Galois actions on the pro-l-unipotent fundamental group^{∗}

J.P.Pridham

March 27, 2010

Contents

Introduction 1

1 The pro-l-unipotent fundamental group 2 1.1 Finite characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Good reduction, l ≠ p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Potentially good reduction, l ≠ p . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Potentially good reduction, l = p . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Global fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 The crystalline fundamental group 11 2.1 Smooth proper schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Smooth non-proper schemes . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 The Hodge filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 p-adic Hodge Theory 19 3.1 Comparison of étale and intermediate homotopy types . . . . . . . . . . 20 3.2 Comparison of crystalline and intermediate homotopy types . . . . . . . 22 3.3 Consequences for fundamental groups . . . . . . . . . . . . . . . . . . . 23

Introduction

In [?], the pro-l-unipotent étale fundamental group π_{1}(X, x̄) ⊗ ℚ_{l} of a variety X was defined, and the Weil conjectures used to prove restrictions on its structure when X is smooth and defined over a finite field. The purpose of this paper is to show that this approach can also be used to describe the Galois action on π_{1}(X, x̄) ⊗ ℚ_{l}, with X now defined over a number field K.

Let G = Gal(K̄/K), and 𝔭 a prime of K̄. The main idea in Section 1 is to define, for each 𝔭, a G_{𝔭}-invariant decomposition on L(π_{1}(X, x̄), ℚ_{l}), the Lie algebra underlying the pro-l-unipotent fundamental group. This can be done for every prime 𝔭 ∤ l at which X is of potentially good reduction. Weight restrictions on cohomology then give restrictions on the possible weight decompositions on this Lie algebra. In particular, when X is proper, the G_{𝔭}-action on π_{1}(X, x̄) ⊗ ℚ_{l} is completely determined by the G_{𝔭}-action on cohomology. The case when 𝔭 | l is more difficult. A suitable weight decomposition can often be defined, but only after tensoring with B_{cris}^{ϕ}. This requires that L(π_{1}(XK̄, x̄), ℚ_{l}) be a potentially crystalline Galois representation, and the remainder of the paper is concerned with proving that this is the case.

∗The author was supported during this research by Trinity College, Cambridge and by the Isle of Man Department of Education.

In Section 2, the pro-unipotent crystalline fundamental group is defined, together with a cosimplicial algebra which can be thought of as the crystalline homotopy type. The cohomology groups of the crystalline homotopy type are the crystalline cohomology groups of the scheme, and the pro-unipotent fundamental group can be recovered from the algebra.

The purpose of Section 3 is to establish a comparison theorem between the crystalline homotopy type and the pro-p étale homotopy type defined in [?]. This just amounts to showing that the chain of isomorphisms of cohomology groups in [?] can be replaced by a chain of quasi-isomorphisms of cosimplicial algebras. As a consequence of this comparison, we can compare fundamental groups (Corollary 3.3) when X is of good reduction:

L(π_{1}^{cris}(X_{k}, x_{s}), K_{0}) ⊗_{K_{0}} B ≅ L(π_{1}^{ét}(XK̄, x̄_{η}), ℚ_{p}) ⊗_{ℚ_{p}} B.

so that L(π_{1}^{ét}(XK̄, x̄_{η}), ℚ_{p}) is indeed a crystalline Galois representation.

1 The pro-l-unipotent fundamental group

As in [?] Section ??, define N_{ℚ_{l}} to be the category of finite-dimensional nilpotent Lie algebras over ℚ_{l}, and N̂_{ℚ_{l}} to be the category of finitely generated (i.e. dim L/[L, L] < ∞) pro-nilpotent Lie algebras over ℚ_{l}.

Recall (ibid. Section ??) that, for a pro-finite group Γ, we define the ℚ_{l} Malcev Lie algebra L(Γ, ℚ_{l}) ∈ N̂_{ℚ_{l}} to pro-represent the functor on N_{ℚ_{l}} given by

𝔤 ↦ Hom_{cts}(Γ, exp(𝔤)).

Define the pro-ℚ_{l}-unipotent completion of Γ by

Γ ⊗ℤ̂ ℚ_{l} := exp(L(Γ, ℚ_{l})).

Given a ground field k, a connected scheme X_{0}/k and a point x ∈ X_{0}, let X = X_{0} ⊗_{k} k̄, and let x̄ ∈ X be the point above x. We wish to study the action of Gal(k̄/k) on the group

π_{1}(X, x̄) ⊗ ℚ_{l},

for l different from the characteristic of k. From ibid. Lemma ??, this fundamental group has tangent space H^{1}(X, ℚ_{l}) and obstruction space H^{2}(X, ℚ_{l}).

Definition 1.1. Given a smooth proper variety X̄ over k̄, with D a divisor locally of normal crossings, let X = X̄ − D, and fix a geometric point x̄ → X. Define the ideal K as the kernel of the map

L(π_{1}(X, x̄), ℚ_{l}) → L(π_{1}(X̄, x̄), ℚ_{l}).

Lemma 1.2. The definition of K is independent of the choice X̄ of compactification of X.

Proof. Consider the Lie algebra L(π_{1}(X, x̄), ℚ_{l})/K. This has tangent space Im (H^{1}(X̄, ℚ_{l}) → H^{1}(X, ℚ_{l})), and obstruction space H^{2}(X, ℚ_{l}). Now, as in [?] 3.2.4.1, and similarly to [?] Corollary VI 5.3, we have the Gysin exact sequence

H^{1}(X̄, ℚ_{l}) → H^{1}(X, ℚ_{l})) → H^{0}(D̃, ℚ_{l}(−1)),

where D̃ is the normalisation of D. If we adopt the same conventions for weights as in [?] (i.e. Frobenius in finite characteristic, and Hodge theory in characteristic zero) then we see that

Im (H^{1}(X̄, ℚ_{l}) → H^{1}(X, ℚ_{l})) = W_{1}H^{1}(X, ℚ_{l}),

the component of weight one. Thus for any two choices of X̄, our Lie algebra has the same tangent and obstruction spaces.

As in [?] 3.2.11 C, we my compare any two compactifications by taking morphisms to a third, and since these morphisms induce isomorphisms of the tangent and obstruction spaces of the corresponding Lie algebras, they must induce isomorphism on the Lie algebras themselves.

Definition 1.3. In the situation above, define the increasing weight filtration on L := L(π_{1}(X, x̄), ℚ_{l}) to be the filtration generated by the following conditions:

W_{−1}L = L, K ⊂ W_{−2}L, [W_{m}L, W_{n}L] ⊂ W_{m+n}L.

Most of this paper is concerned with finding splittings of this weight filtration, i.e. weight decompositions. Note that if X is smooth and proper then the weight filtration is just the commutator filtration W_{−n} = Γ_{n}.

Remark 1.4. There are times when X_{0}/k has no closed points, yet we may still define an outer action of Gal(k̄/k) on π_{1}(X). We may regard π_{1}(X) ⊗ ℚ_{l} as being the hull of the deformation functor 𝔓𝔅(X) defined in [?] Section ??. Alternatively, we may use tangential basepoints, as dealt with in ibid. Remark ??.

1.1 Finite characteristic

Let k = 𝔽_{q} be a finite field, with X variety over k̄, either proper or smooth. There is a natural Frobenius action F on π_{1}(X, x̄), and hence on

L := L(π_{1}(X, x̄), ℚ_{l}).

Let L_{n} = L/(Γ_{n+1}L), recalling that Γ_{n} is the nth commutator (i.e. Γ_{1}L = L, and Γ_{n+1}L = [L, Γ_{n}L]). Since L_{n} is finite dimensional, we may consider Jordan decompositions (over ℚ̄_{l}) given by F.

Lemma 1.5. On L_{n}, all of the eigenvalues of F are algebraic numbers, all of whose conjugates have norm q^{w/2}, for some negative integer w (dependent on α).

Proof. We prove this by induction on n. For n = 1, L_{1} ≅ H^{1}(X, ℚ_{l})^{∨}, so this statement is just [?] Corollaries (3.3.4)–(3.3.6).

Now, note that if v is in the generalised λ-eigenspace for F, and w in the generalised μ-eigenspace, then [v, w] is in the generalised λμ-eigenspace, since

(F − λμ)[v, w] = [(F − λ)v, F w] + [λv, (F − μ)w],

so (F − λμ)^{N}[v, w] = 0 for N ≫ 0.

Hence the statement of the Lemma holds for Γ_{n}(L)/Γ_{n+1}(L) (by multiplicativity), since it holds for L_{1}.

Assume the statement holds for L_{n−1}. Consider the exact sequence

0 → Γ_{n}(L)/Γ_{n+1}(L) → L_{n} → L_{n−1} → 0.

Since the statement holds for the terms on either side, it holds for L_{n}.

Corollary 1.6. There is a weight decomposition L_{n} = ⨁_{i} W_{i}(L_{n}) on the inverse system L_{n}.

Proof. We just decompose by the weights i of the eigenvalues of norm q^{i/2}.

Theorem 1.7. If X is smooth, satisfying the conditions of Definition 1.3, then this weight decomposition is a splitting of the weight filtration W.

