcris}(G)} D_{pcris}^{X}𝕆(G),

while its inverse is

F_{∗}(𝒜) := \underrightarrow{lim}_{K′′} H_{cris}^{0}((X̄_{k′′}, D_{k′′}), 𝒜 ⊗ D_{pcris}^{X}𝕆(G)).

The same formulae define left exact functors F, F_{∗} between the category of finitedimensional D_{pcris}(G)-representations and Isoc((X̄k̄, Dk̄)/K_{0}^{nr}). For any point x ∈ X(K̄),

F (A)_{x} = A ⊗^{D_{pcris}(G)} D_{pcris}(𝕆(G)_{x}) = A ⊗^{D_{pcris}(G)} D_{pcris}(O(G)(x, −)) = A(x),

so F is exact.

For any D_{pcris}(G)-representation A,

F_{∗}F (A) = A ⊗^{D_{pcris}(G)} \underrightarrow{lim}_{K′′} H_{cris}^{0}((X̄_{k′′}, D_{k′′}), D_{pcris}^{X}𝕆(G) ⊗ D_{pcris}^{X}𝕆(G))
= A ⊗^{D_{pcris}(G)} D_{pcris}O(G)
= A.

Moreover, F_{∗} is right adjoint to F, since a morphism A → F_{∗}(𝒜′) is equivalent to a Gequivariant morphism A ⊗ 𝒪_{X,cris} → 𝒜′ ⊗ D_{pcris}^{X}𝕆(G) of isocrystals, which is equivalent to a G-equivariant D_{pcris}^{X}𝕆(G)-linear morphism A ⊗ D_{pcris}^{X}𝕆(G) → 𝒜′ ⊗ D_{pcris}^{X}𝕆(G), which (taking G-invariants) is just a morphism F (A) → 𝒜′. These two statements combine to show that F is full and faithful.

Since F is exact, its essential image is an abelian subcategory. Proposition 7.18 ensures that it contains all potentially admissible objects of MF_{(X̄V̄ ,DV̄)/K̄}^{∇}, so we need only show that anything in the image of F is in the abelian subcategory generated by these potentially admissible objects.

Given any D_{pcris}(G)-representation A, we have a canonical embedding A ↪ A ⊗ D_{pcris}(O(G)), which is a sum of objects of MF_{(Spec V̄ ,∅)/K̄}^{∇}. Thus for some finite-dimensional subobject U, we have an embedding A ↪ U. Replacing A with U/A, we get an embedding U/A ↪ U′, so A = ker(U → U′), and hence F (A) = ker(F (U) → F (U′)). Since F (U) and F (U′) are potentially admissible objects of MF_{(X̄V̄ ,DV̄)/K̄}^{∇}, this completes the proof. □

7.2.4. Crystalline homotopy types. Fix a Galois-equivariant quotient R of ^{pcris}ϖ_{f}^{ét}(XK̄)^{red}, or rather of its full subgroupoid on objects X(K̄)

Definition 7.21. Let ℱ → 𝒞_{cris}^{•}(ℱ) be a choice of functor from isocrystals to cosimplicial sheaves on the log-crystalline site, with the property that 𝒞_{cris}^{•}(ℱ) is a resolution of ℱ, compatible with tensor products, and acyclic for log-crystalline cohomology. Examples of such a functor are given in [Pri1, p.17], or by denormalising the construction DR of [Ols1, 4.29.2]. In both cases, the resolution is given by first choosing a resolution which is acyclic for the derived functor between crystalline and Zariski sites (such as denormalisation of the de Rham complex), then taking a Čech resolution.

Define

C_{cris}^{•}(Y, ℱ) := Γ(Y, 𝒞_{cris}^{•}(ℱ)),

observing that this construction will also be compatible with tensor products.

