CONSTRUCTING DERIVED MODULI STACKS

J.P.PRIDHAM

Abstract. We introduce frameworks for constructing global derived moduli stacks associated to a broad range of problems, bridging the gap between the concrete and abstract conceptions of derived moduli. Our three approaches are via differential graded Lie algebras, via cosimplicial groups, and via quasi-comonoids, each more general than the last. Explicit examples of derived moduli problems addressed here are finite schemes, polarised projective schemes, torsors, coherent sheaves, and finite group schemes.

Introduction

In [Pri6], representability was established for many derived moduli problems involving schemes and quasi-coherent sheaves. However, the derived stacks there were characterised as nerves of ∞-groupoids with very many objects, making it difficult to understand the derived stacks concretely.

By contrast to the indirect approach of satisfying a representability theorem, [CFK2] and [CFK1] construct explicit derived Hilbert and Quot schemes as dg-schemes with the necessary properties, but give no universal family, so the derived moduli spaces lack functorial interpretations. In this paper, we will show how to reconcile these approaches, thereby giving explicit presentations for the derived moduli spaces of [Pri6].

In fact, we go substantially beyond the problems considered in [CFK2] and [CFK1], and give a framework valid in all characteristics (rather than just over ℚ). This is done by working with quasi-comonoid-valued functors, which give a global analogue of the simplicial deformation complexes of [Pri1]. In broad terms, derived moduli constructions over ℚ tend to be based on differential graded Lie algebras (DGLAs), while quasi-comonoids perform the same rôle in much greater generality. Since quasi-comonoids arise naturally from algebraic theories, they are much more general than DGLAs, even in characteristic zero.

Beware that for the purposes of this paper, derived algebraic geometry will mean the theory of [Lur] based on simplicial commutative rings, or on dg algebras when working over ℚ, rather than the more exotic HAG contexts of [TV]. This enables us to apply Lurie’s Representability Theorem in §1, but is also needed in later sections. The key to §3 is that tensoring a commutative algebra with a Lie algebra gives a Lie algebra, but similar constructions could be made with any pair of algebras for Koszul-dual operads. Likewise, the constructions of §§4–5 adapt to give functors on any category of simplicial objects. However, they will not adapt to give functors on symmetric spectra, since they depend on the functor A_{•} ⇝ A_{0}.

The structure of the paper is as follows. Section 1 summarises various results from [Pri7] concerning representability of derived stacks, and gives a few minor generalisations. Section 2 develops some technical results on the pro-Zariski and pro-étale sites. Lemma 2.3 shows that any finitely presented sheaf is a sheaf for the associated pro site, and our main results are Lemmas 2.10 and 2.13, concerning the existence of weakly universal coverings. These are applied in later sections to deal with infinite sums of locally free sheaves, which feature when studying polarised projective varieties.

The author was supported during this research by Trinity College, Cambridge; and by the Engineering and Physical Sciences Research Council [grant number EP/F043570/1].

In Section 3, DGLAs are introduced, together with the Deligne groupoid 𝔇𝔢𝔩(L) associated to any DGLA L with a gauge action. By adapting the techniques of [CFK1], DGLAs are used to construct derived moduli stacks for pointed finite schemes (Proposition 3.16) and for polarised projective schemes (Proposition 3.33). The resulting functors are shown (in Propositions 3.18 and 3.34, respectively) to be equivalent to the corresponding functors in [Pri6], defined as nerves of ∞-groupoids of derived geometric stacks.

DGLAs only tend to work in characteristic 0, and Section 4 shows how to construct derived moduli stacks using cosimplicial groups instead. For any simplicial cosimplicial group G, there is a derived Deligne groupoid \underline{𝔇𝔢𝔩}(G); Proposition 4.15 shows that cosimplicial group-valued functors G give rise to well-behaved derived moduli functors \underline{𝔇𝔢𝔩}(\underline{G}). For any DGLA L with gauge G_{L}, there is an associated cosimplicial group D(exp(L), G_{L}), and Corollary 4.27 shows that the Deligne groupoids associated to L and D(exp(L), G_{L}) are isomorphic. §4.4 defines a kind of sheafification G^{♯} for cosimplicial group-valued functors G, removing the need to sheafify \underline{𝔇𝔢𝔩}(\underline{G}); this gives an immediate advantage of cosimplicial groups over DGLAs. Proposition 4.38 gives a cosimplicial group governing derived moduli of torsors, a problem not easily accessible via DGLAs.

Cosimplicial groups cannot handle all moduli problems, so Section 5 begins by recalling the quasi-comonoids of [Pri4], and the derived Deligne groupoid \underline{𝔇𝔢𝔩}(E) of a simplicial quasi-comonoid E. Corollary 5.43 then shows that quasi-comonoid-valued functors E give rise to well-behaved derived moduli functors \underline{𝔇𝔢𝔩}(\underline{E}). In §5.2.1, we recall basic properties of monads, together with results from [Pri4] showing how these give rise to quasi-comonoids. Monads are ubiquitous, arising whenever there is some kind of algebraic structure. §5.2.2 goes further, by associating quasi-comonoids to diagrams. In particular, this allows derived moduli of morphisms to be constructed for all the examples considered in §6. §5.5 then defines a kind of sheafification E^{♯} for quasi-comonoid-valued functors E, removing the need to sheafify \underline{𝔇𝔢𝔩}(\underline{E}).

For every cosimplicial group G, there is a quasi-comonoid E(G), and Lemma 5.12 shows that \underline{𝔇𝔢𝔩}(E(G)) ≃ \underline{𝔇𝔢𝔩}(G), ensuring consistency between the various approaches. For moduli problems based on additive categories, the associated quasi-comonoid E is linear. This means that its normalisation NE is a DG associative non-commutative algebra (so a fortiori a DGLA), so the techniques of this section give DGLAs for abelian moduli problems. Moreover, Proposition 5.40 gives an equivalence \underline{𝔇𝔢𝔩}(E) ≃ \underline{𝔇𝔢𝔩}(NE), so quasicomonoids and DGLAs give equivalent derived moduli.

Section 6 gives a selection of examples which can be tackled by quasi-comonoids. Derived moduli of finite schemes, of polarised projective schemes, and of finite group schemes are constructed in Propositions 6.4, 6.17 and 6.28, respectively. In Propositions 6.6, 6.18 and 6.32, these are shown to be equivalent to the corresponding functors in [Pri6], defined as nerves of ∞-groupoids of derived geometric stacks. Proposition 6.11 constructs derived moduli of coherent sheaves, and Proposition 6.12 shows that this is equivalent to the nerve of ∞-groupoids of hypersheaves considered in [Pri6].

Contents

Introduction 1 1. Background on representability 3 1.1. Formal quasi-smoothness and homogeneity 4 1.2. Tangent complexes 4 1.3. Representability 6 1.4. Pre-representability 7 2. Sheaves on the pro-Zariski and pro-étale sites 10 2.1. The pro-Zariski topology 10 2.2. The pro-étale topology 12 2.3. Sheaves on derived rings 12 3. Moduli from DGLAs 13 3.1. DGLAs 13 3.2. Moduli of pointed finite schemes 14 3.3. Derived moduli of polarised projective schemes 17 4. Moduli from cosimplicial groups 22 4.1. Cosimplicial groups 23 4.2. Moduli functors from cosimplicial groups 25 4.3. Denormalisation 27 4.4. Sheafification 28 4.5. Derived moduli of G-torsors 30 5. Moduli from quasi-comonoids 32 5.1. Quasi-comonoids 32 5.2. Constructing quasi-comonoids 34 5.3. Linear quasi-comonoids 39 5.4. Moduli functors from quasi-comonoids 40 5.5. Sheafification 42 6. Examples of derived moduli via quasi-comonoids 43 6.1. Finite schemes 43 6.2. Coherent sheaves 45 6.3. Polarised projective schemes 47 6.4. Finite group schemes 48 Bibliography 52 References 52

1. Background on representability

Let 𝕊 be the category of simplicial sets. Denote the category of simplicial commutative rings by sRing, the category of simplicial commutative R-algebras by sAlg_{R}, and the category of simplicial R-modules by sMod_{R}. If ℚ ⊂ R, we let dg_{+}Alg_{R} be the category of differential graded-commutative R-algebras in non-negative chain degrees, and dg_{+}Mod_{R} the category of R-modules in chain complexes in non-negative chain degrees.

Definition 1.1. Given a simplicial abelian group A_{•}, we denote the associated normalised chain complex by N^{s}A (or, when no confusion is likely, by NA). Recall that this is given by N(A)_{n} := ⋂_{i>0} ker(∂_{i} : A_{n} → A_{n−1}), with differential ∂_{0}. Then H_{∗}(NA) ≅ π_{∗}(A).

When ℚ ⊂ R, using the Eilenberg–Zilber shuffle product ([Wei] 8.5.4), normalisation N extends to a right Quillen equivalence

N : sAlg_{R} → dg_{+}Alg_{R},

by [Qui1] §I.4.

Definition 1.2. Define dg_{+}N_{R} (resp. sN_{R}) to be the full subcategory of dg_{+}Alg_{R} (resp. sAlg_{R}) consisting of objects A for which the map A → H_{0}A (resp. A → π_{0}A) has nilpotent kernel. Define dg_{+}N_{R}^{♭} (resp. sN_{R}^{♭}) to be the full subcategory of dg_{+}N_{R} (resp. sN_{R}) consisting of objects A for which A_{i} = 0 (resp. N_{i}A = 0) for all i ≫ 0.

From now on, we will write dN^{♭} (resp. dAlg_{R}, resp. dMod_{R}) to mean either sN_{R}^{♭} (resp. sAlg_{R}, resp. sMod_{R}) or dg_{+}N_{R}^{♭} (resp. dg_{+}Alg_{R}, resp. dg_{+}Mod_{R}), noting that we only use chain algebras in characteristic 0.

Definition 1.3. Say that a surjection A → B in dg_{+}Alg_{R} (resp. sAlg_{R}) is a tiny acyclic extension if the kernel K satisfies I_{A}·K = 0, and K (resp. NK) is of the form cone(M)[−r] for some H_{0}A-module (resp. π_{0}A-module) M. In particular, H_{∗}K = 0.

1.1. Formal quasi-smoothness and homogeneity. The following definitions are mostly taken from [Pri6].

Definition 1.4. Say that a natural transformation η : F → G of functors F, G : dN^{♭} → 𝕊 is homotopic (resp. pre-homotopic) if for all tiny acyclic extensions A → B, the map

F (A) → F (B) ×_{G(B)} G(A)

is a trivial fibration (resp. a surjective fibration). Say that F is homotopic if F → • is so, where • denotes the one-point set.

Definition 1.5. Say that a natural transformation η : F → G of functors F, G : dN^{♭} → 𝕊 is formally quasi-presmooth (resp. formally presmooth) if for all square-zero extensions A → B, the map

F (A) → F (B) ×_{G(B)} G(A)

is a fibration (resp. a surjective fibration).

Say that η is formally quasi-smooth (resp. formally smooth) if it is formally quasipresmooth (resp. formally presmooth) and homotopic.

Definition 1.6. Say that a natural transformation η : F → G of functors on dN^{♭} is formally étale if for all square-zero extensions A → B, the map

F (A) → F (B) ×_{G(B)} G(A)

is an isomorphism.

Definition 1.7. Say that a natural transformation F → G of functors on dN^{♭} is (relatively) homogeneous if for all square-zero extensions A → B, the map

F (A ×_{B} C) → G(A ×_{B} C) ×_{[G(A)×_{G(B)}G(C)]} [F (A) ×_{F (B)} F (C)]

is an isomorphism. Say that F is homogeneous if F → • is relatively homogeneous.

Proposition 1.8. Let α : F → G be a formally étale morphism of functors F, G : dN^{♭} → Set. If G is homogeneous, then so is F. Conversely, if α is surjective and F is homogeneous, then so is G.