Proof. It suffices to show that Gr_{i}^{W} L is pure of weight i (in the sense of Frobenius eigenvalues). Indeed, it is enough for Gr_{Γ}^{r}Gr_{i}^{W} L to be pure of weight i. But now observe that Gr_{Γ}^{r}Gr_{i}^{W} L is spanned by commutators consisting of i − r elements of K/K ∩ Γ_{2} and 2r − i elements of L/(Γ_{2} + K).

Now, L/(Γ_{2} + K) is dual to W_{1}H^{1}(X, ℚ_{l}), which is pure of weight 1, by Lemma 1.2, so L/(Γ_{2} + K) is pure of weight −1. Meanwhile, K/K ∩ Γ_{2} is dual to H^{1}(X, ℚ_{l})/W_{1}H^{1}(X, ℚ_{l}), which is pure of weight 2, so K/K ∩ Γ_{2} is pure of weight −2.

Therefore Gr_{Γ}^{r}Gr_{i}^{W} L is pure of weight i, as required.

Corollary 1.8. For X as above, there is a Frobenius equivariant isomorphism of inverse systems L_{n}(π_{1}(X, x̄), ℚ_{l}) ≅ Gr^{W} L_{n}(π_{1}(X, x̄), ℚ_{l}).

Theorem 1.9. 1. If H^{1}(X, ℚ_{l}) is pure of weight a > 0, then there is a Frobenius

equivariant map

L(H^{1}(X, ℚ_{l})^{∨}) ↠ L(π_{1}(X, x̄), ℚ_{l}),

inducing the identity on tangent spaces H^{1}(X, ℚ_{l}).

2. If moreover H^{2}(X, ℚ_{l}) is mixed of weights ≤ 2a, then there is a Frobenius equivariant isomorphism

L(H^{1}(X, ℚ_{l})^{∨})/(ˇ∪ H^{2}(X, ℚ_{l})^{∨})) ≅ L(π_{1}(X, x̄), ℚ_{l}),

where ˇ∪ is dual to the cup product.

Proof. 1. Since H^{1}(X, ℚ_{l}) is pure of weight a, it follows that Γ_{n}L = W_{−na}L. It then follows that the composition

W_{−a} ↪ L ↠ L/Γ_{2}(L) ≅ H^{1}(X, ℚ_{l})^{∨}

is a Frobenius equivariant isomorphism, so we may take its inverse to give a map H^{1}(X, ℚ_{l})^{∨} → L, extending to give

L(H^{1}(X, ℚ_{l})^{∨}) ↠ L(π_{1}(X, x̄), ℚ_{l}).

2. This is essentially the same as the proof of [?] Theorem ??. Let J be the kernel of the map above, and write H_{1} for H^{1}(X, ℚ_{l})^{∨}. It follows that (J/[H_{1}, J])^{∨} is a universal obstruction space for L, which gives us a unique morphism

(J/[H_{1}, J])^{∨} ↪ H^{2}(X, ℚ_{l})

of obstruction theories. These obstruction theories are both Frobenius equivariant, so this map must be. Now note that J ⊂ Γ_{2} L(H_{1}), so J is of weights ≤ −2a, so (J/[H_{1}, J])^{∨} is of weights ≥ 2a. By hypothesis, H^{2}(X, ℚ_{l}) is of weights ≤ 2a, so the inclusion above implies that (J/[H_{1}, J])^{∨} must be pure of weight 2a.

Now, since [H_{1}, J] is of weights ≤ 3a, the composition

W_{−2a} ∩ J ↪ J ↠ J/[H_{1}, J]

must be an isomorphism, so W_{−2a} ∩ J generates the ideal J, and since H^{1}(X, ℚ_{l}) is pure of weight a,

W_{−2a} = ⋀^{2}H^{1}(X, ℚ_{l})^{∨}.

The inclusion of obstruction spaces therefore corresponds to a Frobenius equivariant map

H^{2}(X, ℚ_{l})^{∨} ↠ W_{−2a} ∩ J = ⋀^{2}H^{1}(X, ℚ_{l})^{∨} ∩ J,

which, by comparing primary obstruction maps, must be dual to the cup product. Hence

J = (ˇ∪ H^{2}(X, ℚ_{l})^{∨}),

so

L(H^{1}(X, ℚ_{l})^{∨})/(ˇ∪ H^{2}(X, ℚ_{l})^{∨})) ≅ L(π_{1}(X, x̄), ℚ_{l}).

Remarks 1.10. 1. If X is smooth and proper, then H^{1}(X, ℚ_{l}) will be pure of weight 1, and H^{2}(X, ℚ_{l}) pure of weight 2, so both conditions will be satisfied.

2. If X = X̄ −D, for X̄ smooth, proper and simply connected, and D a divisor locally of normal crossings, then H^{1}(X, ℚ_{l}) will be pure of weight 2. Since H^{2}(X, ℚ_{l}) is automatically of weights ≤ 4, both conditions of the theorem are satisfied.

3. If X = X̄ − D, with the Gysin map H^{0}(D̃, ℚ_{l}(−1)) → H^{2}(X, ℚ_{l}) injective, then H^{1}(X, ℚ_{l}) will be pure of weight 1, so the first condition is satisfied.

1.2 Good reduction, l ≠ p

In characteristic zero, the easiest case to deal with is when working over a local field. Let V be a complete discrete valuation ring, with fraction field K, and finite residue field k of characteristic p. Denote by K̄ (resp. k̄) the algebraic closure of K (resp. k), and V̄ the integral closure of V in K̄. Write G = Gal(K̄/K). l will be a prime other than p.

Take a smooth proper scheme X̄_{V} over V, with D_{V} ⊂ X̄_{V} a divisor of simple normal crossings, and let X_{V} = X̄_{V} −D_{V}. Let x be a V -valued point of X_{V}, with corresponding points x_{η} ∈ X_{V} (K), x_{s} ∈ X_{V} (k), and the geometric points x̄_{η} ∈ X_{V} (K̄), x̄_{s} ∈ X_{V} (k̄).

We begin with a smooth specialisation theorem for pro-l-unipotent fundamental groups.

Theorem 1.11. There is a G-equivariant isomorphism

π_{1}(XK̄, x̄_{η}) ⊗ ℚ_{l} ≅ π_{1}(Xk̄, x̄_{s}) ⊗ ℚ_{l},

so that, in particular, the G action is unramified, factoring through

G → Gal(k̄/k).

Proof. Consider the maps

π_{1}(XK̄, x̄_{η}) ⊗ ℚ_{l} → π_{1}(XV̄ , x̄_{η}) ⊗ ℚ_{l}
π_{1}(Xk̄, x̄_{s}) ⊗ ℚ_{l} → π_{1}(XV̄ , x̄_{s}) ⊗ ℚ_{l}.

Comparing tangent and obstruction spaces over N_{ℚ_{l}}, we have:

H^{i}(XK̄, ℚ_{l}) ← H^{i}(XV̄ , ℚ_{l})
H^{i}(Xk̄, ℚ_{l}) ← H^{i}(XV̄ , ℚ_{l}),

for i = 1, 2. Since V̄ is strictly Henselian, we may apply the smooth specialisation theorem for geometric fibrations ([?] Propositions 2.3 and 2.5) to see that these are isomorphisms. Since étale maps of pro-representables on N_{ℚ_{l}} must be isomorphisms, it follows that we have isomorphisms on pro-l-unipotent fundamental groups. Note that these isomorphisms are G-equivariant.

It only remains to observe that the embedding V̄ → K̄ provides us with a canonical (G-equivariant) isomorphism

π_{1}(XV̄ , x̄_{s}) → π_{1}(XV̄ , x̄_{η}).

Corollary 1.12. There is a G-invariant weight decomposition on L_{n}(π_{1}(XK̄, x̄_{η}), ℚ_{l}), splitting the weight filtration W. Moreover,

1. If H^{1}(XK̄, ℚ_{l}) is pure of weight a > 0, then there is a Frobenius equivariant map

L(H^{1}(XK̄, ℚ_{l})^{∨}) ↠ L(π_{1}(XK̄, x̄_{η}), ℚ_{l}),

inducing the identity on tangent spaces H^{1}(XK̄, ℚ_{l}).

2. If in addition H^{2}(XK̄, ℚ_{l}) is mixed of weights ≤ 2a, then there is a Frobenius equivariant isomorphism

L(H^{1}(XK̄, ℚ_{l})^{∨})/(ˇ∪ H^{2}(XK̄, ℚ_{l})^{∨})) ≅ L(π_{1}(XK̄, x̄_{η}), ℚ_{l}),

where ˇ∪ is dual to the cup product.

Proof. This follows from Theorems 1.7 and 1.9.

Remark 1.13. As in [?] Remark ??, we see that, if X is smooth and proper, then the ℚ_{l}-homotopy type is formal and the quasi-isomorphism is Galois equivariant.

1.3 Potentially good reduction, l ≠ p

Let K be as before, and K′/K a finite Galois extension, with ring of integers V′. Fix a smooth scheme X/K, with

X = X̄ − D,

such that X̄ is smooth and proper, and D a divisor of simple normal crossings, and fix a point x ∈ X(K).