Definition 7.22. Define the relative crystalline homotopy type X_{k̄,cris}^{D_{pcris}(R),Mal} over D_{pcris}R to be the pro-algebraic homotopy type in Ho(sE(D_{pcris}R)_{∗}) (over K_{0}^{nr}) corresponding under Theorem 2.74 to the D_{pcris}(R)-representation

C_{cris}^{•}((X̄k̄, Dk̄), D_{pcris}^{X}𝕆(R))

in cosimplicial K_{0}^{nr}-algebras, equipped with its natural augmentations to D_{pcris}O(R)(x, −) = C_{cris}^{•}(Spec K_{0}^{nr}, x^{∗}D_{pcris}^{X}𝕆(R)) coming from elements x ∈ X(V̄).

Lemma 7.23. There is a canonical equivalence between representations of ϖ_{f}(Xk̄/K_{0}^{nr})_{cris}^{D_{pcris}(R),Mal} and a full subcategory of Isoc((X̄k̄, Dk̄)/K_{0}^{nr}). Objects of this category are Artinian extensions of those isocrystals corresponding under Proposition 7.20 to D_{pcris}(R)-representations.

Proof. This is [Ols2, Theorem 2.28]. An alternative approach would be to note that the proof of [Pri1, Theorem 2.9] carries over to non-nilpotent torsors. □

Definition 7.24. For a topos T , if 𝒞_{T}^{•}(𝒮) is a canonical cosimplicial T -resolution of a sheaf 𝒮 of algebras on X, with C_{T}^{•} (X, 𝒮) := Γ(X, 𝒞_{T}^{•}(𝒮)), then for any morphism f: X → Y we have a bicosimplicial algebra C_{T}^{•} (Y, f_{∗}𝒞_{T}^{•}(𝒮)), and we define

C_{T}^{•} (f, 𝒮) := τ′′C_{T}^{•} (Y, f_{∗}𝒞_{T}^{•}(𝒮)) ∈ F cAlg,

defined as in Definition 4.33.

Definition 7.25. If we write j for the embedding X ↪ X̄, define the filtered relative crystalline homotopy type (X_{k̄,cris}, j_{k̄,cris})^{D_{pcris}(R),Mal} over D_{pcris}R to be the filtered proalgebraic homotopy type in Ho(sE(D_{pcris}R)_{∗}) (over K_{0}^{nr}) corresponding under Theorem 4.22 to the filtered D_{pcris}(R)-representation

C_{cris}^{•}(j_{k̄,cris}, D_{pcris}^{X}𝕆(R))

in cosimplicial K_{0}^{nr}-algebras, equipped with its natural augmentations to D_{pcris}O(R)(x, −) = C_{cris}^{•}(Spec K_{0}^{nr}, x^{∗}D_{pcris}^{X}𝕆(R)) coming from elements x ∈ X(V̄).

7.2.5. Comparison of homotopy types. From now on, let B := B_{cris}(V) and B̃ := B̃_{cris}(V), from Definition 5.18.

Proposition 7.26. For any Galois-equivariant quotient R of ^{pcris}ϖ_{f}^{ét}(XK̄)^{red}, there is a chain of (ϕ, G^{0})-equivariant quasi-isomorphisms

X_{K̄,ét}^{R,Mal} ⊗_{ℚ_{p}} B̃ ∼ X_{k̄,cris}^{D_{pcris}R,Mal} ⊗_{K_{0}^{nr}} B̃

in sAffB̃(R)_{∗}.

Proof. This amounts to establishing a chain of quasi-isomorphisms

C_{ét}^{•}(XK̄, 𝕆(R)) ⊗_{ℚ_{p}} B̃ ∼ C_{cris}^{•}(Xk̄/K_{0}^{nr}, D_{pcris}^{X}𝕆(R)) ⊗_{K_{0}^{nr}} B̃

in cAlgB̃(R)_{∗}

In the notation of [Ols1, 4.29 and 5.21], C_{cris}^{•}(X_{k}/K_{0}^{nr}, D_{pcris}^{X}𝕆(R)) and C_{ét}^{•}(XK̄, 𝕆(R)) are quasi-isomorphic to the denormalisations of RΓ_{cris}(D_{pcris}^{X}𝕆(R)) and GC(𝕆(R), X(K̄)), since denormalisation and Thom-Sullivan are quasi-inverse up to homotopy (as in Remark 3.31).