Proof. This is [Pri6] Proposition 2.18. □

1.2. Tangent complexes. Given a category C, write Ab(C) for the category of abelian group objects in C.

Definition 1.9. For a homogeneous functor F : dN^{♭} → 𝕊, A ∈ dN^{♭} and M ∈ dMod_{A}, define the tangent space by

T (F, M) := F (A ⊕ M) ∈ 𝕊↓F (A),

noting that this is an abelian group object in this category. Here, 𝕊 ↓ F (A) denotes the category of objects over F (A).

Given a natural transformation α : F → G of homogeneous functors F, G : dN^{♭} → 𝕊, define the relative tangent space by

T (F/G, M) := ker(T (F, M) → T (G, M) ×_{G(A)} F (A)) ∈ Ab(𝕊↓F (A)).

Given x ∈ F (A), define T_{x}(F/G, M) := T (F/G, M) ×_{F (A)} {x} ∈ Ab(𝕊) = sAb.

When α : F → G is formally quasi-presmooth, note that this definition is compatible with [Pri7] Definition 1.8, in the sense that for x ∈ π_{0}F (A), the space T_{x}(F/G)(M) of

[Pri7] is the homotopy fibre of T (F/G, M) → F (A) over x, since T (F/G, M) → F (A) is a fibration.

Definition 1.10. Given a pre-homotopic formally quasi-presmooth transformation F ⟶^{α} G of homogeneous functors F, G : dN^{♭} → 𝕊, an object A ∈ dN^{♭}, a point x ∈ F_{0}(A), and a module M ∈ dMod_{A}, define D_{x}^{i}(F/G, M) as follows, following [Pri7] Definition 3.14.

For i ≤ 0, set

D_{x}^{i}(F/G, M) := π_{−i}(T_{x}(F/G, M)).

For i > 0, set

D_{x}^{i}(F, M) := π_{0}F (T_{x}(F/G, M[−i])/π_{0}(T_{x}(F/G, cone(M)[1 − i])).

Note that homogeneity of F ensures that these are abelian groups for all i, and that the multiplicative action of A on M gives them the structure of A-modules.

If F ⟶^{α} G is formally quasi-smooth, note that [Pri7] Lemma 1.12 gives

D_{x}^{n−i}(F/G, M) = π_{i}(T_{x}(F/G, M[−n])).

The following is immediate.

Lemma 1.11. If X, Y, Z : dN^{♭} → 𝕊 are homogeneous, and X ⟶^{α} Y is formally quasipresmooth, with β : Z → Y any map, set T := X ×_{Y} Z, and observe that T → Z is quasi-presmooth. There is an isomorphism

D_{t}^{∗}(T/Z, M) ≅ D_{x}^{∗}(X/Y, M),

for t ∈ T (A) with image x ∈ X(A).

Proposition 1.12. Let X, Y, Z : dN^{♭} → 𝕊 be homogeneous functors, with X ⟶^{α} Y and Y ⟶^{β} Z formally quasi-smooth. For x ∈ X(A), there is then a long exact sequence

… ⟶^{∂} D_{x}^{j}(X/Y, M) → D_{x}^{j}(X/Z, M) → D_{y}^{j}(Y/Z, M) ⟶^{∂} D_{x}^{j+1}(X/Y, M) → D_{x}^{j+1}(X/Z, M) → …,

where y ∈ Y (A) is the image of x.

Proof. Since T_{x}(X/Y, M) = ker(α : T_{x}(X/Z, M) → T_{y}(Y/Z, M)), we have fibration sequences

… → π_{i}T_{x}(X/Y, M[−n]) → π_{i}T_{x}(X/Z, M[−n]) → π_{i}T_{x}(Y/Z, M[−n]) → …

for all i, n ≥ 0 so the result follows because π_{i}T_{x}(X/Y, M[−n]) = D^{n−i}(X/Y, M), and similarly for X/Z, Y/Z. □

Definition 1.13. Recall that a local coefficient system on S ∈ 𝕊 is an object V of Ab(𝕊↓S) for which the maps ∂_{i} : V_{s} → V_{∂_{i}s} are isomorphisms for all s ∈ S_{n}, where V_{s} := V_{n} ×_{S_{n}} {s}.

Lemma 1.14. If α : X → Y is a formally quasi-smooth morphism between homogeneous functors, take an object A ∈ dN^{♭} and M ∈ dMod_{A}. Then there is a local coefficient system

D^{∗}(X/Y, M)

on X(A), whose stalk at x ∈ X(A) is D_{x}^{∗}(X/Y, M). In particular, D_{x}^{∗}(X/Y, M) depends (up to non-canonical isomorphism) only on the image of x in π_{0}X(A).

Proof. As with [Pri7] Lemma 1.16, this follows straightforwardly from the proof of [Pri7] Lemma 1.9. □

1.2.1. Obstructions.

Proposition 1.15. If F, G : dN^{♭} → 𝕊 are homogeneous, with G pre-homotopic and α : F → G formally quasi-smooth, then for any square-zero extension e : I → A ⟶^{f} B in dN^{♭}, there is a sequence of sets

π_{0}(F A) ⟶^{f_{∗}} π_{0}(F B ×_{GB} GA) ⟶^{o_{e}} Γ(F B, D^{1}(F/G, I)),

where Γ(−) denotes the global section functor. This is exact in the sense that the fibre of o_{e} over 0 is the image of f_{∗} Moreover, there is a group action of D_{x}^{0}(F/G, I) on the fibre of π_{0}(F A) → π_{0}(F B) over x, whose orbits are precisely the fibres of f_{∗}.

For any y ∈ F_{0}A, with x = f_{∗}y, the fibre of F A → F B ×_{GB} GA over x is isomorphic to T_{x}(F/G, I), and the sequence above extends to a long exact sequence

\xymatrix{⋯ \ar[r]^{e_{∗}} & π_{n}(F A, y) \ar[r]^{f_{∗}} & π_{n}(F B ×_{GB} GA, x) \ar[r]^{o_{e}} & D_{y}^{1−n}(F/G, I) \ar[r]^{e_{∗}} & π_{n−1}(F A, y) \ar[r]^{f_{∗}} & ⋯ \\  & ⋯ \ar[r]^{f_{∗}} & π_{1}(F B ×_{GB} GA, x) \ar[r]^{o_{e}} & D_{y}^{0}(F/G, I) \ar[r]^{−∗y} & π_{0}(F A).}

Proof. The proof of [Pri7] Proposition 1.17 carries over to this generality. □

Corollary 1.16. If F, G : dN^{♭} → 𝕊 are homogeneous, with G pre-homotopic and α : F → G pre-homotopic and formally quasi-presmooth, then α is formally presmooth if and only if D_{x}^{i}(F/G, M) = 0 for all i > 0, all discrete rings A, all x ∈ π_{0}F (A) and all A-modules M.

Definition 1.17. Given a functor F on dN_{R}^{♭}, define the functor π^{0}F on Alg_{H_{0}R} by (π^{0}F)(A) := F (A).

Corollary 1.18. Take a morphism α : F → F′ of homogeneous formally quasi-smooth functors F, F′ : dN^{♭} → 𝕊. Then α is a weak equivalence if and only if

(1) π^{0}α : π^{0}F → π^{0}F′ is a weak equivalence of functors Alg_{H_{0}R} → 𝕊, and

(2) the maps D_{x}^{i}(F, M) → D_{αx}^{i}(F′, M) are isomorphisms for all A ∈ Alg_{H_{0}R}, all A-

modules M, all x ∈ F (A)_{0}, and all i > 0.

Proof. For any A ∈ dN^{♭}, we need to show that α_{A} : F (A) → F′(A) is a weak equivalence. By hypothesis, we know that this holds if we replace A with H_{0}A. Now, the map A → H_{0}A is a nilpotent extension; let the kernel be I_{A}. The maps A/I_{A}^{n+1} → A/I_{A}^{n} are squarezero extensions, and their kernels I_{A}^{n}/I_{A}^{n+1} are H_{0}A-modules. This allows us to proceed inductively, using the long exact sequence of Proposition 1.15 to deduce that F (A/I_{A}^{n+1}) → F′(A/I_{A}^{n+1}) is a weak equivalence whenever F (A/I_{A}^{n}) → F′(A/I_{A}^{n}) is so. □

1.3. Representability. For the remainder of this section, R will be a derived G-ring admitting a dualising module (in the sense of [Lur] Definition 3.6.1). In particular, this is satisfied if R is a G-ring admitting a dualising complex in the sense of [Har1] Ch. V. Examples are ℤ, any field, or any Gorenstein local ring.

Theorem 1.19. Take a functor F : dN_{R}^{♭} → 𝕊 satisfying the following conditions: (1) F is formally quasi-smooth.

(2) For all discrete rings A, F (A) is n-truncated, i.e. π_{i}F (A) = 0 for all i > n.

(3) F is homogeneous.

(4) π^{0}F : Alg_{H_{0}R} → 𝕊 is a hypersheaf for the étale topology.

(5) π_{0}π^{0}F : Alg_{H_{0}R} → Set preserves filtered colimits.

(6) For all A ∈ Alg_{H_{0}R} and all x ∈ F (A), the functors π_{i}(π^{0}F, x) : Alg_{A} → Set preserve filtered colimits for all i > 0.

(7) for all finitely generated integral domains A ∈ Alg_{H_{0}R}, all x ∈ F (A)_{0} and all étale morphisms f : A → A′, the maps

D_{x}^{∗}(F, A) ⊗_{A} A′ → D_{fx}^{∗}(F, A′)

are isomorphisms.

(8) for all finitely generated A ∈ Alg_{H_{0}R} and all x ∈ F (A)_{0}, the functors D_{x}^{i}((F/R), −) : Mod_{A} → Ab preserve filtered colimits for all i > 0.

(9) for all finitely generated integral domains A ∈ Alg_{H_{0}R} and all x ∈ F (A)_{0}, the groups D_{x}^{i}(F/R, A) are all finitely generated A-modules.

(10) for any complete discrete local Noetherian H_{0}R-algebra A, with maximal ideal 𝔪,

the map

π^{0}F (A) → \underleftarrow{lim}F (A/𝔪^{r})

is a weak equivalence. Then F is the restriction to dN_{R}^{♭} of an almost finitely presented geometric derived n-

stack F′ : dAlg_{R} → 𝕊. Moreover, F′ is uniquely determined by F (up to weak equivalence).

Proof. This variant of Lurie’s Representability Theorem essentially appears as [Pri7] Theorem 2.17, which takes a homotopy-preserving, homotopy-homogeneous functor instead of a formally quasi-smooth homogeneous functor. However, every homotopic functor is homotopy-preserving (by [Pri6] Lemma 2.24), while every formally quasi-presmooth homogeneous functor is homotopy-homogeneous (by [Pri6] Lemma 2.27). Finally, note that formal quasi-presmoothness allows us to replace homotopy limits with limits. □

Remark 1.20. For the definition of hypersheaves featuring in (4) above, see [Pri7] Definition 1.29. For all the applications in this paper, the following observation suffices. Given a groupoid-valued functor Γ : Alg_{H_{0}R} → Gpd, the nerve BΓ : Alg_{π_{0}R} → 𝕊 is a hypersheaf if and only if Γ is a stack (in the sense of [LMB]).

Remark 1.21. Note that there are slight differences in terminology between [TV] and [Lur]. In the former, only disjoint unions of affine schemes are 0-representable, so arbitrary schemes are 2-geometric stacks, and Artin stacks are 1-geometric stacks if and only if they have affine diagonal. In the latter, algebraic spaces are 0-stacks. A geometric n-stack is called n-truncated in [TV], and it follows easily that every n-geometric stack in [TV] is n-truncated. A weak converse is that every geometric n-stack is (n + 2)-geometric.

Theorem 1.19 follows the convention from [Lur], so “geometric derived n-stack” means “n-truncated derived geometric stack”.

Beware, however, that Condition (2) of the theorem only applies to discrete rings. In general, if A ∈ dN_{R}^{♭} with H_{i}A = 0 for i > m, then a geometric derived n-stack F will have the property that π_{j}F (A) = 0 for all j > m + n.