Theorem 1.14. If there exists some Y/V′ smooth and proper, with Z ⊂ Y a divisor of simple normal crossings, such that Y_{K′} = X̄_{K′} and Z_{K′} = D_{K′}, then there is a Ginvariant weight decomposition on L_{n}(π_{1}(XK̄, x̄), ℚ_{l}), splitting the weight filtration W. Moreover,

1. If H^{1}(XK̄, ℚ_{l}) is pure of weight a > 0, then there is a G-equivariant map

L(H^{1}(XK̄, ℚ_{l})^{∨}) ↠ L(π_{1}(XK̄, x̄), ℚ_{l}),

inducing the identity on tangent spaces H^{1}(XK̄, ℚ_{l}).

2. If in addition H^{2}(XK̄, ℚ_{l}) is of weight ≤ 2a, then there is a G-equivariant isomor-

phism

L(H^{1}(XK̄, ℚ_{l})^{∨})/(ˇ∪ H^{2}(XK̄, ℚ_{l})^{∨})) ≅ L(π_{1}(XK̄, x̄), ℚ_{l}),

where ˇ∪ is dual to the cup product.

Proof. Observe that if we replace G by G′ = Gal(K̄/K′), then this is just Theorem 1.2. Take the G′-invariant weight decomposition W_{i} from that theorem. It will suffice to show that this decomposition is in fact G-invariant.

Fix g ∈ G, and note that gW_{∗} is the decomposition due to the Frobenius operator gF̃g^{−1}. Since W_{i} is independent of the choice of Frobenius, if follows that gW_{i} = W_{i}. The remainder of the proof is now identical to Theorem 1.2.

1.4 Potentially good reduction, l = p

Having established the structure of π_{1}(XK̄, x̄_{η}) ⊗ ℚ_{l}, it is natural to wish to understand π_{1}(XK̄, x̄_{η}) ⊗ ℚ_{p}. By passing to a global field, and then localising at a different prime, we may describe the structure, but not the Galois action. If we could describe the Galois action on π_{1}(XK̄, x̄_{η}) ⊗ ℚ_{p}, then we would also be able to describe the action on its tangent space, H^{1}(XK̄, ℚ_{p}), but this requires p-adic Hodge theory.

Let K be as before, and K′/K a finite Galois extension, with residue field k′ = 𝔽_{q}, of characteristic p. Let W′ be the ring of Witt vectors W = W (k′), with fraction field K′_{0}. Let G = Gal(K̄/K), and G′ = Gal(K̄/K′). Fix a smooth scheme X/K, with

X = X̄ − D,

such that X̄ is smooth and proper and D a divisor locally of normal crossings, and fix a point x ∈ X(K). Assume that X̄ is of good reduction over K′.

We wish to study the Galois structure of L := L(π_{1}(XK̄, x̄), ℚ_{p}). We can no longer split the weight filtration here, since there is no appropriate Frobenius. Instead we have to use p-adic periods.

Consider the K′_{0}-Lie algebra

D′_{cris}(L) := \underleftarrow{lim} D′_{cris}(L_{n}) = \underleftarrow{lim}(L_{n} ⊗_{ℚ_{p}} B_{cris})^{G′},

where B_{cris} is defined as in [?].

This will automatically have a σ-linear Frobenius operator ϕ, where σ is the Frobenius operator on K′_{0}. It also has a Hodge filtration Fil^{∗} on D′_{cris}(L) ⊗_{K′_{0}} K′, and an action of G, factoring through the finite group G/G′. These structures all respect the Lie bracket, while ϕ preserves the Hodge filtration.

The weight filtration given in Definition 1.3 extends naturally to give a weight filtration on D′_{cris}(L). Our aim is to show that this weight filtration can be split, using a weight decomposition as defined in Section 1.2.

Assume that L is an inverse system of crystalline G′-modules. Section 3 gives conditions for this hypothesis to hold. Having made this assumption, we are able to proceed as before.

Now, observe that

Gr_{i}^{W} Gr_{Γ}^{1}D′_{cris}(L) = Gr_{i}^{W} D′_{cris}((H^{1}(XK̄, ℚ_{p}))^{∨}),

and recall that

H^{1}(X̄K̄, ℚ_{p}) ↠ Gr_{1}^{W} H^{1}(XK̄, ℚ_{p}),
Gr_{2}^{W} H^{1}(XK̄, ℚ_{p}) ↪ H^{0}(D̃K̄, ℚ_{p}(−1)),

with all other graded weights being zero.

It follows automatically that D′_{cris}(H^{0}(D̃K̄, ℚ_{p}(−1))) is a direct sum of vector spaces K′_{0}[−1], on which ϕ = pσ, and G acts as on K′_{0}, the Hodge filtration being concentrated in degree one. From the crystalline conjecture (proved in [?] or [?]), it also follows that

D′_{cr}(H^{1}(X̄K̄, ℚ_{p})) ≅ H_{cris}^{1}(X̄_{k′}/W′) ⊗ K′_{0}.

Next, consider the action of Frobenius on D′_{cris}(L). Although ϕ is only σ-linear, we will have ϕ^{r} linear, where q = p^{r}. We may therefore consider the eigenvalues of Frobenius. It is clear that this action is pure of weight −2 on Gr_{−2}^{W}Gr_{Γ}^{1}D′_{cris}(L).

From [?], for any smooth projective curve Y/k′, the Frobenius action on

H_{cris}^{1}(Y/W′) ⊗ K′_{0}

is pure of weight 1. Therefore, for any smooth projective curve Y/K′ of good reduction, we see that the Frobenius action on

D_{cris}(H_{ét}^{1}(YK̄, ℚ_{p}))

is pure of weight 1.

By considering the Albanese variety, we see that this must hold for any smooth proper variety Y/K′ of good reduction, so in particular holds for X̄. Now,

Gr_{−1}^{W}Gr_{Γ}^{1}D′_{cris}(L) ↪ D′_{cris}((H_{ét}^{1}(X̄K̄, ℚ_{p}))^{∨}),

so the Frobenius action on

Gr_{−1}^{W}Gr_{Γ}^{1}D′_{cris}(L)

is pure of weight −1.

Lemma 1.15. There is a (G, ϕ)-invariant weight decomposition W_{∗} on D′_{cris}(L_{n}), splitting the weight filtration W.

Proof. The proof follows almost exactly as for Theorem 1.2, one minor difference being that the Frobenius decomposition is defined by ϕ^{r}, not ϕ. Since ϕ commutes with ϕ^{r}, it preserves the weight decomposition. The action of G commutes with that of ϕ^{r} (though not ϕ), so preserves W.

Remark 1.16. In general, W_{∗} is incompatible with the Hodge filtration Fil^{∗}, as the latter is not preserved by ϕ.

Theorem 1.17. If the pair (X̄_{K′}, D_{K′}) is of good reduction for some finite Galois extension K′/K, and {L_{n}} is a system of potentially crystalline Galois representations, then there is a G-invariant weight decomposition on L_{n}(π_{1}(XK̄, x̄), ℚ_{p}) ⊗_{ℚ_{p}} B_{cris}^{ϕ}, splitting the weight filtration W, where B_{cris}^{ϕ} is the ϕ-invariant subring of B_{cris}. Moreover,

1. If D′_{cris}H^{1}(XK̄, ℚ_{p}) is pure of weight a > 0, then there is a canonical G-equivariant

map

L(H^{1}(XK̄, ℚ_{p})^{∨}) ⊗_{ℚ_{p}} B_{cris}^{ϕ} ↠ L(π_{1}(XK̄, x̄), ℚ_{p}) ⊗_{ℚ_{p}} B_{cris}^{ϕ},

inducing the identity on tangent spaces H^{1}(XK̄, ℚ_{p}) ⊗_{ℚ_{p}} B_{cris}^{ϕ}.

2. If in addition D′_{cris}H^{2}(XK̄, ℚ_{p}) is of weight ≤ 2a, then there is G-equivariant isomorphism

B_{cris}^{ϕ} ⊗_{ℚ_{p}} L(H^{1}(XK̄, ℚ_{p})^{∨})/(ˇ∪ H^{2}(XK̄, ℚ_{p})^{∨})) ≅ B_{cris}^{ϕ} ⊗_{ℚ_{p}} L(π_{1}(XK̄, x̄), ℚ_{p}),

where ˇ∪ is dual to the cup product.

Proof. This is very similar to Theorem 1.2. The weight decomposition is just (W_{∗} ⊗_{K′_{0}} B_{cris})^{ϕ}.

Letting L := L(π_{1}(XK̄, x̄), ℚ_{p}) and L^{ab} := L/[L, L], the first condition amounts to saying that D′_{cris}(L^{ab}) is pure of weight −a. This gives a unique ϕ-equivariant section of D′_{cris}(L) → D′_{cris}(L^{ab}), and hence a canonical (G, ϕ)-equivariant section of

L ⊗_{ℚ_{p}} B_{cris} → L^{ab} ⊗_{ℚ_{p}} B_{cris},

which proves the first part.