Since the affine group schemes R/ℚ_{p} and D_{pcris}(R)/K_{0}^{nr} are associated by an isomorphism

B ⊗_{ℚ_{p}} O(R) ≅ B ⊗_{K_{0}^{nr}} D_{pcris}O(R),

the required result is then ibid. 6.15.1, combined with the observation in ibid. Proposition 6.19 that pullback preserves associations, thus ensuring that these associations are compatible with the augmentation maps coming from basepoints.

The proof of ibid. 6.15.1 proceeds by adapting the isomorphisms on cohomology groups from [Fal, 5.6] to quasi-isomorphisms of DG algebras. Since the latter proves that the cohomological isomorphisms respect cup products, an alternative approach would be to extend the isomorphisms to quasi-isomorphisms of the minimal E_{∞}-algebras they underlie. Remark 2.54 would then imply that the corresponding objects in dgN̂ (R) are weakly equivalent. □

Remark 7.27. When L is a crystalline étale sheaf on X_{K} and R is the Zariski closure of the image of π_{1}^{ét}(XK̄, x̄) → GL(Lx̄) with nilpotent monodromy around each component of the divisor, then Proposition 7.26 is effectively [Ols1, Theorem 1.7] (replacing “crystalline” with “potentially crystalline” throughout). The nilpotent hypothesis was needed for Tannakian considerations, which in our case are obviated by Proposition 7.8.

Theorem 7.28. Given a Galois-equivariant quotient R of ^{pcris}ϖ_{f}^{ét}(XK̄), the Galois action on X_{K̄,ét}^{RMal} is algebraic and potentially crystalline.

Proof. In the notation of §5.3, we need to show that the map G → Aut^{h}(X_{K̄,ét}^{R,Mal}) factors through G^{pcris}. Apply Proposition 5.20 to Proposition 7.26, taking

Y = Aut^{h}(X_{K̄,ét}^{R,Mal}) ×_{Aut(R)} G^{pcris,0}

with the G^{0} action on Y given by left multiplication.

Now, note that D_{pcris}(G^{pcris,0} × R) = D_{pcris}(G^{pcris,0}) × R, giving a K_{0}^{nr}-linear map f: D_{pcris}(G^{pcris,0}) × R → D_{pcris}R. In fact, D_{pcris}(G^{pcris,0}) = Spec B^{ker(G^{0}→G^{pcris,0})}, so this map just comes from the isomorphism (D_{pcris}O(R)) ⊗_{K_{0}^{nr}} B ≅ O(R) ⊗_{ℚ_{p}} B.

We now define Z over D_{pcris}G^{pcris,0} to be the affine scheme given by

Z(A) = Iso_{Ho(dgAff_{A}(R)_{∗})}(X_{K̄,ét}^{R,Mal} ⊗_{ℚ_{p}} A, f^{♯}(X_{cris}^{D_{pcris}R,Mal} ⊗_{K_{0}^{nr}} A)),

for D_{pcris}O(G^{pcris,0})-algebras A.

Since G^{pcris,0} is potentially crystalline, we have an isomorphism α: G^{pcris,0} × Spec B̃ → (D_{pcris}G^{pcris,0}) ×_{Spec K_{0}^{nr}} Spec B̃, so the scheme Z ×_{Spec K_{0}^{nr}} Spec B̃ can be regarded as a scheme over G^{pcris,0} × Spec B̃

The G^{0}-equivariant isomorphism of Proposition 7.26 then gives, for any D_{pcris}O(G^{pcris,0}) ⊗_{K_{0}^{nr}} B̃-algebra A, a G^{0}-equivariant isomorphism