1.4. Pre-representability. Concrete approaches to derived moduli can naturally produce functors F: dN_{R}^{♭} → 𝕊 with the property that π_{i}F (A) = 0 for all i > n and all A. Such functors are not geometric derived n-stacks, since they cannot be both homotopic and homogeneous. The purpose of this section is to establish weaker conditions which can be satisfied by such functors, and still allow us to associate a geometric derived n-stack \underline{F} to F. In particular, all of the examples in §§3–6 will work by constructing derived geometric 1-stacks \underline{F} from groupoid-valued functors F.

Definition 1.22. Define a simplicial enrichment of sN_{R}^{♭} as follows. For A ∈ sN_{R}^{♭} and a finite simplicial set K, A^{K} ∈ sN_{R}^{♭} is defined by

(A^{K})_{n} := Hom_{𝕊}(K × Δ^{n}, A).

Spaces \underline{Hom}(A, B) ∈ 𝕊 of morphisms are then given by

\underline{Hom}_{sN_{R}^{♭}}(A, B)_{n} := Hom_{sN_{R}^{♭}}(A, B^{Δ^{n}}).

Definition 1.23. Define a simplicial enrichment of dg_{+}N_{R}^{♭} as follows. First set Ω_{n} to be the differential graded algebra

ℚ[t_{0}, t_{1}, …, t_{n}, dt_{0}, dt_{1}, …, dt_{n}]/(∑ t_{i} − 1, ∑ dt_{i})

of rational differential forms on the n-simplex Δ^{n}, where t_{i} is of degree 0. These fit together to form a simplicial complex Ω_{•} of graded-commutative DG-algebras, and we define A^{Δ^{n}} as the good truncation A^{Δ^{n}} := τ_{≥0}(A ⊗ Ω_{n}). [Note that this construction commutes with finite limits, so extends to define A^{K} for finite simplicial sets K.]

Spaces \underline{Hom}(A, B) ∈ 𝕊 of morphisms are then given by

\underline{Hom}_{dg_{+}N_{R}^{♭}}(A, B)_{n} := Hom_{dg_{+}N_{R}^{♭}}(A, B^{Δ^{n}}).

Definition 1.24. Given a functor F : dN^{♭} → 𝕊, we define \underline{F} : dN^{♭} → s𝕊, (for s𝕊 the category of bisimplicial sets), by

\underline{F} (A)_{n} := F (A^{Δ^{n}}).

Definition 1.25. Define W̄ : s𝕊 → 𝕊 to be the right adjoint to Illusie’s total Dec functor given by DEC (X)_{mn} = X_{m+n+1}. Explicitly,

W̄_{p}(X) = {(x_{0}, x_{1}, …, x_{p}) ∈ ∏_{i=0}^{p} X_{i,p−i}|∂_{0}^{v}x_{i} = ∂_{i+1}^{h}x_{i+1}, ∀0 ≤ i < p}

with operations

∂_{i}(x_{0}, …, x_{p}) = (∂_{i}^{v}x_{0}, ∂_{i−1}^{v}x_{1}, …, ∂_{1}^{v}x_{i−1}, ∂_{i}^{h}x_{i+1}, ∂_{i}^{h}x_{i+2}, …, ∂_{i}^{h}x_{p}),
σ_{i}(x_{0}, …, x_{p}) = (σ_{i}^{v}x_{0}, σ_{i−1}^{v}x_{1}, …, σ_{0}^{v}x_{i}, σ_{i}^{h}x_{i}, σ_{i}^{h}x_{i+1}, …, σ_{i}^{h}x_{p}).

In [CR2], it is established that the canonical natural transformation

diag X → W̄ X

from the diagonal is a weak equivalence for all X.

Lemma 1.26. For a homotopic functor F : dN^{♭} → 𝕊, the natural transformation F → W̄ F is a weak equivalence.

Proof. This is [Pri7] Lemma 3.13. □

Proposition 1.27. If a formally quasi-presmooth homogeneous functor F : dN^{♭} → 𝕊 is pre-homotopic, then the functor W̄ \underline{F} : dN^{♭} → 𝕊 is homogeneous and formally quasismooth.

Proof. This is essentially the same as [Pri7] Corollaries 3.10 and 3.12 (replacing weak equivalences with isomorphisms, and homotopy fibre products with fibre products), using the result from [CR1] that diagonal fibrations are W̄ -fibrations. □

Lemma 1.28. Given a formally quasi-presmooth pre-homotopic homogeneous functor F : dN_{R}^{♭} → 𝕊, an object A ∈ dN^{♭}, a point x ∈ F_{0}(A), and a module M ∈ dMod_{A}, there are canonical isomorphisms

D_{x}^{i}(F, M) ≅ D_{x}^{i}(W̄ \underline{F} , M).

Proof. The proof of [Pri7] Lemma 3.15, which deals with the case when A and M are discrete, carries over to this generality. □

Corollary 1.29. If a formally quasi-presmooth homogeneous functor F : dN^{♭} → 𝕊 is prehomotopic, and admits a morphism α : F → G to a formally quasi-smooth homogeneous functor, then α induces a functorial weak equivalence

W̄ \underline{F} ≃ G

if and only if (1) π^{0}α : π^{0}F → π^{0}G is a weak equivalence of functors Alg_{H_{0}R} → 𝕊, and

(2) the maps D_{x}^{i}(F, M) → D_{αx}^{i}(G, M) are isomorphisms for all A ∈ Alg_{H_{0}R}, all A-

modules M, all x ∈ F (A)_{0}, and all i > 0.

Proof. By Lemma 1.28, the map from F to W̄ \underline{F} induces isomorphisms on D^{i}, so the maps

D_{x}^{i}(W̄ \underline{F} , M) → D_{αx}^{i}(W̄ \underline{G}, M)

are isomorphisms. Proposition 1.27 shows that W̄ \underline{F} and W̄ \underline{G} are formally quasi-smooth homogeneous functors. Since F ⇝ W̄ \underline{F} does not change π^{0}F, Lemma 1.28 and Corollary 1.18 imply that the map

W̄ \underline{F} → W̄ \underline{G}

is a weak equivalence.

Since G is also homogeneous and formally quasi-smooth, Corollary 1.18 gives a weak equivalence G → W̄ \underline{G}. Combining this with the weak equivalence above, we see that W̄ \underline{F} and G are canonically weakly equivalent. □

Remark 1.30. By replacing Proposition 1.15 with [Pri7] Proposition 1.17, the proof of Corollary 1.29 works just as well if F is homotopy-homogeneous and homotopy-surjecting, while G is homotopy-homogeneous and homotopy-preserving. In particular, this holds if G is any presentation of a derived geometric n-stack.

Theorem 1.31. Take a functor F : dN_{R}^{♭} → 𝕊 satisfying the following conditions. (1) F is pre-homotopic.

(2) F is formally quasi-presmooth.

(3) For all discrete rings A, F (A) is n-truncated, i.e. π_{i}F (A) = 0 for all i > n .

(4) F is homogeneous.

(5) π^{0}F : Alg_{H_{0}R} → 𝕊 is a hypersheaf for the étale topology.

(6) π_{0}π^{0}F : Alg_{H_{0}R} → Set preserves filtered colimits.

(7) For all A ∈ Alg_{H_{0}R} and all x ∈ F (A), the functors π_{i}(π^{0}F, x) : Alg_{A} → Set preserve filtered colimits for all i > 0.

(8) for all finitely generated integral domains A ∈ Alg_{H_{0}R}, all x ∈ F (A)_{0} and all étale morphisms f : A → A′, the maps

D_{x}^{∗}(F, A) ⊗_{A} A′ → D_{fx}^{∗}(F, A′)

are isomorphisms.

(9) for all finitely generated A ∈ Alg_{H_{0}R} and all x ∈ F (A)_{0}, the functors D_{x}^{i}(F, −) : Mod_{A} → Ab preserve filtered colimits for all i > 0.

(10) for all finitely generated integral domains A ∈ Alg_{H_{0}R} and all x ∈ F (A)_{0}, the groups D_{x}^{i}(F, A) are all finitely generated A-modules.

(11) for all complete discrete local Noetherian H_{0}R-algebras A, with maximal ideal 𝔪,

the map

π^{0}F (A) → \underleftarrow{lim}F (A/𝔪^{r})

is a weak equivalence. Then W̄ \underline{F} is the restriction to dN_{R}^{♭} of an almost finitely presented geometric derived nstack F′ : sAlg_{R} → 𝕊 (resp. F′ : dg_{+}Alg_{R} → 𝕊). Moreover, F′ is uniquely determined by F (up to weak equivalence).

Proof. This essentially appears as [Pri7] Theorem 3.16, which takes a homotopy-surjecting, homotopy-homogeneous functor instead of a formally quasi-smooth pre-homotopic homogeneous functor. However, every pre-homotopic functor is automatically homotopysurjecting, while every formally quasi-presmooth homogeneous functor is homotopyhomogeneous (by [Pri6] Lemma 2.27).

Alternatively, note that Proposition 1.27 ensures that W̄ \underline{F} is homogeneous and formally quasi-smooth, so we may apply Theorem 1.19. □

2. Sheaves on the pro-Zariski and pro-étale sites

Our primary motivation for this section is the following. In general, an infinite direct sum M = ⨁_{i} M_{i} of locally free A-modules is not locally free for the étale topology, in the sense that there need not exist any faithfully flat étale morphism A → A′ with M ⊗_{A} A′ free. However, for all maximal ideals 𝔪 of A, the A_{𝔪}-module M ⊗_{A} A_{𝔪} is free. Indeed, for any set S of maximal ideals, the ∏_{𝔪∈S} A_{𝔪}-module M ⊗_{A} ∏_{𝔪∈S} A_{𝔪} is free. As we will show below, this amounts to saying that M is locally free for the pro-Zariski topology, and hence for the pro-étale topology.

Definition 2.1. A presheaf ℱ : Alg_{R} → Set is said to be locally of finite presentation if for any filtered direct system {A_{i}}_{i}, the map

\underrightarrow{lim}_{i} ℱ (A_{i}) → ℱ (\underrightarrow{lim}_{i} A_{i})

is an isomorphism.

Definition 2.2. Given a property P of morphisms of affine schemes, we say that f : X → Y is pro-P if it can be expressed as the limit X = \underleftarrow{lim}_{i} X_{i} of a filtered inverse system {X_{i}}_{i} of P-morphisms X_{i} → Y, in which all structure maps X_{i} → X_{j} are P-morphisms. Likewise, we say that a map A → B of rings is ind-P if Spec B → Spec A is pro-P.

Lemma 2.3. If ℱ : Alg_{R} → Set is locally of finite presentation and a sheaf for a class P of covering morphisms, then ℱ is also a sheaf for the class pro(P).

Proof. Given any finite (possibly empty) set {A_{s}}_{s∈S} of objects of Alg_{R}, we automatically have an isomorphism

ℱ (∏_{s∈S} A_{s}) → ∏_{s∈S} ℱ (A_{s}),

so we need only check that for any ring homomorphism A → B in ind(P), the diagram

ℱ (A) → ℱ (B) ⟹ ℱ (B ⊗_{A} B)

is an equaliser diagram.

Now, we can express A → B as a direct limit B = \underrightarrow{lim}_{i} B_{i} of P-morphisms A → B_{i}, so

ℱ (B) ≅ \underrightarrow{lim}_{i} ℱ (B_{i}), ℱ (B ⊗_{A} B) ≅ \underrightarrow{lim}_{i} ℱ (B_{i} ⊗_{A} B_{i}),

ℱ being locally of finite presentation. Since ℱ is a P-sheaf, the diagram

ℱ (A) → ℱ (B_{i}) ⟹ ℱ (B_{i} ⊗_{A} B_{i})

is an equaliser, and the required result now follows from the observation that finite limits commute with filtered direct limits. □

We will now construct weak universal covers for the topologies which concern us.

2.1. The pro-Zariski topology.

Definition 2.4. A morphism A → B of commutative rings is said to be conservative if the map

A^{×} → A ×_{B} B^{×}

is an isomorphism, where A^{×} denotes units in A. Say that a morphism Spec B → Spec A of affine schemes is conservative if A → B is so.

Definition 2.5. Say that a map A → B of commutative rings is a localisation if B ≅ A[S^{−1}], for some subset S ⊂ A.