Now write V for D′_{cris}(H^{1}(Xk̄, ℚ_{l}))^{∨}, and let J be the kernel of the map L(V) → D′_{cris}(L) defined above. Considering the extension

e : (L(V)/[V, J]) ⊗_{K′_{0}} B_{cris} ↠ L ⊗_{ℚ_{p}} B_{cris},

it follows that (J/[V, J])^{∨} ⊗_{K′_{0}} B_{cris} ≅ H^{2}(L, ℚ_{l}) ⊗_{ℚ_{p}} B_{cris} is the universal obstruction space for L ⊗_{ℚ_{p}} B_{cris} over B_{cris}, which gives us a unique (G, ϕ)-equivariant morphism

(J/[V, J])^{∨} ⊗_{K′_{0}} B_{cris} ↪ H^{2}(Xk̄, ℚ_{l}) ⊗_{ℚ_{p}} B_{cris}

of obstruction theories. Taking G′-invariants gives the ϕ-equivariant map

(J/[V, J])^{∨} ↪ D′_{cris}(H^{2}(Xk̄, ℚ_{l})).

The remainder of the proof is as for Theorem 1.9.

Remarks 1.18. In Corollary 3.4 it will be shown that, when D is a divisor of simple normal crossings, L is a potentially crystalline representation. This then gives more information about the Galois action on L(π_{1}^{ét}(XK̄, x̄_{η}), ℚ_{p}) than Olsson’s later results ([?] Theorem 1.13) obtained independently, which show that the Galois action on L ⊗ ⊗_{ℚ_{p}}B_{cris} is determined by the action on H^{∗}(X, ℚ_{p}) ⊗_{ℚ_{p}} B_{cris} when X is proper, without giving an explicit description.

In particular, note that all conditions of the theorem are satisfied by X = ℙ^{1} − {0, 1, ∞}, as studied in [?] and [?], since its cohomology is pure.

1.5 Global fields

We are now in a position to consider global fields. Let K/ℚ be a number field. Let

X = X̄ − D,

where X̄/K is a smooth proper variety, and D is a divisor locally of normal crossings. Let x ∈ X(k). For each prime l, we wish to consider the action of G = Gal(K̄/K) on the pro-l-unipotent fundamental group

π_{1}(XK̄, x̄) ⊗ ℚ_{l}.

As in [?], we have a weight decomposition for each prime:

Now assume that the divisor D ⊂ X̄ is of simple normal crossings.

Theorem 1.19. For each prime 𝔭 ∤ l of K̄ at which (X̄, D) has potentially good reduction, there is a weight decomposition _{𝔭}W_{∗}(L) of L := L(π_{1}(XK̄, x̄), ℚ_{l}), splitting the weight filtration of Definition 1.3. These decompositions are conjugate under the action of G, i.e.

g(_{𝔭}W_{∗}(L)) = _{g𝔭}W_{∗}(L).

For each prime 𝔭 | l at which (X̄, D) has potentially good reduction and L is a potentially crystalline G_{𝔭} representation, there is a weight decomposition _{𝔭}W_{∗}(L ⊗_{ℚ_{p}} B_{cris}^{ϕ}) of L ⊗_{ℚ_{p}} B_{cris}^{ϕ}, splitting the weight filtration. These decompositions are conjugate.

Proof. We prove this for 𝔭 ∤ l; the case 𝔭 | l is similar. Localisation at 𝔭 gives a G_{𝔭}-equivariant isomorphism

π_{1}(XK̄, x̄) ⊗ ℚ_{l} ⟵^{θ_{𝔭}} π_{1}(XK̄_{𝔭}, x̄_{𝔭}) ⊗ ℚ_{l},

preserving the weight filtration. Since (X̄, D) is of potentially good reduction at 𝔭, by Theorem 1.14 (or Theorem 1.17 when 𝔭 | l) there is a G_{𝔭}-invariant splitting W_{i} of the weight filtration on the Lie algebra associated to the latter fundamental group. Set

𝔭W_{i}(L) := θ_{𝔭}(W_{i}(L(π_{1}(XK̄_{𝔭}, x̄_{𝔭}), ℚ_{l}))).

To see that these are conjugate, consider the commutative diagram

π_{1}(XK̄, x̄) ⊗ ℚ_{l} ⟵^{θ_{𝔭}} π_{1}(XK̄_{𝔭}, x̄_{𝔭}) ⊗ ℚ_{l}
g↓  ^{g}↓
π_{1}(XK̄, x̄) ⊗ ℚ_{l} ⟵^{θ_{g𝔭}} π_{1}(XK̄_{g𝔭}, x̄_{g𝔭}) ⊗ ℚ_{l}.

By uniqueness of the weight decomposition over local fields, we must have g(W_{i}) = W_{i} on the right, so

g(_{𝔭}W_{∗}(L)) = _{g𝔭}W_{∗}(L).

2 The crystalline fundamental group

In this section, we set about defining a pro-unipotent crystalline fundamental group, analogous to the pro-unipotent étale fundamental group defined in [?].

2.1 Smooth proper schemes

Given a perfect field k, let W be the ring of Witt vectors W = W (k), with DP-structure γ and fraction field K. We wish to define, for any connected scheme X/k, and any point x ∈ X(k), a pro-nilpotent crystalline fundamental group

π_{1}^{cris}(X, x) ⊗ K.

Definition 2.1. Given a smooth group scheme H/W, let H_{n} = H ⊗_{W} W_{n}. Define a constructible H-torsor 𝔻 on X to be an inverse system of principal H-torsors 𝔻_{n} on the crystalline sites X/W_{n}, such that

𝔻_{n} = i_{n}^{−1}𝔻_{n+1},

for i_{n} : (X/W_{n})_{cris} → (X/W_{n+1})_{cris}, so we may regard 𝔻 as

𝔻 = \underleftarrow{lim} 𝔻_{n}.

The principal H-torsor is then given by the sheaf

(U ↪ T) ↦ H(T),

for U ⊂ X, and U → T a DP-thickening.

Definition 2.2. Given a smooth group scheme G/K, define a constructible G-torsor 𝔹 to be

𝔹 = 𝔻 ⊗_{W} K,

for some group scheme H/W with G = H ⊗_{W} K, and some constructible H-torsor 𝔻. To make this explicit, a constructible G-torsor is a pair (H, 𝔻), with H/W as above, and if we have a map H_{1} → H_{2} of smooth group schemes over W, becoming the identity map on G when tensored with K, then we regard the H_{2}-torsor

T ↦ 𝔻_{1}(T) ×^{H_{1}(T)} H_{2}(T)

as being equivalent to the H_{1}-torsor (H_{1}, 𝔻_{1}).

We are now in a position to define a functor on N_{K} which π_{1}^{cris}(X, x) ⊗ K should pro-represent. Given any 𝔤 ∈ N_{K}, observe that exp(𝔤) is represented by a smooth affine group scheme over K. Explicitly, we may construct this group scheme by considering the polynomial algebra over K generated by 𝔤^{∨}, with comultiplication coming from the Campbell-Baker-Hausdorff formula, which will converge, since 𝔤 is nilpotent.

Definition 2.3. Define the functor

𝔗(X) : N_{K} → Grpd

to associate to 𝔤 the groupoid of constructible exp(𝔤)-torsors on X.

Define the groupoid

𝔗_{x}(X)(𝔤)

to be the fibre of

𝔗(X)(𝔤) → 𝔗(x)(𝔤)

over (exp(𝔤 ⊗ 𝒪_{X/W}), id).

The crucial difference between the groupoids 𝔗_{x}(X)(𝔤) and 𝔗(X)(𝔤) is that only those morphisms which give the identity on 𝔹_{x} are permitted in the former.

Definition 2.4. Define

L(π_{1}^{cris}(X, x), K)

to pro-represent the functor 𝔗_{x}(X). Note that this functor is clearly homogeneous, and has trivial automorphisms, so will be pro-representable provided the relevant tangent space (which will be H_{cris}^{1}(X/W) ⊗ K) is finite dimensional. Define

π_{1}^{cris}(X, x) ⊗ K := exp(L(π_{1}^{cris}(X, x), K)).

Remark 2.5. Since exponentiation gives an equivalence between N_{K} and the category of unipotent algebraic K-groups, Tannakian duality implies that linear representations of π_{1}^{cris}(X, x)⊗K correspond to nilpotent isocrystals on X. This means that our definition of π_{1}^{cris} agrees with [?] Definition 4.1.5.

Definition 2.6. Let

𝔉ℌ𝔖(X) : N_{ℚ_{l}} → Cat

associate to 𝔤 the category of faithful constructible homogeneous exp(𝔤)-sheaves on X. Observe that 𝔗(X) is the fibre of 𝔉ℌ𝔖(X) over the trivial exp(0)-sheaf 1 (the constant sheaf of singleton sets).

In order to find a suitable SDC to govern this problem, we will follow [?] 5.28 and 5.26. Assume that we have a smooth formal scheme 𝔛/W, and a closed immersion i : X → 𝔛.

Take an affine cover 𝔘_{α} of 𝔛, let

U_{α} = 𝔘_{α} ⊗ k, X′ = ∐_{α} U_{α}

and, for each n, form the W_{n}-PD envelope

U_{α} ↪ U_{α}^{DP,n}

associated to the embedding

U_{α} ↪ 𝔘_{α} ⊗ W_{n},

and let

X′^{DP,n} = ∐_{α} U_{α}^{DP,n}.

Similarly, form the W_{n}-PD envelope

X ↪ X^{DP,n},

associated to the embedding X ↪ 𝔛 ⊗ W_{n}.