Z(A) ≅ Iso_{Ho(dgAff_{A}(R)_{∗})}(X_{K̄,ét}^{R,Mal} ⊗_{ℚ_{p}} A, α^{♯}X_{K̄,ét}^{R,Mal} ⊗_{ℚ_{p}} A),

but the right-hand side is just Y (A), giving a G^{0}-equivariant isomorphism

Z ×_{K_{0}^{nr}} Spec B̃_{cris} ≅ Y ×_{ℚ_{p}} Spec B̃_{cris},

as required. □

Corollary 7.29. For x, y ∈ X(K̄), the G^{0}-actions on

ϖ_{n}(X_{K̄,ét}^{R,Mal}, x) and ϖ_{f}(X_{K̄,ét}^{R,Mal})(x, y)

are potentially crystalline.

Proof. This is just the observation that the map Aut(X_{K̄,ét}) → Aut(ϖ_{n}(X_{K̄,ét}^{R,Mal}, x))(ℚ_{p})

factors through Aut^{h}(X_{K̄,ét}^{R,Mal}). □

Note that if we set R = 1 and look at the fundamental group, this recovers the comparison theorem of [Shi] and [Vol] between pro-unipotent étale and crystalline fundamental groups.

In fact, we may extend Proposition 7.26 to a filtered version:

Proposition 7.30. For any Galois-equivariant quotient R of ^{pcris}ϖ_{f}^{ét}(XK̄)^{red} and for j: X → X̄, there is a chain of canonical (ϕ, G^{0})-equivariant quasi-isomorphisms

(X_{K̄,ét}, j_{K̄,ét})^{R,Mal} ⊗_{ℚ_{p}} B̃ ∼ (X_{k̄,cris}, j_{k̄,cris})^{D_{pcris}(R),Mal} ⊗_{K_{0}^{nr}} B̃

in F sAffB̃(R)_{∗}.

Proof. The proof of Proposition 7.26 adapts. □

Lacking a suitable p-adic analogue of Lafforgue’s Theorem (although [Ked, Theorem 6.3.4] might provide a viable replacement in some cases), we now impose a purity hypothesis.

Assumption 7.31. Assume that D_{pcris}^{X}𝕆(R) is an ind-object in the category of ι-pure overconvergent F -isocrystals. Like Definition 6.2, this is equivalent to saying that for every R-representation V, the corresponding sheaf 𝕍 on XK̄ can be embedded in the pullback of a crystalline étale sheaf 𝕌 on X_{K′′}, associated to an ι-pure overconvergent F -isocrystal on (X̄_{k′′}, D_{k′′})/K′′, for some finite extension K′ ⊂ K′′. Also note that this implies that the Frobenius action on D_{pcris}O(R) is ι-pure.

Example 7.32. To see how the hypotheses of Assumption 7.31 arise naturally, assume that f: Y_{K} → X_{K} is a geometric fibration (in the sense of [Fri, Definition 11.4], for instance any smooth proper morphism) with connected components, for Y of potentially good reduction. Let G(x̄, z̄) be the Zariski closure of the map

π_{f}^{ét}(XK̄)(x̄, z̄) → ∏_{n} Iso((R^{n}f_{K̄,∗}^{ét}ℚ_{p})x̄, (R^{n}f_{K̄,∗}^{ét}ℚ_{p})z̄),

so G is a pro-algebraic groupoid on objects X(K̄), and then set R = G^{red}. By [Fal], R^{n}f_{K̄,∗}^{ét}ℚ_{p} is associated to R^{n}f_{k̄,∗}^{cris}𝒪_{Yk̄,cris}, which by [Ked, Theorem 6.6.2] is ι-pure (or if f is not proper, globally ι-mixed). Thus the semisimplifications of the G-representations x̄ ↦ (R^{n}f_{K̄,∗}^{ét}ℚ_{p})x̄ are direct sums of ι-pure representations. Since these generate the Tannakian category of R-representations, the hypotheses are satisfied.

For x̄ ∈ X(K̄), we may write F := Y ×_{f,X,x̄} Spec K̄, and Theorem 3.32 then shows that the homotopy fibre of

(Y ét ; K̄)^{R,Mal} → (Xét ; K̄)^{R,Mal}

over x̄ is (F ét ; K̄)^{1,Mal}.