Note that Spec D → Spec C is an open immersion if and only if D ≅ C[S^{−1}] for some finite set S. Thus A → B is a localisation if and only if Spec B → Spec A is a pro-(open immersion).

Lemma 2.6. Any commutative ring homomorphism f : A → B has a unique factorisation A → C → B as a localisation followed by a conservative map.

Proof. This is [Ane] Proposition 52. The factorisation is given by setting S := {a ∈ A : f(a) ∈ B^{×}}, then letting C := A[S^{−1}]. □

In order to study the Zariski topology, we wish to use local isomorphisms rather than open immersions. Likewise, for the pro-Zariski topology, we want pro-(local isomorphisms) rather than pro-(open immersions).

Definition 2.7. A morphism A → B of commutative rings is said to be strongly conservative if it is conservative, and the map id(A) → id(B) on sets of idempotent elements is an isomorphism. Say that a morphism Spec B → Spec A of affine schemes is strongly conservative if A → B is so.

Remark 2.8. The set id(A) just consists of ring homomorphisms ℤ^{2} → A. If A is finitely generated, then Spec A has a finite set π(Spec A) of components. Since an arbitrary ring A can be expressed as a filtered colimit A = \underrightarrow{lim}_{i} A_{i} of finitely generated rings, we can then define π(A) to be the profinite set \underrightarrow{lim}_{i} π(Spec A_{i}). Thus a conservative morphism Spec B → Spec A is strongly conservative if and only if π(Spec B) → π(Spec A) is an isomorphism of profinite sets.

Lemma 2.9. Every morphism f : X → Y of affine schemes has a unique factorisation X → (X/Y)^{loc} → Y as a strongly conservative map followed by a pro-(local isomorphism).

Proof. This is remarked at the end of [Ane] §4.2, where strongly conservative maps are denoted by Conv, and pro-(local isomorphisms) by Zet. Explicitly, we first factorise f as X → Y ×_{π(Y)} π(X) → Y, and then apply Lemma 2.6 to the first map, obtaining X → (X/Y)^{loc} → Y ×_{π(Y)} π(X) → Y. If X = Spec B and Y = Spec A, note that

Y ×_{π(Y)} π(X) = Spec (A ⊗_{ℤ.id(A)} ℤ.id(B)).

Note that we would get the same construction by applying Lemma 2.6 to X → Y × π(X). □

Lemma 2.10. For any commutative ring A, the category of pro-Zariski covers of Spec A has a weakly initial object Spec C. In other words, for any covering pro-(local isomorphism) Y → Spec A, there exists a map Spec C → Y over Spec A, although the map need not be unique.

Moreover, every covering pro-(local isomorphism) Z → Spec C has a section.

Proof. Let S be the set of maximal ideals of A, and set C := (A/ ∏_{𝔪∈S}(A/𝔪))^{loc}, as constructed in Lemma 2.9. Explicitly, we first form the subring A′ of A^{S} consisting of functions f : S → A with finite image. To form C, we then invert any element f ∈ A′ whenever for all s ∈ S, f(s) ∉ 𝔪_{s}.

Now, given any covering pro-(local isomorphism) Spec B → Spec A, use the covering property to lift the closed points of A to closed points of B; this gives us a map

g : B → ∏_{𝔪} A/𝔪.

Properties of unique factorisation systems then give a unique map

B → (A/ ∏_{𝔪∈S}(A/𝔪))^{loc}

compatible with g.

For the second part, take a covering pro-(local isomorphism) Z = Spec D → Spec C, and choose a lift D → A/𝔪 of each canonical map C → A/𝔪. This gives a diagram A ⟶^{h} D → ∏_{𝔪∈S} A/𝔪 with h opposite to a pro-(local isomorphism), so the universal property of C then gives a unique factorisation D → C → ∏_{𝔪∈S} A/𝔪. The composition C → D → C must then be the identity, since C → ∏_{𝔪∈S} A/𝔪 is strongly conservative. □

2.2. The pro-étale topology.

Definition 2.11. A morphism f : A → B is said to be Henselian if any factorisation A → A′ → B, with A → A′ étale, has a section A′ → A over B. Say that a morphism Spec B → Spec A of affine schemes is Henselian if A → B is so.

Lemma 2.12. Every morphism f : X → Y of affine schemes has a unique factorisation X → (X/Y)^{hen} → Y as a Henselian map followed by a pro-étale morphism.

Proof. This is an immediate consequence of [Ane] Proposition 64, which shows that indétale morphisms and Henselian morphisms form the left and right classes of a unique factorisation system on the category of commutative rings. Explicitly, if Y = Spec A and X = Spec B, then

(A/B)^{hen} := \underrightarrow{lim} A_{i},

where A_{i} runs over all factorisations A → A_{i} → B of f^{♯} with A → A_{i} étale. Then (X/Y)^{hen} := Spec (A/B)^{hen}. □

Lemma 2.13. For any commutative ring A, there is a weakly initial object Spec C in the category of pro-étale coverings of Spec A.

Moreover, every pro-étale covering Z → Spec C has a section.

Proof. For each point x of Spec A, choose a geometric point x̄ over x, so k(x̄) is a separably closed field, and let the set of all these points be S. Now, use Lemma 2.12 to construct the unique factorisation

A → (A/ ∏_{x̄∈S} k(x̄))^{hen} → ∏_{x̄∈S} k(x̄)

of A → ∏_{x̄∈S} k(x̄). The arguments of Lemma 2.10 now adapt to show that Spec C := [(Spec ∏_{x̄∈S} k(x̄))/Spec A]^{hen} is weakly initial in the category of pro-étale coverings of Spec A, and that every covering of Spec C has a section. □

2.3. Sheaves on derived rings.

Definition 2.14. Given a subclass P of flat morphisms of commutative rings, closed under pushouts and composition, say that a morphism f : A → B in sRing is

(1) homotopy-P if π_{0}f : π_{0}A → π_{0}B is in P, and the maps

π_{n}(A) ⊗_{π_{0}A} π_{0}B → π_{n}B

are isomorphisms for all n;

(2) strictly P if f_{0} : A_{0} → B_{0} is in P, and the maps

A_{n} ⊗_{A_{0}} B_{0} → B_{n}

are isomorphisms for all n.

Definition 2.15. Given P as above, say that a morphism f : A → B in dg_{+}Alg_{ℚ} is (1) homotopy-P if H_{0}f : H_{0}A → H_{0}B is in P, and the maps

H_{n}(A) ⊗_{H_{0}A} H_{0}B → H_{n}B

are isomorphisms for all n;

(2) strictly P if f_{0} : A_{0} → B_{0} is in P, and the maps
A_{n} ⊗_{A_{0}} B_{0} → B_{n}

are isomorphisms for all n.

Lemma 2.16. Every strictly P morphism in sRing or dg_{+}Alg_{ℚ} is homotopy-P.

Proof. We first prove this in the simplicial case. Take a strictly P morphism f : A → B; taking homotopy groups gives π_{n}(B) ≅ π_{n}(A) ⊗_{A_{0}} B_{0}, by flat base change. We then have isomorphisms

π_{n}(B) ≅ π_{n}(A) ⊗_{A_{0}} B_{0}
≅ π_{n}(A) ⊗_{π_{0}A} (π_{0}A ⊗_{A_{0}} B_{0})
≅ π_{n}(A) ⊗_{π_{0}A} π_{0}B,

as required. For the chain algebra case, replace π_{n} with H_{n}. □

Definition 2.17. On sRing^{opp} and dg_{+}Alg_{ℚ}^{opp}, we define topologies for every class P as above by setting P_{c} to be the intersection of P with faithfully flat morphisms, and saying that f : A → B is a homotopy-P covering (resp. a strict P covering) if f is homotopy-P_{c} (resp. strictly P_{c}).

In this way, we define both homotopy and strict sites for the étale, Zariski, pro-étale and pro-Zariski topologies.

3. Moduli from DGLAs

3.1. DGLAs.

Definition 3.1. A differential graded Lie algebra (DGLA) is a graded ℚ-vector space L = ⨁_{i∈ℕ_{0}} L^{i}, equipped with operators [−, −] : L × L → L bilinear and d : L → L linear, satisfying:

(1) [L^{i}, L^{j}] ⊂ L^{i+j}.

(2) [a, b] + (−1)^{āb̄}[b, a] = 0.

(3) (−1)^{c̄ā}[a, [b, c]] + (−1)^{āb̄}[b, [c, a]] + (−1)^{b̄c̄}[c, [a, b]] = 0.

(4) d(L^{i}) ⊂ L^{i+1}.

(5) d ∘ d = 0.

(6) d[a, b] = [da, b] + (−1)ā[a, db] Here ā denotes the degree of a, mod 2, for a homogeneous.

3.1.1. Maurer–Cartan.

Definition 3.2. Given a DGLA L^{•}, define the Maurer–Cartan set by

MC(L) := \{ω ∈ L^{1} | dω + \frac{1}{2}[ω, ω] = 0 ∈ L^{2}\}

Lemma 3.3. If a map e : L ↠ M of DGLAs has kernel K, with [K, K] = 0, then for any ω ∈ MC(M), the obstruction to lifting ω to MC(L) lies in

H^{2}(K, d + [ω, −]).

Proof. This is well-known. Given ω ∈ MC(M), choose a lift ω̃ ∈ L^{1}, and look at u(ω̃) := dω̃ + \frac{1}{2}[ω̃, ω̃]. Since [a, [a, a]] = 0 for any a ∈ L^{1}, we get

du + [ω̃, u(ω̃)] = [dω̃, ω̃] + [ω̃, dω̃] = 0,

so u ∈ Z^{2}(K, d + [ω, −]). Another choice for ω̃ is of the form ω̃ + a, for a ∈ K^{1}, and then

u((ω̃ + a) = u((ω̃) + da + [ω̃, a],

so the obstruction is

o_{e}(ω) := [u(ω̃)] ∈ Z^{2}(K, d + [ω, −])/(d + [ω, −])K^{1} = H^{2}(K, d + [ω, −]).
□

3.1.2. The gauge action.

Definition 3.4. Given a DGLA L, we say that a group G_{L} is a gauge group for L if it is equipped with the following extra data:

(1) group homomorphisms ad : G_{L} → GL(L^{n}) for all n, and

(2) a map D : G_{L} → L^{1}, satisfying the following conditions for g, h ∈ G_{L}, v, w ∈ L:

(1) ad_{g}([v, w]) = [ad_{g}v, ad_{g}w],

(2) D(gh) = Dg + ad_{g}(Dh),

(3) d(Dg) = \frac{1}{2}[Dg, Dg],

(4) d(ad_{g}(v)) = [Dg, ad_{g}(v)] + ad_{g}(dv).

Examples 3.5. If the DGLA L is nilpotent, then a canonical choice for G_{L} is the group exp(L^{0}), with D(g) = (dg) · g^{−1}.

When L^{0} is finite-dimensional, G_{L} will typically be an algebraic group integrating L^{0}, again with D(g) = (dg) · g^{−1}.

Definition 3.6. Given a gauge group G_{L} for a DGLA L, define the gauge action of G_{L} on MC(L) by

g ⋆ ω := ad_{g}(ω) − Dg

for g ∈ G_{L} and ω ∈ MC(L), noting that the conditions on ad_{g} and D ensure that this is well-defined and a group homomorphism.

Definition 3.7. Given a DGLA L with gauge group G_{L}, define the Deligne groupoid by 𝔇𝔢𝔩(L) := [MC(L)/G_{L}] In other words, 𝔇𝔢𝔩(L) has objects MC(L), and morphisms from ω to ω′ consist of {g ∈ G_{L} : g ⋆ ω = ω′}.

Define Del(L) ∈ 𝕊 to be the nerve B𝔇𝔢𝔩(L) of 𝔇𝔢𝔩(L).

3.2. Moduli of pointed finite schemes. For a fixed r ∈ ℕ, we now construct a DGLA governing moduli of pointed finite schemes of rank r+1. For any commutative ℚ-algebra A, our moduli groupoid consists of non-unital commutative A-algebras B, with the A-module underlying B being locally free of rank r. Our approach is analogous to the treatment of finite subschemes in [CFK2] §3.