Consider the maps (as in [?] 5.26):

[figure: X′_{cris}|_{X′}_{DP,n}; ϕ; j; v; X′_{Zar}^{DP,n}; v; X′_{cris}; u_{X′/Wn}; v; X′_{Zar}; v; X_{cris}|_{X^{DP,n}}; ϕ; j; X_{Zar}^{DP,n}; X_{cris}; u_{X/Wn}; X_{Zar},]

Now, adopting suitable p-adic conventions, we have a functorial comonadic adjunction

\xymatrix{𝔉ℌ𝔖(X_{cris}) \ar[r]^{w^{−1}} & 𝔉ℌ𝔖(X′_{cris}|_{X′}_{DP}), \ar[l]^{w_{∗}}|{⊥}}

where w = uj. The adjunction j^{−1} ⊣ j_{∗} is comonadic, since X_{cris}|_{X^{DP}} ⟶^{j} X_{cris} is a covering (as in [?] 5.28). The adjunction v^{−1} ⊣ v_{∗} is also comonadic, since the maps X′ → X are coverings. Note that w = vj = jv, and that the maps v^{−1}, v_{∗}, j^{−1}, j_{∗} all commute with each other (whenever the composition can be defined), since v^{−1} and j^{−1} always commute with everything, and direct images commute with each other.

Next, observe that the maps

\xymatrix{𝔉ℌ𝔖(X′_{cris}|_{X′}_{DP}) \ar[r]^{φ_{∗}} & 𝔉ℌ𝔖(X′_{Zar}) \ar[l]^{φ^{∗}}}

give an equivalence of categories. The second category above has uniformly trivial deformation theory, as exp(𝔤) is smooth, and torsors on an affine scheme are trivial.

Hence deformations are described by the SDC

S_{X}^{n}(𝔤) = Hom_{𝔉ℌ𝔖(X′_{cris}|_{_{X′}DP})(𝔤)}(w^{−1} exp(𝔤 ⊗ 𝒪_{X/W}), (w^{−1}w_{∗})^{n}w^{−1} exp(𝔤 ⊗ 𝒪_{X/W})).

Now, write

C^{n}(X, ℱ) := Γ(X′_{cris}|_{X′}^{DP}, (w^{−1}w_{∗})^{n}w^{−1}ℱ),

for sheaves ℱ on (X/W)_{cris}, and

𝒞^{n}(ℱ) := (w_{∗}w^{−1})^{n+1}ℱ ,

so that C^{n}(X, ℱ) = Γ(X, 𝒞^{n}(ℱ)).

We then have isomorphisms

S^{n}(𝔤) ≅ Γ(X′_{cris}|_{X′}_{DP}, (w^{−1}w_{∗})^{n}w^{−1} exp(𝔤 ⊗ 𝒪_{X/W})) ≅ C^{n}(X, exp(𝔤 ⊗ 𝒪_{X/W})),
(b ↦ g · b) ↤ g,

the latter being a cosimplicial complex of groups, becoming an SDC via the Alexander-Whitney cup product. Finally, note that

C^{n}(X, exp(𝔤 ⊗ 𝒪_{X/W})) ≅ exp(C^{n}(X, 𝒪_{X/W}) ⊗_{W} 𝔤),

so following [?] Remark ??, we regard

C^{n}(X, 𝒪_{X/W}) ⊗_{W} K

as being the crystalline homotopy type of X. Remark 2.7. This is essentially the same as the crystalline homotopy type defined in [?], which is taken as the Thom-Whitney algebra of the cosimplicial algebra C_{Zar}^{•}(X, Ω_{X/W}^{•}) ⊗_{W} K. If D denotes the denormalisation functor from DG algebras to cosimplicial algebras, we have quasi-isomorphisms

C^{•}(X, 𝒪_{X/W}) ⊗_{W} K → C^{•}(X, DΩ_{X/W}^{•}) ⊗_{W} K ← C_{Zar}^{•}(X, DΩ_{X/W}^{•}) ⊗_{W} K,

giving the required equivalence. Since the homotopy categories of cosimplicial and DG algebras are equivalent, for convenience we work exclusively with cosimplicial algebras.

Lemma 2.8. If H_{cris}^{i}(X/W) and H_{cris}^{i−1}(X/W) are finitely generated, then the cohomology groups of S_{X}^{•} are isomorphic to

H_{cris}^{∗}(X/W) ⊗_{W} K.

Proof. As for [?] Lemma ??.

Theorem 2.9. Assume that x ∈ X(k) lifts to x̃ ∈ 𝔛(W). This enables us to define the morphism

S_{X}^{•} ⟶^{i_{x}^{∗}} S_{x}^{•}

of SDCs. Define the SDC

S_{X,x}^{n} ⊂ S_{X}^{n}

to be the fibre of i_{x}^{∗} over the unique 0-valued point of S_{x}^{n}, then

𝔇𝔢𝔣_{S_{X,x}}

is equivalent to the discrete groupoid pro-represented by L(π_{1}^{cris}(X, x), K).

The cohomology of S_{X,x}^{•} is

H_{cris}^{i}((X, x)/W) ⊗ K ≅ \begin{cases}0 & i = 0 \\ H_{cris}^{i}(X/W) ⊗ K & i ≠ 0, \end{cases}

provided that the H_{cris}^{i}(X/W) are finitely generated.

Observe that, since these constructions were functorial, the absolute Frobenius endomorphism F on X gives rise to a Frobenius endomorphism on S_{X}^{•}. If k is the finite field 𝔽_{q}, with q = p^{r}, then F^{r} is a morphism of W -schemes, and we have the commutative diagram

Spec k ⟶^{x} X
↓  ↓^{F^{r}}
Spec k ⟶^{x} X.

This then provides a Frobenius endomorphism of

S_{X,x}^{n},

which will agree with the corresponding Frobenius endomorphism on

π_{1}^{cris}(X, x) ⊗ K.

Assume that all the eigenvalues of Frobenius F^{r} acting on the cohomology group H_{cris}^{1}(X/W) ⊗ K are algebraic numbers α, and for each α, there exists a weight n, such that all complex conjugates of α have norm q^{n/2}. This provides us with a weight decomposition

H_{cris}^{1}(X/W) ⊗ K = ⨁_{n} W_{n}H_{cris}^{1}(X/W) ⊗ K.

In particular, this will be true if X is smooth and projective (by [?]). This will enable the definition of a weight decomposition on L(π_{1}^{cris}(X, x), K).

Theorem 2.10. Under the above hypotheses on eigenvalues of Frobenius, there is an isomorphism

L(π_{1}^{cris}(X, x), K) ≅ L(H_{cris}^{1}(X/W) ⊗ K^{∨})/(f(H_{cris}^{2}(X/W) ⊗ K^{∨})),

where L(V) is the free pro-nilpotent Lie algebra (as in [?] ??), and

f : H_{cris}^{2}(X/W) ⊗ K^{∨} → Γ_{2}L(H_{cris}^{1}(X/W) ⊗ K^{∨})

preserves the (Frobenius) weight decompositions. The isomorphism also preserves the Frobenius decompositions of the finite quotients of L(π_{1}^{cris}(X, x), K).

Moreover, the quotient map

f : H_{cris}^{2}(X/W) ⊗ K^{∨} → Γ_{2}/Γ_{3} ≅ ⋀^{2}(H_{cris}^{1}(X/W) ⊗ K^{∨})

is dual to half the cup product

H_{cris}^{1}(X/W) ⊗ K × H_{cris}^{1}(X/W) ⊗ K ⟶^{\frac{1}{2} ∪} H_{cris}^{2}(X/W) ⊗ K.

Proof. As for [?] Theorem ??.

Corollary 2.11. If X is smooth and projective, then

L(π_{1}^{cris}(X, x), K)

is quadratically presented.

In fact,

L(π_{1}^{cris}(X, x), K) ≅ L(H_{cris}^{1}(X/W) ⊗ K^{∨})/(ˇ∪ H_{cris}^{2}(X/W) ⊗ K^{∨})),

where ˇ∪ is dual to the cup product. This isomorphism is, moreover, Frobenius equivariant.

Proof. This follows as for Theorem 1.9, using the purity results of [?].

Remark 2.12. Of course, in this case the Frobenius decomposition provides a decomposition on the weight filtration (which in this case is just the commutator filtration). The same result will hold for any smooth proper scheme whose Frobenius action on H_{cris}^{1} is pure of weight i.

2.2 Smooth non-proper schemes

There is a similar approach for the log-crystalline fundamental group. Adopting the conventions of [?], let X^{o} = X − D, for X/k a smooth proper scheme, and D a divisor locally of normal crossings. We wish to define, for x ∈ X^{o}(k), a pro-nilpotent logcrystalline fundamental group

π_{1}^{cris}(X^{o}, x) ⊗ K.

We define constructible torsors as before, replacing the crystalline site by the logcrystalline site throughout.

Definition 2.13. Define the functor

𝔗(X^{o}) : exp(𝔤) → Grpd

to associate to 𝔤 the groupoid of constructible exp(𝔤)-torsors on the log-crystalline site of X^{o}.

Define the groupoid

𝔗_{x}(X^{o})(𝔤)

to be the fibre of

𝔗(X^{o})(𝔤) → 𝔗(x)(𝔤)

over (exp(𝔤), id).