Example 7.33. A more comprehensive example would be to let G(x̄, z̄) be the Zariski closure of the map π_{f}^{ét}(XK̄)(x̄, z̄) → ∏_{n,f} Iso((R^{n}f_{K̄,∗}^{ét}ℚ_{p})x̄, (R^{n}f_{K̄,∗}^{ét}ℚ_{p})z̄), where f ranges over all geometric fibrations of potentially good reduction with connected components, and then to set R := G^{red}. The resulting homotopy type (Xét ; K̄)^{R,Mal} would be very close to possible conceptions of a pro-algebraic motivic homotopy type.

Theorem 7.34. Given a Galois-equivariant quotient R of ^{pcris}ϖ_{f}^{ét}(XK̄) satisfying Assumption 7.31, the Galois action on X_{K̄,ét}^{R,Mal} is ι-mixed in the sense of Definition 5.22, giving a

canonical weight decomposition on X_{K̄,ét}^{R,Mal} ⊗ B^{σ}.

Proof. This is essentially the same as Proposition 6.6. Frobenius gives a canonical element of Aut^{h}(X_{cris}^{D_{pcris}R,Mal}). We first show that this is ι-mixed of integral weights. By Lemma 4.9, we need only consider the Frobenius action on cohomology

H_{cris}^{∗}((X̄k̄, Dk̄), D_{pcris}^{X}𝕆(R)).

The Leray spectral sequence gives

H_{cris}^{2a+b}(X̄k̄, R_{cris}^{−a}j_{∗}D_{pcris}^{X}𝕆(R)) ⟹ H_{cris}^{a+b}((X̄k̄, Dk̄), D_{pcris}^{X}𝕆(R)).

If we write D^{(n)} for the normalisation of the n-fold intersection of the local components of D, and i_{n}: D^{(n)} → X̄ for the embedding, then as in [Del1, 3.2.4.1], there is an isomorphism

H_{cris}^{2a+b}(X̄k̄, R_{cris}^{−a}j_{∗}D_{pcris}^{X}𝕆(R)) ≅ H_{cris}^{2a+b}(D(−a) ; k̄, i_{n}^{∗}j_{∗}D_{pcris}^{X}𝕆(R)(a)),

since j_{∗}D_{pcris}^{X}𝕆(R) is associated to a locally constant sheaf on X.

Now, [Ked, Theorem 6.6.2] combined with Poincaré duality proves that H_{cris}^{2a+b}(D(−a) ; k̄, i_{n}^{∗}j_{∗}D_{pcris}^{X}𝕆(R)(a)) is ι-pure of weight b. Thus Lemma 4.9 implies that the

Frobenius element of Aut^{h}(X_{cris}^{D_{pcris}R,Mal}) is ι-mixed of integral weights.

We need to show that the composite morphism

ℤ^{alg,0} → G^{pcris} ⊗_{ℚ_{p}} B^{σ} → Aut^{h}(X_{K̄,ét}^{R,Mal}) ⊗_{ℚ_{p}} B^{σ}

factors through M_{ι}^{0}. By Proposition 7.26,

Aut^{h}(X_{K̄,ét}^{R,Mal}) ⊗_{ℚ_{p}} B̃^{σ} ≅ Aut^{h}(X_{cris}^{D_{pcris}R,Mal}) ⊗_{K_{0}^{nr}} B̃^{σ},

so the map

ℤ^{alg,0} → G^{pcris} ⊗_{ℚ_{p}} B^{σ} → Aut^{h}(X_{K̄,ét}^{R,Mal}) ⊗_{ℚ_{p}} B̃^{σ}

factors through M_{ι}^{0}. Since B^{σ} ⊂ B̃^{σ}, this completes the proof. □

Theorem 7.35. For R as in Theorem 7.34, the filtered homotopy type (X_{K̄,ét}, j_{K̄,ét})^{R,Mal}⊗ B^{σ} is quasi-formal, corresponding to the E_{2}-term

JE_{1}^{a,b}(X_{K̄,ét}^{R,Mal}) ⊗ B^{σ} = ⨁_{a,b} H^{2a+b}(X̄K̄, R^{−b}j_{∗}𝕆(R)) ⊗ B^{σ} ∈ F DGAlg_{B^{σ}}(R),

of the Leray spectral sequence for the immersion j: X → X̄, and the formality isomorphism is equivariant with respect to the Galois action.