Definition 3.8. Given a graded vector space V over ℚ, let CL(V) be the free (indconilpotent) graded Lie coalgebra ⨁_{n≥1} CL_{n}(V) cogenerated by V. Note that CL_{n}(V) is a quotient of V^{⊗n} by graded shuffle permutations.

Definition 3.9. Given a graded-commutative chain algebra A, define β(A) to be the dg Lie coalgebra CL(A[−1]), with coderivation d_{C} given on cogenerators by

d_{C}(a_{1} ⊗ a_{2} … ⊗ a_{n}) = \begin{cases}da_{1} & n = 1 \\ a_{1}a_{2} & n = 2 \\ 0 & n > 2. \end{cases}

Definition 3.10. Define a DGLA L by

L^{n} := Hom_{ℚ}(CL_{n+1}(ℚ^{r}[−1]), ℚ^{r}[−1]);

this can be identified with the space of degree −n Lie coalgebra derivations of β(ℚ^{r}), and this latter description allows us to define differential and bracket as

d_{L}(f) = d_{β} ∘ f ± f ∘ d_{β} [f, g] = f ∘ g ∓ g ∘ f.

Define a gauge group for L by setting G_{L} = GL(ℚ^{r}) = GL_{r}(ℚ). This has a canonical action on β(ℚ^{r}), so we set ad : G_{L} → GL(L^{n}) to be the adjoint action on derivations. Finally, D : G_{L} → L^{1} is given by D(g) = d_{β} − β(g) ∘ d_{β} ∘ β(g)^{−1}.

Definition 3.11. Given a differential graded (chain) Lie coalgebra C, define the gradedcommutative chain algebra β^{∗}(C) to be the free graded-commutative algebra on generators C[1], with derivation given on generators by

d_{β^{∗}(C)} = d_{C} + Δ : C[1] → C ⊕ S^{2}(C[1])[−1],

where Δ : C[1] → S^{2}(C[1])[−1] = CL_{2}(C)[1] is the cobracket.

Note that β^{∗} is left adjoint to the functor β from graded-commutative chain algebras to ind-conilpotent chain Lie coalgebras.

Lemma 3.12. If we set G_{L⊗A} := GL_{r}(A), then for any commutative ℚ-algebra A, 𝔇𝔢𝔩(L⊗A) is canonically isomorphic to the groupoid of non-unital -commutative A-algebra structures on the A-module A^{r}.

Proof. This is standard. Square-zero A-linear degree −1 derivations on β(ℚ^{r}) ⊗_{ℚ} A are all of the form d_{β(ℚ^{r})} + ω, for ω ∈ MC(L ⊗ A). Given g ∈ GL(A), the derivation d_{β(ℚ^{r})} + g ⋆ ω is then β(g) ∘ (d_{β(ℚ^{r})} + ω) ∘ β(g)^{−1}.

An element ω ∈ MC(L ⊗ A) is just an associative multiplication S^{2}(A^{r}) → A^{r}, so corresponds to a non-unital commutative A-algebra structure. □

Definition 3.13. Given A ∈ dg_{+}N_{ℚ}^{♭}, define L ⊗ A to be the DGLA

(L ⊗ A)^{n} := ⨁_{i} L^{n+i} ⊗ A_{i},

with differential d_{L} ± d_{A} and bracket given by [v ⊗ a, w ⊗ b] = ±[v, w] ⊗ (ab), where signs follow the usual graded conventions.

Definition 3.14. For the DGLA L of Definition 3.10, define the groupoid-valued functor G : dg_{+}N_{ℚ}^{♭} → Gpd to be the stackification of the groupoid presheaf

A ↦ [MC(L ⊗ A)/GL_{r}(A_{0})]

in the strict Zariski topology of Definition 2.17.

Explicitly, objects of G(A) are pairs (ω, g) ∈ MC(L ⊗ A ⊗_{A_{0}} B) × GL_{r}(B ⊗_{A_{0}} B), for A_{0} → B a faithfully flat local isomorphism (so Spec B → Spec A_{0} is an open cover), satisfying the following conditions:

(1) g ⋆ (pr_{1}^{∗}ω) = pr_{0}^{∗}ω ∈ MC(L ⊗ A ⊗_{A_{0}} B ⊗_{A_{0}} B),

(2) pr_{02}^{∗}g = (pr_{01}^{∗}g) · (pr_{12}^{∗}g) ∈ GL_{r}(B ⊗_{A_{0}} B ⊗_{A_{0}} B).

An isomorphism from (B, ω, g) to (C, ν, h) is a local isomorphism B ⊗_{A_{0}} C → D with A_{0} → D faithfully flat, together with an element α ∈ GL_{r}(D) such that α ⋆ ω = ν ∈ MC(L ⊗ A ⊗_{A_{0}} D), with (pr_{0}^{∗}α) · g = h · (pr_{1}^{∗}α) ∈ GL_{r}(D ⊗_{A_{0}} D).

Definition 3.15. As in [DK], given a simplicial object C in the category of categories, we define the simplicial set W̄ C by first forming the nerve BC (a bisimplicial set), then applying the functor W̄ of Definition 1.25, giving

W̄ C := W̄ BC.

Explicitly,

(W̄ Γ)_{n} = \{(\underline{x}, \underline{g}) : \underline{x} ∈ Ob Γ_{n} × Ob Γ_{n−1} × … × Ob Γ_{0},
\underline{g} ∈ Γ_{n−1}(∂_{0}x_{n}, x_{n−1}) × Γ_{n−2}(∂_{0}x_{n−1}, x_{n−2}) × … × Γ_{0}(∂_{0}x_{1}, x_{0})\},

with operations giving ∂_{i}(x_{n}, …, x_{0}; g_{n−1}, …, g_{0}) as

\begin{cases}(\begin{matrix}x_{n−1}, …, x_{0}; \\ g_{n−2}, …, g_{0} \end{matrix}) & i = 0, \\ (\begin{matrix}∂_{i}x_{n}, ∂_{i−1}x_{n−1}, …, ∂_{1}x_{n−i+1}, x_{n−i−1}, …, x_{0}; \\ ∂_{i−1}g_{n−1}, …, ∂_{1}g_{n−i+1}, (∂_{0}g_{n−i})g_{n−i−1}, g_{n−i−2}, …, g_{0} \end{matrix}) & 0 < i < n, \\ (\begin{matrix}∂_{n}x_{n}, …, ∂_{1}x_{1}; \\ ∂_{n−1}g_{n−1}, …, ∂_{1}g_{1} \end{matrix}) & i = n, \end{cases}

and σ_{i}(x_{n}, …, x_{0}; g_{n−1}, …, g_{0}) as

(\begin{matrix}σ_{i}x_{n}, σ_{i−1}x_{n−1}, …, σ_{0}x_{n−i}, x_{n−i}, …, x_{0}; \\ σ_{i−1}g_{n−1}, …, σ_{0}g_{n−i}, id_{x_{n−i}}, g_{n−i−1}, …, g_{0} \end{matrix}) .

Proposition 3.16. The functor W̄ \underline{G} : dg_{+}N_{ℚ}^{♭} → 𝕊 is representable by an almost finitely presented derived geometric 1-stack.

Proof. We verify the conditions of Theorem 1.31 for BG : dg_{+}N_{ℚ}^{♭} → 𝕊. Homogeneity follows immediately, because both MC(L ⊗ −) and GL_{r} preserve finite limits. Lemma 3.3 implies that MC(L⊗−) is pre-homotopic, since for any tiny acyclic extension A → B with kernel I, it gives the obstruction space as

H^{2}(L ⊗ I, d + [ω, −]) = ⨁_{n} H^{2+n}(L ⊗ H_{0}B, d + [ω, −]) ⊗_{H_{0}B} H_{n}(I) = 0.

It follows immediately that BG is pre-homotopic, and formal quasi-presmoothness is a consequence of the smoothness of GL_{r}.

Now, for A ∈ Alg_{ℚ}, Lemma 3.12 implies that G(A) is equivalent to the groupoid of rank r commutative algebras over A. This implies that π^{0}G is a stack, so π^{0}BG is a hypersheaf, and it also guarantees that the other conditions relating to G hold, so we need only verify the cohomological conditions.

For an A-algebra B corresponding to an object [B] of G(A), the results of [CFK2] §2 imply that

D_{[B]}^{i}(BG, M) ≅ Ext_{A⊕B}^{i+1}(𝕃_{•}^{A⊕B/A}, M ⊗_{A} B),

which has all the properties we require. Here, 𝕃^{A⊕B/A} denotes the cotangent complex (in the sense of [Ill1]) of the unital algebra A ⊕ B over A. This corresponds to the cotangent complex 𝕃^{B/A} in the category of non-unital commutative rings, defined using the formalism of [Qui2]. □

Remark 3.17. Alternatively, we can describe the associated derived geometric 1-stack explicitly. The functor A ↦ MC(L ⊗ A) is an affine dg scheme, and W̄ \underline{G} is just the hypersheafification of the quotient B[MC(L)/GL_{r}] in the homotopy-Zariski (and indeed homotopy-étale) topologies. In the terminology of [Pri8], the simplicial affine dg scheme B[MC(L)/GL_{r}] is a derived Artin 1-hypergroupoid representing W̄ \underline{G}.

Proposition 3.18. For A ∈ dg_{+}N_{ℚ}^{♭}, the space W̄ \underline{G}(A) is functorially weakly equivalent to the nerve W̄ 𝔊(A) of the ∞-groupoid 𝔊(A) of non-unital graded-commutative chain Aalgebras B in non-negative degrees for which B ⊗_{A}^{L} H_{0}A is weakly equivalent to a locally free module rank r over H_{0}A.

Proof. The data (ω, g) ∈ G(A) amount to giving a locally free A_{0}-module M of rank r (defined by the descent datum g), and a closed degree −1 differential δ on the free chain Lie A-coalgebra CL_{A_{0}}(M[−1]) ⊗_{A_{0}} A. Note that in the notation of [CFK2] 3.5, RCA(ℚ^{r}) is the dg scheme representing MC(L ⊗ −).

The functor β^{∗} from Definition 3.11 maps from dg Lie A-coalgebras to non-unital gradedcommutative chain A-algebras, giving us a chain algebra

β^{∗}(CL_{A_{0}}(M[−1]) ⊗_{A_{0}} A, δ).

over A. Thus we have defined a functor β^{∗} : G(A) → 𝔊(A), and Lemma 3.12 implies that this is a weak equivalence when A ∈ Alg_{ℚ}, so π^{0}G ≃ π^{0}𝔊(A).

Now, for A ∈ Alg_{ℚ}, ω ∈ MC(L ⊗ A) and an A-module N, a standard calculation gives

D_{(ω,id)}^{i}(BG, N) ≅ H^{i+1}(L ⊗ A, d + [ω,]),

which by [CFK2] §2 is just Ext_{A⊕B}^{i+1}(𝕃^{A⊕B/A}, N ⊗_{A} B), where B is the non-unital Aalgebra corresponding to ω. By faithfully flat descent, we deduce that if an A-algebra B is associated to (ω, g) ∈ G(A), then

D_{(ω,id)}^{i}(BG, N) ≅ Ext_{A⊕B}^{i+1}(𝕃^{A⊕B/A}, N ⊗_{A} B).

Adapting [Pri6] Corollary 3.10 and Example 3.11 to non-unital algebras,

D_{[B]}^{i}(W̄ 𝔊, N) ≅ Ext_{A⊕B}^{i+1}(𝕃^{A⊕B/A}, N ⊗_{A} B),

so f induces isomorphisms on the cohomology groups D^{i}.

As [Pri6] Example 3.11 adapts to non-unital algebras, the functor W̄ 𝔊 is also representable by a derived geometric 1-stack, so the weak equivalence follows from Remark 1.30. □

3.3. Derived moduli of polarised projective schemes. Fix a numerical polynomial h ∈ ℚ[t], with h(i) ≥ 0 for i ≫ 0. We will now study the moduli of polarised projective schemes (X, 𝒪_{X}(1)) over an affine base, with 𝒪_{X}(1) ample, for which Γ(X, 𝒪_{X}(n)) is locally free of rank h(n) for n ≫ 0. As in [Mum] Lecture 7 Corollary 3, such a polynomial h exists for every flat projective scheme over a connected Noetherian base.