Definition 2.14. Define

L(π_{1}^{cris}(X^{o}, x), K)

to pro-represent the functor 𝔗_{x}(X^{o}).

We now wish to find a suitable SDC to govern this problem. Assume that we have a smooth formal scheme 𝔛/W, with a smooth formal divisor 𝔇/W and closed immersions i : X → 𝔛, D → 𝔇.

Take an affine cover 𝔘_{α} of 𝔛, and, as before, form the schemes

X^{DP,n},  D^{DP,n},  U_{α}^{DP,n} and X′^{DP,n}.

In the notation of the previous section, We now have the functorial comonadic adjunction

\xymatrix{𝔉ℌ𝔖(X_{cris}^{o}) \ar[r]^{w^{−1}} & 𝔉ℌ𝔖((X^{o})′_{cris}|_{(X^{o})′}_{DP}), \ar[l]^{w_{∗}}|{⊥}}

the second category having uniformly trivial deformation theory.

Hence deformations are described by the SDC

S_{X^{o}}^{n}(𝔤) = Hom_{𝔉ℌ𝔖((X^{o})′_{cris}|_{_{(X^{o})′}DP})(𝔤)}(w^{−1} exp(𝔤⊗𝒪_{X^{o}/W}), (w^{−1}w_{∗})^{n}w^{−1} exp(𝔤⊗𝒪_{X^{o}/W})).

Now, write

C^{n}(X^{o}, ℱ) := Γ((X^{o})′_{cris}|_{(X^{o})′}_{DP}, (w^{−1}w_{∗})^{n}w^{−1}ℱ),

for sheaves ℱ on (X^{o}/W)_{cris}, and

𝒞^{n}(ℱ) := (w_{∗}w^{−1})^{n+1}ℱ ,

so that C^{n}(X^{o}, ℱ) = Γ(X^{o}, 𝒞^{n}(ℱ)).

We then have isomorphisms

S^{n}(𝔤) ≅ Γ((X^{o})′_{cris}|_{(X^{o})′}_{DP}, (w^{−1}w_{∗})^{n}w^{−1} exp(𝔤 ⊗ 𝒪_{X^{o}/W})),

and

S^{n}(𝔤) ≅ C^{n}(X^{o}, exp(𝔤 ⊗ 𝒪_{X^{o}/W})),
(b ↦ g · b) ↤ g,

the latter being a cosimplicial complex of groups, becoming an SDC via the Alexander-Whitney cup product. Since

C^{n}(X^{o}, exp(𝔤 ⊗ 𝒪_{X^{o}/W})) ≅ exp(C^{n}(X^{o}, 𝒪_{X^{o}/W}) ⊗_{W} 𝔤),

we regard

C^{n}(X^{o}, 𝒪_{X/W}) ⊗_{W} K

as being the log-crystalline homotopy type of X^{o}.

Lemma 2.15. If H_{cris}^{i}(X^{o}/W) and H_{cris}^{i−1}(X^{o}/W) are finitely generated, then the cohomology groups of S_{X^{o}}^{•} are isomorphic to

H_{cris}^{∗}(X^{o}/W) ⊗_{W} K.

Proof. As for [?] Lemma ??.

Theorem 2.16. Assume that x ∈ X^{o}(k) lifts to x̃ ∈ 𝔛(W) − 𝔇(W). This enables us to define the morphism

S_{X^{o}}^{•} ⟶^{i_{x}^{∗}} S_{x}^{•}

of SDCs. Define the SDC

S_{X^{o},x}^{n} ⊂ S_{X^{o}}^{n}

to be the fibre of i_{x}^{∗} over the unique 0-valued point of S_{x}^{n}, then

𝔇𝔢𝔣_{S_{X^{o},x}}

is equivalent to the discrete groupoid pro-represented by L(π_{1}^{cris}(X^{o}, x), K). The cohomology of S_{X^{o},x}^{•} is

H_{cris}^{i}((X^{o}, x)/W) ⊗ K ≅ \begin{cases}0 & i = 0 \\ H_{cris}^{i}(X^{o}/W) ⊗ K & i ≠ 0, \end{cases}

provided that the H_{cris}^{i}(X^{o}/W) are finitely generated.

Once again, if k is finite, 𝔽_{q} say, then there is a Frobenius endomorphism of

S_{X^{o},x}^{n},

which will agree with the corresponding Frobenius endomorphism on

π_{1}^{cris}(X^{o}, x) ⊗ K.

From now on, assume that 𝔛 is the formal completion of a smooth, proper scheme X_{W} /W, with 𝔇 the formal completion of a divisor D_{W} of simple normal crossings relative to W. Assume moreover that the Frobenius action on

H_{cris}^{1}(X/W) ⊗ K

is pure of weight 1, noting in particular that this will hold if X is projective (by [?]).

If we now consider the Gysin sequence (as in Lemma 1.2):

H^{1}(XK̄, ℚ_{p}) → H^{1}(Xo ; K̄, ℚ_{p})) → H^{0}(D̃K̄, ℚ_{p}(−1)),

then using [?] Theorem 5.6 (the crystalline conjecture), we see that the following sequence is exact:

H_{cris}^{1}(X/W) ⊗ K → H_{cris}^{1}(X^{o}/W) ⊗ K → H_{cris}^{0}(D̃/W) ⊗ K[−1],

so that we have a weight decomposition (due to Frobenius) on H_{cris}^{1}(X^{o}/W)⊗K, splitting the weight filtration.

Theorem 2.17. The Frobenius action on L(π_{1}^{cris}(X^{o}, x), K) provides a splitting of weight filtration. Moreover, if H_{cris}^{1}(X^{o}/W) ⊗ K is pure of weight a > 0, then there is a Frobenius equivariant map

L((H_{cris}^{1}(X^{o}/W) ⊗ K)^{∨}) ↠ L(π_{1}^{cris}(X^{o}, x), K),

inducing the identity on tangent spaces H_{cris}^{1}(X^{o}/W) ⊗ K.

Proof. As for Theorems 1.7 and 1.9.

2.3 The Hodge filtration

Considering the commutative diagram on page 13, and given a crystal ℱ on (X/W)_{cris}, define the sheaf

𝒞_{Hdg}^{n}(ℱ) := (j^{−1}j_{∗})^{n+1}ℱ .

These give rise to a natural cosimplicial complex of sheaves, which is a resolution of the crystal ℱ, i.e. is exact for the functor u_{X^{o}/W ∗}. When ℱ is a group-valued sheaf, we can define a Hodge filtration on this cosimplicial complex, by

Fil^{i}𝒞_{Hdg}^{n}(ℱ) = \begin{cases}⋂_{j_{1},...j_{n+1−i}} ker(σ^{j_{1}} ⋯ σ^{j_{n+1−i}} : 𝒞_{Hdg}^{n}(ℱ) → 𝒞_{Hdg}^{i−1}(ℱ)) & n ≥ i \\ 1 & n < i. \end{cases}

Note that each Fil^{i}𝒞_{Hdg}^{•}(ℱ) is then a cosimplicial complex of groups, and that this agrees with the usual Hodge filtration on the normal complex, so that if ℱ is abelian, this will induce the Hodge filtration on cohomology.

Now, observe that, if we take C_{cris} as defined on pages 14 and 17, then

S_{X^{o},x}^{n}(𝔤) = C_{cris}^{n}(X_{cris}^{o}, exp(𝔤 ⊗ 𝒪_{X^{o}/W}))
= Γ(X_{cris}^{o}, (w_{∗}w^{−1})^{n+1} exp(𝔤 ⊗ 𝒪_{X^{o}/W}))
= Γ(X_{cris}^{o}, (v_{∗}v^{−1})^{n+1}(j^{−1}j_{∗})^{n+1} exp(𝔤 ⊗ 𝒪_{X^{o}/W}))
= Γ(X_{cris}^{o}, (v_{∗}v^{−1})^{n+1}𝒞_{Hdg}^{n}(exp(𝔤 ⊗ 𝒪_{X^{o}/W})))
= Γ(X_{Zar}^{o}, u_{X^{o}/W ∗}(v_{∗}v^{−1})^{n+1}𝒞_{Hdg}^{n}(exp(𝔤 ⊗ 𝒪_{X^{o}/W})))
= Γ(X_{Zar}^{o}, (v_{∗}v^{−1})^{n+1}u_{X^{o}/W ∗}𝒞_{Hdg}^{n}(exp(𝔤 ⊗ 𝒪_{X^{o}/W})))
= Č(U^{o}, u_{X^{o}/W ∗}𝒞_{Hdg}^{n}(exp(𝔤 ⊗ 𝒪_{X^{o}/W}))),

on which the Hodge filtration is induced from that on 𝒞_{Hdg}^{n}.

3 p-adic Hodge Theory

In this section we will establish a comparison theorem between the étale and crystalline homotopy types. This will imply that the pro-p-unipotent fundamental group is a crystalline representation, thus satisfying the hypotheses of Section 1.4.

Let V be a complete discrete valuation ring, with fraction field K, and finite residue field k of characteristic p. Denote by K̄ (resp. k̄) the algebraic closure of K (resp. k), and V̄ the integral closure of V in K̄. Write G = Gal(K̄/K). Let W = W (k), the ring of Witt vectors over k, and K_{0} the fraction field of W.