The filtered homotopy type (X_{K̄,ét}, j_{K̄,ét})^{R,Mal} is also quasi-formal, but the formality isomorphism is not in general Galois-equivariant or canonical.

Proof. Since the Galois action is ι-mixed in the sense of Definition 5.22, there is a Galoisequivariant weight decomposition 𝔾_{m} → RAut_{J}(X_{K̄,ét}^{R,Mal} ⊗B^{σ}), using Lemma 5.21 and the observation after Definition 5.3. The argument of Corollary 6.15 now adapts to show that X_{K̄,ét}^{R,Mal} ⊗ B^{σ} is quasi-formal, with the formality quasi-isomorphism equivariant under the Galois action, proving the first part.

In particular this implies that

RAut_{J}(X_{K̄,ét}^{R,Mal})(B^{σ}) → Aut(_{J}E_{1}^{∗,∗}(XR,Mal ; K̄))(B^{σ})

is a pro-unipotent extension. Thus the corresponding morphism of pro-algebraic groups is surjective, allowing us to lift the weight decomposition on E_{1}^{∗,∗}(XR,Mal ; K̄) non-canonically to X_{K̄,ét}. This decomposition need not be compatible with the canonical decomposition on X_{K̄,ét}^{R,Mal} ⊗ B^{σ}. The argument of Corollary 6.15 adapted to this decomposition now shows

that X_{K̄,ét}^{R,Mal} is quasi-formal. □

Corollary 7.36. For X and R as above, we can describe the homotopy groups ϖ_{n}^{ét}(XR,Mal ; K̄, x) ×_{Spec ℚ_{p}} Spec B^{σ} explicitly in terms of the Leray spectral sequence as

ϖ_{n}^{ét}(XR,Mal ; K̄, x)^{∨} ⊗_{ℚ_{p}} B^{σ} = H^{n−1}(G(_{J}E_{1}^{∗,∗}(X_{K̄,ét}^{R,Mal}))^{∨}) ⊗_{ℚ_{p}} B^{σ},

for G as in Definition 4.20. Of course, if the conditions of Theorem 3.40 hold (including goodness), then this also calculates π_{n}^{ét}(XK̄, x) ⊗ℤ̂ B^{σ} as a Galois representation.

Remarks 7.37. (1) In the case when X is projective and R is a quotient of ^{Gal}ϖ_{f}(Xk̄), this is essentially the main formality result of [Ols2, §4], which has since been extended to the general projective case in [Ols1, Theorem 7.22], although Frobeniusequivariance is not made explicit there. The proofs also differ in that they work with minimal algebras, rather than minimal Lie algebras.

(2) Although at first sight Theorem 7.35 is weaker than Theorem 7.3, it is more satisfactory in one important respect. Theorem 7.3 effectively shows that relative Malcev ℓ-adic homotopy types carry no more information than cohomology, whereas to recover a relative Malcev p-adic homotopy type from Theorem 7.35, we still need to identify (X_{K̄,ét}, j_{K̄,ét})^{R,Mal} ⊂ (X_{K̄,ét}, j_{K̄,ét})^{R,Mal} ⊗ B^{σ}. This must be done by describing the Hodge filtration on (X_{cris}^{D_{pcris}R,Mal}, j_{k̄,cris}), which is not determined by cohomology (since it is not Frobenius-equivariant). Thus the Hodge filtration is the only really new structure on the relative Malcev homotopy type. This phenomenon is similar to the formality results for mixed Hodge structures in [Pri5, §2].

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