Note that a 𝔾_{m}-representation M in A-modules is equivalent to an A-linear decomposition

M = ⨁_{n∈ℤ} M{n},

with λ ∈ 𝔾_{m}(A) acting on M{n} as multiplication by λ^{n}. The functors β^{∗} and β of the previous section both extend naturally to 𝔾_{m}-equivariant objects.

Definition 3.19. Given p ≫ 0 and q ≥ p, define a DGLA L_{[p,q]} over ℚ by

L_{[p,q]}^{n} := Hom_{ℚ}^{𝔾_{m}}(CL_{n+1}( ⨁_{q≥r≥p} ℚ^{h(r)}{r}[−1]), ⨁_{q≥r≥p} ℚ^{h(r)}{r}[−1]);

this can be identified with the space of 𝔾_{m}-equivariant degree −n Lie coalgebra derivations of β(⨁_{q≥r≥p} ℚ^{h(r)}{r}), and this latter description allows us to define differential and bracket as

d_{L_{[p,q]}}(f) = d_{β} ∘ f ± f ∘ d_{β} [f, g] = f ∘ g ∓ g ∘ f.

Definition 3.20. Given p ≫ 0, define the pro-DGLA L_{p} over ℚ to be the inverse system L_{p} = \{L_{[p,q]}\}_{q}, so the underlying DGLA is \underleftarrow{lim}_{q} L_{[p,q]}, given by

L_{p}^{n} = Hom_{ℚ}^{𝔾_{m}}(CL_{n+1}(⨁_{r≥p} ℚ^{h(r)}{r}[−1]), ⨁_{r≥p} ℚ^{h(r)}{r}[−1]),

which can be identified with the space of 𝔾_{m}-equivariant degree −n Lie coalgebra derivations of β(⨁_{r≥p} ℚ^{h(r)}\{r\}), the latter regarded as the colimit \underrightarrow{lim}_{q} β(⨁_{q≥r≥p} ℚ^{h(r)}\{r\}).

Given a ℚ-vector space V, we then define L_{p} ˆ⊗V to be the completed tensor product

L_{p} ˆ⊗V := \underleftarrow{lim}_{q}(L_{[p,q]} ⊗ V),

so

(L_{p} ˆ⊗V)^{n} := Hom_{ℚ}^{𝔾_{m}}(CL_{n+1}(⨁_{r≥p} ℚ^{h(r)}{r}[−1]), ⨁_{r≥p} V^{h(r)}{r}[−1]).

Definition 3.21. Given A ∈ Alg_{ℚ}, define a gauge group for L_{p} ˆ⊗A by setting G_{L_{p}}(A) := ∏_{r≥p} GL_{h(r)}(A). This has a canonical action on β(⨁_{r≥p} ℚ^{h(r)}(r)) ⊗_{ℚ} A, so we set ad : G_{L_{p}} → GL(L_{p}^{n}) to be the adjoint action on derivations. Finally, D : G_{L_{p}} → L^{1} is given by D(g) = d_{β} − β(g) ∘ d_{β} ∘ β(g)^{−1}.

Lemma 3.22. For any commutative ℚ-algebra A, the groupoid [MC(L_{p} ˆ⊗A)/G_{L_{p}}(A)] is naturally equivalent to the groupoid of 𝔾_{m}-equivariant non-unital commutative A-algebra structures on

⨁_{r≥p} A^{h(r)}{r}.

Proof. This is just a graded version of Lemma 3.12. □

Definition 3.23. Given A ∈ dg_{+}N_{ℚ}^{♭}, define L_{p}(A) to be the DGLA

L_{p}(A)^{n} := ⨁_{i} L_{p}^{n+i} ˆ⊗A_{i}

with differential d_{L} ± d_{A} and bracket given by [v ⊗ a, w ⊗ b] = ±[v, w] ⊗ (ab), where signs follow the usual graded conventions.

Definition 3.24. For the DGLA L_{p} of Definition 3.20, define the groupoid-valued functor G_{p} : dg_{+}N_{ℚ}^{♭} → Gpd to be the stackification of the groupoid presheaf

A ↦ [MC(L_{p}(A))/G_{L_{p}}(A_{0})]

in the strict pro-Zariski topology of Definition 2.17.

Making use of Lemma 2.10, we can describe this explicitly by setting A′_{0} := (A_{0}/ ∏_{𝔪} A_{0}/𝔪)^{loc}, where 𝔪 runs over all maximal ideals of A_{0}, and then setting A′ := A′_{0} ⊗_{A_{0}} A. Objects of G_{p}(A) are then pairs (ω, g) ∈ MC(L_{p}(A′)) × G_{L_{p}}(A′_{0} ⊗_{A_{0}} A′_{0}), satisfying the following conditions:

(1) g ⋆ (pr_{1}^{∗}ω) = (pr_{0}^{∗}ω) ∈ MC(L(A′ ⊗_{A} A′)),

(2) pr_{02}^{∗}g = (pr_{01}^{∗}g) · (pr_{12}^{∗}g) ∈ G_{L_{p}}(A′_{0} ⊗_{A_{0}} A′_{0} ⊗_{A_{0}} A′_{0}).

An isomorphism from (ω_{1}, g_{1}) to (ω_{2}, g_{2}) is an element α ∈ G_{L_{p}}(A′_{0}) such that α ⋆ ω_{1} = ω_{2} ∈ MC(L_{p}(A′)), with (pr_{0}^{∗}α) · g_{1} = g_{2} · (pr_{1}^{∗}α) ∈ G_{L_{p}}(A′ ⊗_{A} A′)

Lemma 3.25. For A ∈ Alg_{ℚ}, the groupoid G_{p}(A) is canonically equivalent to the groupoid of non-unital 𝔾_{m}-equivariant commutative A-algebras

B = ⨁_{r≥p} B{r},

with each A-module B{r} locally free of rank h(r).

Proof. If we set A′ := (A/ ∏_{𝔪} A/𝔪)^{loc}, where 𝔪 runs over all maximal ideals of A, then Lemma 2.10 shows that any Zariski cover Spec B → Spec A′ must have a section. Hence locally free A′-modules are free. Lemma 3.22 then implies that [MC(L_{p}(A′))/G_{L_{p}}(A′)] is equivalent to the groupoid of non-unital 𝔾_{m}-equivariant commutative A′-algebras B′ = ⨁_{r≥p} B′{r}, with each A′-module B′{r} locally free of rank h(r).

Given an object (ω, g) of G_{p}(A), it thus follows that ω corresponds to such an A′-algebra B′, while g is a descent datum. This determines a unique A-algebra B with B′ = B ⊗_{A} A′, and isomorphisms behave as required. □

Definition 3.26. For A ∈ Alg_{ℚ}, define M_{p}(A) to be the full subgroupoid of G_{p}(A) whose objects correspond under Lemma 3.25 to finitely generated commutative A-algebras.

Lemma 3.27. The morphism M_{p} → π^{0}G_{p} of groupoid-valued functors on Alg_{ℚ} is formally étale.

Proof. For any square-zero extension A ↠ B of commutative ℚ-algebras, we need to show that

M_{p}(A) → M_{p}(B) ×_{G_{p}(B)} G_{p}(A)

is an isomorphism. This follows because any flat A-algebra C is finitely generated if and only if C ⊗_{A} B is finitely generated as a B-algebra, since any lift of a generating set for C ⊗_{A} B must give a generating set for C. □

Lemma 3.28. The functor M_{p} : Alg_{ℚ} → Gpd is locally of finite presentation, in the sense that for any filtered direct system \{A_{i}\}_{i} of commutative ℚ-algebras with A = \underrightarrow{lim}_{i} A_{i}, the map

\underrightarrow{lim}_{i} M_{p}(A_{i}) → M_{p}(A)

is an equivalence of groupoids.

Proof. We first show essential surjectivity. Take an object B ∈ M_{p}(A). Since B is finitely generated, we can choose homogeneous generators x_{j} of degree d_{j} for 1 ≤ j ≤ n, giving us a surjection

f : A[x_{1}, …, x_{n}] ↠ B.

If we let S := ℤ[x_{1}, …, x_{n}], then I := ker f is a graded ideal of S ⊗ A. In the notation of [HS], we have a degree functor deg : ℕ^{n} → ℕ given by (a_{1}, …, a_{n}) ↦ ∑_{i} a_{i}d_{i}, and the Hilbert polynomial is given by h_{I} = h. By [HS] Corollary 1.2, there is a projective scheme H_{S}^{h} over ℤ with H_{S}^{h}(A) the set of all graded ideals of S with Hilbert function

h_{p}(i) := \begin{cases}h(i) & i ≥ p \\ 0 & i < p. \end{cases}

[Note that we do not use Grothendieck’s construction from [Gro2], since that only describes A-valued points of the Hilbert scheme for A Noetherian.]

In particular, H_{S}^{h_{p}} is of finite presentation, so H_{S}^{h_{p}}(A) = \underrightarrow{lim}_{i} H_{S}^{h_{p}}(A_{i}). Therefore there

exists B_{i} ∈ H_{S}^{h_{p}}(A_{i}) with B ≅ B_{i} ⊗_{A_{i}} A. The forgetful functor H_{S}^{h_{p}} → M_{p} then ensures that B_{i} ∈ M_{p}(A_{i}).

It only remains to show that \underrightarrow{lim}_{i} M_{p}(A_{i}) → M_{p}(A) is full and faithful. Now, [HS] Proposition 3.2 shows that the ideal I above is finitely generated, so B is finitely presented over A. Likewise, any objects B_{i}, B′_{i} ∈ M_{p}(A_{i}) will be finitely presented, which implies that

Hom_{M_{p}(A)}(B_{i} ⊗_{A_{i}} A, B′_{i} ⊗_{A_{i}} A) ≅ \underrightarrow{lim}_{j} Hom_{M_{p}(A_{j})}(B_{i} ⊗_{A_{i}} A_{j}, B′_{i} ⊗_{A_{i}} A_{j}),

completing the proof. □

Definition 3.29. Define M : Alg_{ℚ} → Gpd by M(A) := \underrightarrow{lim}_{p} M_{p}(A). Likewise, define

G := \underrightarrow{lim}_{p} G_{p} : dg_{+}N_{ℚ}^{♭} → Gpd

and M̃ : dg_{+}N_{ℚ}^{♭} → Gpd by

M̃(A) := G(A) ×_{G(H_{0}A)} M(H_{0}A).

Proposition 3.30. For A ∈ Alg_{ℚ}, M(A) is equivalent to the groupoid of flat polarised schemes (X, 𝒪_{X}(1)) of finite type over A, with 𝒪_{X}(1) ample and the A-modules Γ(X, 𝒪_{X}(n)) locally free of rank h(n) for all n ≫ 0.

Proof. This is fairly standard — the analogue for subschemes is [HS] Lemma 4.1. Given an object B ∈ M(A), there exists p with B lifting to B ∈ M_{p}(A). Therefore we can define

(X, 𝒪_{X}(1)) := Proj (A ⊕ B).

Replacing B with its image in M_{q}(A) (for q > p) does not affect Proj (A ⊕ B), so we have a functor Proj (A ⊕ −) from M(A) to polarised projective schemes over A.

For the quasi-inverse functor, take a polarised scheme (X, 𝒪_{X}(1)) and some p for which Γ(X, 𝒪_{X}(n)) is locally free of rank h(n) for all n ≥ p. Then define B ∈ M_{p}(A) by

B := ⨁_{n≥p} Γ(X, 𝒪_{X}(n)).

□

Remark 3.31. Note that the hypothesis that Γ(X, 𝒪_{X}(n)) be locally free for n ≫ 0 ensures that X is flat over A. If A is Noetherian, then the proof of [Har2] III.9.9 shows that the converse holds, and indeed that if A is connected, then there exists a Hilbert polynomial h with Γ(X, 𝒪_{X}(n)) locally free of rank h(n) for all n ≫ 0.