X will be a smooth and proper scheme over V, and x a V -valued point of X, with corresponding points x_{η} ∈ X(K), x_{s} ∈ X(k), and the geometric points x̄_{η} ∈ X(K̄), x̄_{s} ∈ X(k̄).

We adopt the conventions that

H_{cris}^{∗}(X) := H_{cris}^{∗}(X_{k}/W) ⊗_{W} K_{0},

where

H_{cris}^{∗}(X_{k}/W) = \underleftarrow{lim} H_{cris}^{∗}(X_{k}/W_{n}),

Following [?] Remark ??, it ought to be true that every cohomology theory with coefficients in a field of characteristic zero should have an associated homotopy type, arising from a standard cosimplicial algebra employed to calculate the cohomology. The SDC arising from this cosimplicial algebra should govern the pro-nilpotent fundamental group.

In the crystalline case, the homotopy type is the complex

C_{cris}^{•}(X, K_{0}),

defined on page 14.

From [?], there is an isomorphism

H_{cris}^{∗}(X) ⊗ B_{cris} ≅ H^{∗}(XK̄, ℚ_{p}) ⊗ B_{cris},

preserving various structures. It is thus natural to expect that there should be a quasiisomorphism

C_{cris}^{•}(X) ⊗ B_{cris} ∼ C^{•}(XK̄, ℚ_{p}) ⊗ B_{cris}

preserving the various structures. This would imply an isomorphism

π_{1}^{cris}(X_{k}, x_{s}) ⊗ B_{cris} ≅ π_{1}(XK̄, x̄_{η}) ⊗ B_{cris}.

If X is smooth and projective, then Corollary 2.11 provided a Frobenius equivariant isomorphism

L(π_{1}^{cris}(X_{k}, x_{s}), K_{0}) ≅ L(H_{cris}^{1}(X)^{∨})/(ˇ∪ H_{cris}^{2}(X)^{∨})),

where ˇ∪ is dual to the cup product.

This will then provide a Galois equivariant isomorphism

L(π_{1}(XK̄, x̄_{η}), ℚ_{p}) ≅ L(H^{1}(XK̄, ℚ_{p})^{∨})/(ˇ∪H^{2}(X ⊗ K̄, ℚ_{p})^{∨})).

Similarly, there will be a log-crystalline version. All working will be given for the open case, since the extra generality is no harder to handle.

Let X^{o} = X − D, for X smooth and proper over V, and D a divisor with simple normal crossings over V.

3.1 Comparison of étale and intermediate homotopy types

In this section, we will establish the quasi-isomorphism between the étale homotopy type, and the homotopy type corresponding to Faltings’ intermediate cohomology ([?]). Given a constructible ℚ_{p}-sheaf ℱ on X, we denote by

C_{ét}^{•}(X, ℱ) and 𝒞_{ét}^{•}(ℱ)

the complexes defined in [?] Section ??.

Given a scheme X, an affine cover U = {U_{α}} of X, and a Zariski sheaf ℱ on X, define the associated Čech complex

Č^{n}(U, ℱ) := Γ(X, (u_{∗}u^{∗})^{n+1}ℱ),

where ∐_{α} U_{α} ⟶^{u} X. This has a canonical cosimplicial structure, associated to the adjunction u^{∗} ⊣ u_{∗}.

Given a topological group Δ, and a topological field F, define the complex computing group cohomology by

C_{Gp}^{n}(Δ, F) := Hom_{Set,cts}(Δ^{n+1}, F),

with the usual cosimplicial structure.

In the situation above, take an affine cover U of X, and denote by U^{o} the corresponding cover of X^{o}. We then have a quasi-isomorphism

C_{ét}^{•}(X^{o} ⊗ K̄, ℚ_{p}) → Č^{•}(U^{o} ⊗ K̄, 𝒞_{ét}^{•}(ℚ_{p})),  (1)

where, on the right, we mean the diagonal complex associated to the double complex shown, a convention to be adopted throughout this section. That this is a quasiisomorphism follows from the observation that 𝒞_{ét}^{n}(ℚ_{p}) is acyclic for Zariski cohomology (since it is for étale cohomology), forcing the second spectral sequence (associated to the bigrading) to degenerate.

As in [?] p.46, define the étale presheaf of geometric fundamental groups Δ on X^{0} by

Δ(U) = π_{1}(U ⊗ K̄),

this only being a presheaf, since there is a certain ambiguity about the isomorphisms, due to inner automorphisms. This gives rise to the étale presheaf

𝒞_{Gp}^{•}(Δ, ℚ_{p}),

of cosimplicial complexes.

Since 𝒞_{Gp}^{•}(Δ, ℚ_{p}) is a resolution of ℚ_{p}, there is a quasi-isomorphism:

Č^{•}(U^{o} ⊗ K̄, 𝒞_{ét}^{•}(ℚ_{p})) → Č^{•}(U^{o} ⊗ K̄, 𝒞_{ét}^{•}(𝒞_{Gp}^{•}(Δ, ℚ_{p}))),  (2)

where on the right we are taking the diagonal complex associated to a triple cosimplicial complex.

Now, consider the map

Č^{•}(U^{o} ⊗ K̄, 𝒞_{Gp}^{•}(Δ, ℚ_{p})) → Č^{•}(U^{o} ⊗ K̄, 𝒞_{ét}^{•}(𝒞_{Gp}^{•}(Δ, ℚ_{p}))).  (3)

In general, this will not be a quasi-isomorphism. However, it will be whenever the U_{α} are such that the U_{α}^{o} ⊗ K̄ are K(π, 1) spaces.

Recall (for instance, from [?] Ch.II §2) the definition of a K(π, 1) space. Suppose Z is a smooth irreducible scheme over an algebraically closed field of characteristic zero, with fundamental group π. If F is a finite abelian group with a continuous π-action, corresponding to a locally constant sheaf F. There is a natural transformation

H_{Gp}^{∗}(π, F) → H_{ét}^{∗}(X, F)

(corresponding, in fact to the morphism (3) of complexes above). Z is said to be a K(π, 1) if this map is an isomorphism for all such F.

By [?] Ch.II Lemma 2.1, there exist affine covers U of X for which each U_{α}^{o} ⊗ K̄ is a K(π, 1). It then follows that the map (3) is a quasi-isomorphism for such covers.

We thus obtain a complex computing intermediate cohomology by

Č^{•}(U^{o} ⊗ K̄, 𝒞_{Gp}^{•}(Δ, ℚ_{p})),

for U^{o} ⊗ K̄ a cover by open affine K(π, 1)’s, which we have hence shown to be quasiisomorphic (as a cosimplicial algebra) to

C_{ét}^{•}(X^{o} ⊗ K̄, ℚ_{p}).

Note that since all maps were canonical, this quasi-isomorphism is Gal(K̄/K)-invariant.

3.2 Comparison of crystalline and intermediate homotopy types

Recall from Section 3.1 that the intermediate homotopy type is

C_{int}^{•}(X^{o}, ℚ_{p}) = Č^{•}(U^{o} ⊗ K̄, 𝒞_{Gp}^{•}(Δ, ℚ_{p})),

while, from Section 2.3, the crystalline homotopy type is

C_{cris}^{•}(X_{k}^{o}, K_{0}) ≅ Č(U_{k}^{o}, u_{X_{k}^{o}/W ∗}𝒞_{Hdg}^{n}(𝒪_{X_{k}^{o}/W} ⊗ K_{0})).

As in [?] p.62, we may use the crystal 𝒞_{Hdg}^{•}(𝒪_{X^{o}/W_{n}}) to define sheaves

𝒞_{Hdg}^{•}(𝒪_{X_{k}^{o}/W_{n}}) ⊗_{B}^{+} /(I^{[n]} + p^{n}B^{+})

on the site X^{o} defined in [?], by associating to each U = Spec R, the module

u_{X_{k}^{o}/W_{n}∗}𝒞_{Hdg}^{•}(𝒪_{X_{k}^{o}/W_{n}})(R) ⊗_{R} B^{+}(R̂)/(I^{[n]} + p^{n}B^{+}).

This then gives us a morphism

Č(U_{k}^{o}, u_{X^{o}/W_{n}∗}𝒞_{Hdg}^{n}(𝒪_{X_{k}^{o}/W_{n}})) ⊗ B^{+}(V)/(I^{[n]} + p^{n}B^{+})
→ Č^{•}(U^{o} ⊗ K̄, 𝒞_{Gp}^{•}(Δ, u_{X_{k}^{o}/W_{n}∗}𝒞_{Hdg}^{•}(𝒪_{X_{k}^{o}/W}) ⊗_{B}^{+} /(I^{[n]} + p^{n}B^{+}))),

which, by taking inverse limits and tensoring (over B^{+}(V)) with B(V), gives a morphism

C_{cris}^{•}(X^{o}, K_{0})⊗_{K_{0}}B(V) → \underleftarrow{lim} C_{int}^{•}(X^{o}, u_{X_{k}^{o}/W_{n}∗}𝒞_{Hdg}^{•}(𝒪_{X_{k}^{o}/W})⊗_{B}^{+}/(I^{[n]}+p^{n}B^{+}))⊗B(V) which by [?] is a filtered quasi-isomorphism, as is the natural morphism

C_{int}^{•}(X^{o}, ℚ_{p})⊗_{ℚ_{p}}B(V) → \underleftarrow{lim} C_{int}^{•}(X^{o}, u_{X_{k}^{o}/W_{n}∗}𝒞_{Hdg}^{•}(𝒪_{X_{k}^{o}/W})⊗_{B}^{+}/(I^{[n]}+p^{n}B^{+}))⊗B(V).