Proposition 3.32. If A ∈ Alg_{ℚ} and X = Proj (A ⊕ C) for C ∈ M_{p}(A), then for any A-module M, there are canonical isomorphisms

D_{[C]}^{i}(BM̃, M) ≅ 𝔼xt_{X}^{i+1}(𝕃^{X/B𝔾_{m}⊗A}, 𝒪_{X} ⊗_{A} M),

where 𝕃 is the cotangent complex of [Ill1].

Proof. Given C ∈ M_{p_{0}}(A), first let X̃ := Spec (A ⊕ C) − {0}, where {0} denotes the copy of Spec A defined by the ideal C. X̃ inherits a 𝔾_{m}-action from C (with trivial action on A), and in fact

X̃ = Spec_{X}(⨁_{n∈ℤ} 𝒪_{X}(n)),

with X = X̃/𝔾_{m} and X̃ = X ×_{B𝔾_{m}⊗A}^{h} Spec A. Writing π : X̃ → X for the projection, base change gives π^{∗}𝕃^{X/B𝔾_{m}⊗A} ≃ 𝕃^{X̃/A}. Since j : X̃ → Spec (A ⊕ C) is an open immersion, it is étale, so 𝕃^{X̃/A} ≃ j^{∗}𝕃^{(A⊕C)/A}.

For any C-module N, the associated quasi-coherent sheaf N^{♯} on X is given by N^{♯} = (π_{∗}j^{∗}N)^{𝔾_{m}}. Now, Lemma 3.28 implies that there exists a finitely generated ℚ-subalgebra A^{0} ⊂ A and an object C^{0} ∈ M_{p_{0}}(A^{0}) with C = C^{0} ⊗_{A^{0}} A. Since both D^{i} and 𝔼xt are compatible with base change (the former by [Pri7] Lemma 1.15), it suffices to show that

D_{[C^{0}]}^{i}(BM̃, M) ≅ 𝔼xt_{X_{0}}^{i+1}(𝕃^{X_{0}/B𝔾_{m}⊗A^{0}}, 𝒪_{X_{0}} ⊗_{A^{0}} M),

where X_{0} = Proj (A^{0} ⊕ C^{0}). Replacing A and C with A^{0} and C^{0}, we may therefore reduce to the case where A is a finitely generated ℚ-algebra (hence Noetherian). Because both expressions above commute with filtered colimits of the modules M, we may assume that M is a finitely generated A-module.

Since M → π^{0}G is formally étale by Lemma 3.27,

D_{[C]}^{i}(BM̃, M) ≅ D_{[C]}^{i}(BG, M).

As A ⊕ β^{∗}β(B) is a cofibrant resolution of A ⊕ B, we have

D_{[C]}^{i}(BG, M) ≅ \underrightarrow{lim}_{p≥p_{0}} Ext_{A⊕B\{≥p\}}^{i+1}(𝕃^{(A⊕B\{≥p\})/A}, B\{≥ p\} ⊗_{A} M)^{𝔾_{m}}.

Now, the proof of Serre’s Theorem ([Ser] §59) still works over any Noetherian base, so shows that for a finitely generated C-module N and any n ∈ ℤ,

Ext_{X}^{i}(𝒪_{X}(n), N^{♯}) ≅ \underrightarrow{lim}_{p} Ext_{C}^{i}(C(n)\{≥ p\}, N\{≥ p\})^{𝔾_{m}}.

Indeed, a spectral sequence argument shows that the same is true if we replace C(n) with any finitely generated C-module L, since L will then admit a resolution by finite sums of C(n)’s. In fact, another spectral sequence argument allows us to take a chain complex L whose homology groups H_{i}(L) are finite and bounded below, giving

𝔼xt_{X}^{i}(L^{♯}, N^{♯}) ≅ \underrightarrow{lim}_{p} 𝔼xt_{C}^{i}(L\{≥ p\}, N\{≥ p\})^{𝔾_{m}}.

Thus for any fixed p ≥ p_{0},

Ext_{X}^{i}(𝕃^{X̃/B𝔾_{m}⊗A}, 𝒪_{X} ⊗_{A} M) ≅ \underrightarrow{lim}_{q≥p} Ext_{A⊕B\{≥p\}}^{i}((𝕃^{(A⊕B\{≥p\})/A})\{≥ q\}, B\{≥ p\} ⊗_{A} M)^{𝔾_{m}}.

We can then take the colimit over the poset of pairs (p, q) with q ≥ p ≥ p_{0}. Since the set of pairs (p, p) is cofinal in this poset, we get

Ext_{X}^{i}(𝕃^{X̃/B𝔾_{m}⊗A}, 𝒪_{X} ⊗_{A} M) ≅ \underrightarrow{lim}_{p≥p_{0}} Ext_{A⊕B\{≥p\}}^{i}(𝕃^{(A⊕B\{≥p\})/A}, B\{≥ p\} ⊗_{A} M)^{𝔾_{m}},

as required. □

Proposition 3.33. The functor BM̃ : dg_{+}N_{ℚ}^{♭} → 𝕊 satisfies the conditions of Theorem 1.31, so the associated functor W̄ \underline{M̃} : dg_{+}N_{ℚ}^{♭} → 𝕊 is representable by an almost finitely presented derived geometric 1-stack.

Proof. We apply Theorem 1.31 to M̃. First, note that π^{0}M̃ = M, which is a stack, locally of finite presentation by Lemma 3.28. Proposition 3.30 and Grothendieck’s formal existence theorem ([Gro1] 5.4.5) ensure that for any complete local Noetherian ℚ-algebra Λ, the map

M(Λ) → \underleftarrow{lim}_{n} M(Λ/𝔪^{n})

is surjective on objects. That the map is an isomorphism is then a consequence of ([Gro1] 5.1.4).

Now, homogeneity of BM̃ is immediate, and Lemma 3.3 gives pre-homotopicity. All the remaining conditions follow from Proposition 3.32, with the same reasoning as for Proposition 3.16. □

Proposition 3.34. For A ∈ dg_{+}N_{ℚ}^{♭}, the space W̄ \underline{M̃}(A) is functorially weakly equivalent to the nerve W̄ 𝔐(A) of the ∞-groupoid 𝔐(A) of derived geometric 0-stacks 𝔛 over B𝔾_{m}× Spec A for which X := 𝔛⊗_{A}^{L}H_{0}A is weakly equivalent to a flat projective scheme over H_{0}A, with the polarisation X → B𝔾_{m} ⊗ H_{0}A ample with Hilbert polynomial h.

Proof. We adapt the proof of Proposition 3.18. An object of G_{p}(A) corresponds to a locally free 𝔾_{m}-equivariant A_{0}-module N{≥ p}, with N{r} locally free of rank h(r), together with a closed degree −1 differential δ on the free chain Lie coalgebra CL_{A_{0}}(N[−1]) ⊗_{A_{0}} A. We may therefore form the DG-scheme

𝔛 := Proj (A ⊕ β^{∗}(CL_{A_{0}}(N[−1]) ⊗_{A_{0}} A, δ)).

As in [Pri8] §6.4, there is a canonical derived geometric 1-stack associated to 𝔛. To give an explicit map from this to B𝔾_{m}, we first let 𝔛̃ := Spec (A ⊕ β^{∗}(CL_{A_{0}}(N[−1]) ⊗_{A_{0}} A, δ)) − {0}, and then form the simplicial scheme

𝔛̃ ×^{𝔾_{m}} E𝔾_{m},

which is a simplicial resolution of X, and has a canonical map to the simplicial scheme B𝔾_{m}. Here, E𝔾_{m} is the universal 𝔾_{m}-space over B𝔾_{m}, given by the simplicial 0-coskeleton E𝔾_{m} = cosk_{0}𝔾_{m}, so (E𝔾_{m})_{n} = (𝔾_{m})^{n+1}. For an explicit Artin hypergroupoid representation of X, we could go further and replace 𝔛̃ with its Čech nerve associated to any open affine cover.

Now, if our object C lies in M̃_{p} ⊂ G_{p}(A), then C ⊗_{A} H_{0}A lies in M_{p}(H_{0}A), so C ⊗_{A} H_{0}A = β(B), for a finitely generated commutative algebra structure B on N. Since the map

β^{∗}β(B) → B

is a quasi-isomorphism, this means that

𝔛 ⊗_{A}^{L} H_{0}A ≃ Proj (H_{0}A ⊕ B),

which is a polarised projective scheme with Hilbert polynomial h.

Since Proj is unchanged on replacing N with N{≥ q} for q > p, we have defined a functor

α_{A} : M̃(A) → 𝔐(A).

By [Pri6] Example 3.39, the functor W̄ 𝔐 is also representable by a derived geometric 1-stack, so we just need to check that W̄ M̃ → W̄ 𝔐 satisfies the conditions of Remark 1.30.

If A ∈ Alg_{ℚ}, then Proposition 3.30 implies that α_{A} is an equivalence of groupoids. Combining Proposition 3.32 with [Pri6] Corollary 3.32 and Example 3.39, we have isomorphisms

D_{[C]}^{i}(BM̃, M) ≅ Ext_{X}^{i+1}(𝕃^{X/𝔾_{m}⊗A}, 𝒪_{X} ⊗_{A} M) ≅ D_{[C]}^{i}(W̄ 𝔐̃, M),

so Remark 1.30 applies. □

Remark 3.35. Replacing the DGLA L_{p} with the finite-dimensional DGLA L_{[p,q]} in the definitions above gives us a functor M̃_{[p,q]}. Since L_{p} = \underleftarrow{lim}_{q} L_{[p,q]}, we will have M̃_{p} =

\underleftarrow{lim}_{q} M̃_{[p,q]}, and hence

M̃ = \underrightarrow{lim}_{p} \underleftarrow{lim}_{q} M̃_{[p,q]}.

It is natural to seek an open substack of M on which these limits stabilise. If we define M^{(k)} ⊂ M to be the open substack consisting of polarised schemes (X, 𝒪_{X}(1)) for which 𝒪_{X}(k) is very ample, then we may regard X as a subscheme of ℙ^{h(k)}, so [CFK1] Theorem 1.2.3(b) and Theorem 1.4.1 imply that for q ≫ p ≫ 0, the maps

M^{(k)} ← M_{p}^{(k)} → M_{[p,q]}^{(k)}

are equivalences of underived stacks.

Moreover, for fixed i, [CFK2] Theorem 4.1.1 implies that for q ≫ p ≫ 0, the maps

D_{[C]}^{i}(BM̃^{(k)}, M) ← D_{[C]}^{i}(BM̃_{p}^{(k)}, M) → D_{[C]}^{i}(BM̃_{[p,q]}^{(k)}, M)

are isomorphisms for all [C] and M. This does not give give suitable p, q for all i simultaneously.

However, if we restrict further to the open substack M^{(k),LCI} ⊂ M^{(k)} of local complete intersections, then the cotangent complex 𝕃^{X/𝔾_{m}⊗A} will be concentrated in chain degrees [0, 1]. Thus D_{[C]}^{i}(BM̃^{(k),LCI}, M) = 0 for i ∉ [−1, deg h], so for q ≫ p ≫ 0, we have weak equivalences

W̄ \underline{M̃}^{(k),LCI} ← W̄ \underline{M̃}_{p}^{(k),LCI} → W̄ \underline{M̃}_{[p,q]}^{(k),LCI}

of derived stacks, by applying Remark 1.30.

4. Moduli from cosimplicial groups

Since suitable DG Lie algebras can usually only be constructed in characteristic 0, we now work with cosimplicial groups, which form the first step towards a more general construction.

4.1. Cosimplicial groups.

Definition 4.1. Let cGp be the category of cosimplicial groups, and csGp the category of cosimplicial simplicial groups.

4.1.1. Maurer–Cartan.

Definition 4.2. Define MC : cGp → Set by

MC(G) := Z^{1}(G) = {ω ∈ G^{1} : σ^{0}ω = 1 ∂^{1}ω = ∂^{2}ω · ∂^{0}ω}.

Definition 4.3. Define \underline{MC} : csGp → 𝕊 by setting \underline{MC}(G) ⊂ ∏_{n≥0}(G^{n+1})^{Δ^{n}} to consist of elements (ω_{n})_{n≥0} satisfying

∂_{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.