We therefore have a quasi-isomorphism

C_{cris}^{•}(X_{k}^{o}, K_{0}) ⊗_{K_{0}} B(V) ∼ C_{int}^{•}(XK̄^{o}, ℚ_{p}) ⊗_{ℚ_{p}} B(V),

respecting the Hodge filtration, and the Frobenius and Galois actions.

3.3 Consequences for fundamental groups

Combining the two previous sections, we have a quasi-isomorphism

C_{cris}^{•}(X_{k}^{o}, K_{0}) ⊗_{K_{0}} B(V) ∼ C_{ét}^{•}(Xo ; K̄, ℚ_{p}) ⊗_{ℚ_{p}} B(V),

preserving the various structures.

If x ∈ X^{o}(V), let x_{η} ∈ X^{o}(K) and x_{s} ∈ X^{o}(k) be the corresponding closed points, with x̄_{η} ∈ X^{o}(K̄) the geometric point.

Recall that

S_{ét}^{•}(𝔤) := exp(C_{ét}^{•}(Xo ; K̄, x̄_{η}; ℚ_{p}) ⊗_{ℚ_{p}} 𝔤)

is the SDC over N_{ℚ_{p}} giving rise to the Lie algebra

L^{ét} := L(π_{1}^{ét}(Xo ; K̄, x̄_{η}), ℚ_{p}),

while the SDC

S_{cris}^{•}(𝔤) := exp(C_{cris}^{•}(X_{k}^{o}, x_{s}; K_{0}) ⊗_{K_{0}} 𝔤)

over N_{K_{0}} gives rise to the Lie algebra

L^{cris} := L(π_{1}^{cris}(X_{k}^{o}, x_{s}), K_{0}).

We wish to compare these two.

Definition 3.1. Given a field κ of characteristic zero, define the category N_{κ}^{∞} to consist of nilpotent Lie algebras over κ, not necessarily finite dimensional. Note that any functor F on N_{κ} which commutes with filtered direct limits can be extended uniquely to N_{κ}^{∞} by setting

F (𝔤) = \underrightarrow{lim} F (𝔤_{i}),

where the 𝔤_{i} run over all finitely generated (and hence finite dimensional) sub-Lie algebras of 𝔤.

Recall that, on N_{ℚ_{p}}, we have functorial isomorphisms

HomN̂_{ℚp}(L^{ét}, 𝔤) ≅ Def_{S_{ét}}(𝔤).

Now, the functor S_{ét}^{n} extends naturally to N_{ℚ_{p}}^{∞}, and commutes with direct limits since tensor products do. The functor Hom(L^{ét}, −) also extends naturally to N_{ℚ_{p}}^{∞}, and commutes with direct limits since the Lie algebra is pro-finitely presented.

Note that there is a canonical map

N_{K_{0}}^{∞} → N_{ℚ_{p}}^{∞},

given by the forgetful functor, so that the isomorphism of functors above can be used to give an isomorphism of functors on N_{K_{0}}^{∞}, namely

HomN̂_{K_{0}}^{∞} (L^{ét} ⊗_{ℚ_{p}} K_{0}, 𝔤) ≅ Def_{S_{ét}⊗_{ℚp}K_{0}}(𝔤),

where

S_{ét}^{n} ⊗_{ℚ_{p}} K_{0}(𝔤) := exp(C_{ét}^{•}(Xo ; K̄, x̄_{η}; ℚ_{p}) ⊗_{ℚ_{p}} 𝔤).

Similarly, we have an isomorphism

HomN̂_{K_{0}}^{∞} (L^{cris}, 𝔤) ≅ Def_{S_{cris}}(𝔤).

Theorem 3.2. There is a canonical isomorphism

HomN̂_{K_{0}}^{∞} (L^{cris}, 𝔤 ⊗_{K_{0}} B) ≅ HomN̂_{K_{0}}^{∞} (L^{ét} ⊗_{ℚ_{p}} K_{0}, 𝔤 ⊗_{K_{0}} B),

functorial in 𝔤 ∈ N_{K_{0}}^{∞}, and respecting the Galois and Frobenius actions. In fact, this is functorial not only in 𝔤, but also in 𝔤 ⊗_{K_{0}} B, meaning that the isomorphism respects B-linear Lie algebra maps 𝔤 ⊗_{K_{0}} B → 𝔥 ⊗_{K_{0}} B.

Proof. The isomorphisms established above imply that it suffices to show

Def_{S_{cris}}(𝔤 ⊗_{K_{0}} B) ≅ Def_{S_{ét}⊗_{ℚp}K_{0}}(𝔤 ⊗_{K_{0}} B),

functorially in 𝔤 ⊗_{K_{0}} B.

Now, observe that

Def_{S_{cris}}(𝔤 ⊗_{K_{0}} B) ≅ Def_{S_{cris}⊗_{K_{0}}B}(𝔤),

where

S_{cris}^{n} ⊗_{K_{0}} B(𝔤) := exp((C_{cris}^{•}(X_{k}^{o}, x_{s}; K_{0}) ⊗_{K_{0}} B) ⊗_{K_{0}} 𝔤).

Similarly

Def_{S_{ét}⊗_{ℚp}K_{0}}(𝔤 ⊗_{K_{0}} B) ≅ Def_{S_{ét}⊗_{ℚp}B}(𝔤),

where

S_{ét}^{n} ⊗_{ℚ_{p}} B(𝔤) := exp((C_{ét}^{•}(Xo ; K̄, x̄_{η}; ℚ_{p}) ⊗_{ℚ_{p}} B) ⊗_{K_{0}} 𝔤).

Now the quasi-isomorphism

C_{cris}^{•}(X_{k}^{o}, x_{s}; K_{0}) ⊗_{K_{0}} B ∼ C_{ét}^{•}(Xo ; K̄, x̄_{η}; ℚ_{p})⊗_{ℚ_{p}}

established in the previous sections provides us with an isomorphism of the associated deformation functors

Def_{S_{cris}⊗_{K_{0}}B}(𝔤) ≅ Def_{S_{ét}⊗_{ℚp}B}(𝔤),

for 𝔤 ∈ N_{K_{0}}. Now, since tensor products commute with direct limits, both functors do, giving us isomorphisms

Def_{S_{cris}⊗_{K_{0}}B}(𝔤) ≅ Def_{S_{ét}⊗_{ℚp}B}(𝔤),

for all 𝔤 ∈ N_{K_{0}}^{∞}, as required.

Corollary 3.3. There is a Galois and Frobenius equivariant isomorphism

L(π_{1}^{cris}(X_{k}^{o}, x_{s}), K_{0}) ⊗_{K_{0}} B ≅ L(π_{1}^{ét}(Xo ; K̄, x̄_{η}), ℚ_{p}) ⊗_{ℚ_{p}} B.

Proof. From the theorem, we have an isomorphism

HomN̂_{K_{0}}^{∞} (L^{cris}, 𝔤 ⊗_{K_{0}} B) ⟶^{θ} HomN̂_{K_{0}}^{∞} (L^{ét} ⊗_{ℚ_{p}} K_{0}, 𝔤 ⊗_{K_{0}} B),

or equivalently an isomorphism

Hom_{B}(L^{cris} ⊗_{K_{0}} B, 𝔤 ⊗_{K_{0}} B) ⟶^{θ} Hom_{B}(L^{ét} ⊗_{ℚ_{p}} B, 𝔤 ⊗_{K_{0}} B),

functorial not only in 𝔤, but also in 𝔤 ⊗_{K_{0}} B. Define

α = θ(id) : L^{ét} ⊗_{ℚ_{p}} B → L^{cris} ⊗_{K_{0}} B,

and define

β = θ^{−1}(id) : L^{cris} ⊗_{K_{0}} B → L^{ét} ⊗_{ℚ_{p}} B.

Now, for f : 𝔤 ⊗_{K_{0}} B → 𝔥 ⊗_{K_{0}} B, and a : L^{cris} ⊗_{K_{0}} B → 𝔤 ⊗_{K_{0}} B, we must have

θ(f ∘ a) = f ∘ θ(a),

by functoriality, and similarly for θ^{−1}.

Therefore

id = θ(β) = θ(β ∘ id) = β ∘ α
id = θ^{−1}(α) = θ^{−1}(α ∘ id) = α ∘ β.

We now have the following corollary, showing that in this case the hypotheses of Section 1.4 are satisfied:

Corollary 3.4. If X^{o} = X − D, for X smooth and proper over V, and D a divisor with simple normal crossings over V, then L(π_{1}^{ét}(Xo ; K̄, x̄_{η}), ℚ_{p}) is a crystalline Galois representation.

Remark 3.5. A generalisation of this comparison result to relative Malcev completions has since been independently obtained by Olsson ([?] Theorem 1.9).