Define MC : csGp → Set by MC(G) = \underline{MC}(G)_{0}, noting that this agrees with Definition 4.2 when G ∈ cGp.

Remark 4.4. Note that by the proof of [Pri2] Lemma 3.3,

MC(G) ≅ Hom_{c𝕊}(Δ, W̄ G),

for W̄ as in Definition 3.15, where the cosimplicial simplicial set Δ is given by the nsimplex Δ^{n} in cosimplicial level n. Thus MC(G) = Tot_{0}W̄ G, for Tot : c𝕊 → 𝕊 the total space functor of [GJ] Ch. VIII, originally defined in [BK] Ch. X.

In fact, W̄ has a left adjoint G (the loop group functor), and \underline{MC}(G) ≅ \underline{Hom}_{cGp}(G(Δ), W̄ G). However, W̄ is not simplicial right Quillen, so this does not equal \underline{Hom}_{c𝕊}(Δ, W̄ G) = Tot (W̄ G).

Definition 4.5. Given a cosimplicial group G, define the nth matching object M^{n}G to be the group

M^{n}G = {(g_{0}, g_{1}, …, g_{n−1}) ∈ (G^{n−1})^{n} | σ^{i}g_{j} = σ^{j−1}g_{i} ∀i < j}.

The Reedy matching map G^{n} → M^{n}G sends g to (σ^{0}g, σ^{1}g, …, σ^{n−1}g).

There is then a Reedy model structure on scGp (analogous to [GJ] §VII.4) in which a morphism f : G → H is a (trivial) fibration whenever the canonical maps

G^{n} → H^{n} ×_{M^{n}H} M^{n}G

are (trivial) fibrations in sGp for all n ≥ 0.

Definition 4.6. Given G ∈ cGp, define the cosimplicial normalisation by N_{c}^{n}G := G^{n} ∩ ⋂_{i=0}^{n−1} ker σ^{i}. If G is abelian, then we make N_{c}G into a cochain complex by setting

d_{c} := ∑_{i=0}^{n}(−1)^{i}∂^{i} : N_{c}^{n−1}G → N_{c}^{n}G.

Lemma 4.7. A morphism f : G → H in scGp is a (trivial) fibration whenever the maps

f^{n} : G^{n} → H^{n}

are all (trivial) fibrations in sGp.

Proof. First note that N_{c}^{n}G = ker(G^{n} → M^{n}G). Given (g_{0}, g_{1}, …, g_{n−1}) ∈ M^{n}G, we can functorially construct a pre-image.

First, set g(1) := ∂^{1}g_{0}; this has σ^{0}g(1) = g_{0}. Proceeding by induction, assume that we have constructed g(r) ∈ G^{n} with σ^{i}g(r) = g_{i} for all i < r. Set g_{i}(r) := σ^{i}g(r)^{−1} · g_{i}, so (g_{0}(r), g_{1}(r), …, g_{n−1}(r)) ∈ M^{n}G, with g_{i}(r) = 1 for all i < r. Now let g(r + 1) := g(r) · ∂^{r+1}g_{r}(r), noting that this satisfies the inductive hypothesis.

Thus we have an isomorphism G^{n} ≅ N^{n}G × M^{n}G as simplicial sets, and f : G → H is therefore a (trivial) fibration whenever N^{n}f : N^{n}G → N^{n}H is a (trivial) fibration in 𝕊 for all n. Since N^{n}f is a retraction of f^{n}, the result follows. □ Lemma 4.8. If f : G → H is a (trivial) fibration in scGp, then the map

\underline{MC}(f) : \underline{MC}(G) → \underline{MC}(H)

is a (trivial) fibration in 𝕊. In particular, if f : G → H is a trivial fibration, then MC(f) is surjective. Proof. In the proof of [Pri4] Proposition 6.7, a cofibrant object Φ is constructed in scGp, with the property that

\underline{MC}(G) ≅ \underline{Hom}(Φ, G),

where the simplicial sets \underline{Hom} come from a simplicial model structure. Since Φ is cofibrant, \underline{Hom}(Φ, −) is right Quillen, so has the properties claimed. □

Definition 4.9. Define the total complex functor Tot^{Π} from chain cochain complexes (i.e. bicomplexes) to chain complexes by

(Tot^{Π}V)_{n} := ∏_{a−b=n} V_{a}^{b},

with differential d := d^{s} + (−1)^{a}d_{c} on V_{a}^{b}. Lemma 4.10. If A ∈ csGp is abelian, then

MC(A) ≅ Z_{−1}(Tot^{Π}N^{s}N_{c}A)

and

π_{n}\underline{MC}(A) ≅ H_{n−1}(Tot^{Π}σ^{≥1}N^{s}N_{c}A),

where σ^{≥1} denotes brutal truncation in cochain degrees ≥ 1. Proof. This is a fairly straightforward application of the simplicial and cosimplicial Dold– Kan correspondences. Alternatively, we could appeal to Proposition 4.23, noting that A = exp(DN_{c}A). □

4.1.2. The gauge action. Definition 4.11. For G ∈ scGp, there is an action of the simplicial group G^{0} on the simplicial set \underline{MC}(G), called the gauge action, and given by writing

(g ⋆ ω)_{n} = ((∂^{1})^{n+1}(σ_{0})^{n}g) · ω_{n} · (∂^{0}(∂^{1})^{n}(σ_{0})^{n}g^{−1}),

as in [Pri2] Definition 3.8, with (σ_{0})^{n} denoting the canonical map G → G^{Δ^{n}}. Definition 4.12. Given G ∈ scGp, define the Deligne groupoid by 𝔇𝔢𝔩(G) := [MC(G)/G_{0}^{0}] In other words, 𝔇𝔢𝔩(G) has objects MC(G), and morphisms from ω to ω′ consist of {g ∈ G_{0}^{0} : g ⋆ ω = ω′}.

Define the derived Deligne groupoid to be the simplicial object in groupoids given by \underline{𝔇𝔢𝔩}(G) := [\underline{MC}(G)/G^{0}], so 𝔇𝔢𝔩(G) = \underline{𝔇𝔢𝔩}(G)_{0}.

Define the simplicial sets Del(G), \underline{Del}(G) ∈ 𝕊 to be the nerves B𝔇𝔢𝔩(G) and W̄ \underline{𝔇𝔢𝔩}(G), respectively. Lemma 4.13. If A ∈ csGp is abelian, then

π_{n}\underline{Del}(A) ≅ H_{n−1}(Tot^{Π}N^{s}N_{c}A),

whereas π_{1}Del(A) ≅ H^{0}(A_{0}), with

π_{0}Del(A) ≅ Z_{−1}(Tot^{Π}N^{s}N_{c}A)/d_{c}(A_{0}^{0}).

Proof. This is a straightforward consequence of Lemma 4.10. □

4.2. Moduli functors from cosimplicial groups.

Proposition 4.14. If G : Alg_{R} → cGp is a homogeneous functor, with each G^{n} formally smooth, then the functor

\underline{MC}(\underline{G}) : dN_{R}^{♭} → 𝕊

is homogeneous and formally quasi-smooth, so

MC(\underline{G}) : dN_{R}^{♭} → Set

is homogeneous and pre-homotopic.

Proof. Homogeneity is automatic, as \underline{MC} preserves arbitrary limits.

We can extend G^{n} to a functor G^{n} : dN_{R}^{♭} → Gp, given by G^{n}(A) := G^{n}(A_{0}). Then formal smoothness of G^{n} implies that the extended G^{n} is pre-homotopic. It is automatically formally quasi-presmooth, as all discrete morphisms are fibrations. Thus Proposition 1.27 implies that \underline{G}^{n} : dN_{R}^{♭} → 𝕊 is formally smooth, and hence formally quasi-smooth.

Lemma 4.7 therefore implies that \underline{G}(A) → \underline{G}(B) is a (trivial) fibration in scGp for all (acyclic) square-zero extensions A → B, and Lemma 4.8 then implies that \underline{MC}(\underline{G})(A) → \underline{MC}(\underline{G})(B) is a (trivial) fibration, as required. □

Proposition 4.15. If G : Alg_{R} → cGp is a homogeneous functor, with each G^{n} formally smooth, then the functor

\underline{Del}(\underline{G}) : dN_{R}^{♭} → 𝕊

is homogeneous and formally quasi-smooth, while

Del(\underline{G}) : sN_{R}^{♭} → 𝕊

is homogeneous, pre-homotopic and formally quasi-presmooth.

Proof. Homogeneity is immediate, combining Proposition 4.14 with the fact that \underline{G}^{0} is homogeneous. Now, take a square-zero (acyclic) extension f : A → B in dN_{R}^{♭}. Since \underline{MC}(\underline{G}) is formally quasi-smooth, the map \underline{MC}(\underline{G}(A)) → \underline{MC}(\underline{G}(B)) is a (trivial) fibration.

Combining [GJ] Lemma IV.4.8 with [CR1], this means that

W̄ [\underline{MC}(\underline{G}(A))/\underline{G}^{0}(A)] → W̄ [\underline{MC}(\underline{G}(B))/\underline{G}^{0}(A)]

is a (trivial) fibration in 𝕊. Now, the map

W̄ [\underline{MC}(\underline{G}(B))/\underline{G}^{0}(A)] → W̄ [\underline{MC}(\underline{G}(B))/\underline{G}^{0}(B)]

is a pullback of W̄ \underline{G}^{0}(A) → W̄ \underline{G}^{0}(B), which is a (trivial) fibration as \underline{G}^{0} is formally smooth. Composing the two morphisms above, we see that

\underline{Del}(\underline{G})(A) → \underline{Del}(\underline{G})(B)

is a (trivial) fibration, so \underline{Del}(\underline{G}) is formally quasi-smooth.

Meanwhile, pre-homotopicity of Del(\underline{G}) follows immediately from pre-homotopicity of MC(\underline{G}), while formal quasi-presmoothness of Del(\underline{G}) is an immediate consequence of formal smoothness of G^{0}, by [GJ] Ch.V. □

4.2.1. Cohomology.

Definition 4.16. Given a homogeneous, levelwise formally smooth functor G : Alg_{R} → cGp, a ring A ∈ Alg_{R}, an A-module M and ω ∈ MC(G(A)), define the cosimplicial module C_{ω}^{•}(G, M) to be the tangent space

C_{ω}^{n}(G, M) := T_{1}(G^{n}, M)

with operations on a ∈ C_{ω}^{n}(G, M) given by

σ^{i}a = σ_{G}^{i}a
∂^{i}a = \begin{cases}((∂_{G}^{2})^{n}ω)(∂_{G}^{0}a)((∂_{G}^{2})^{n}ω^{−1}) & i = 0 \\ ∂_{G}^{i}a & i ≥ 1. \end{cases}

Definition 4.17. For G, A, M, ω as above, define

H_{ω}^{i}(G, M) := H^{i}C_{ω}^{•}(G, M).

Lemma 4.18. Given a homogeneous, levelwise formally smooth functor G : Alg_{R} → scGp, A ∈ Alg_{R}, M ∈ dMod_{A} and ω ∈ MC(G_{0}(A)), the fibre of MC(G(A ⊕ M)) → MC(G(A)) over ω is canonically isomorphic to MC(C_{ω}^{•}(G, M)).

Proof. Given α ∈ MC(C_{ω}^{•}(G, M))), the associated element β ∈ MC(G(A ⊕ M)) is given by

β_{n} := α_{n}(∂_{G}^{2})^{n}ω ∈ T_{(∂_{G}^{2})^{n}ω}(G_{n}^{n+1}, M).
□

Lemma 4.19. If G : Alg_{R} → cGp is a homogeneous, levelwise formally smooth functor, with A ∈ Alg_{R} and M an A-module, then

D_{ω}^{i}(\underline{MC}(\underline{G}), M) ≅ D_{ω}^{i}(MC(\underline{G}), M) ≅ \begin{cases}H_{ω}^{i+1}(G, M) & i > 0 \\ Z^{1}C_{ω}^{•}(G, M) & i = 1. \end{cases}

Proof. By Lemma 4.10, for any L ∈ dMod_{A},

D_{ω}^{i}(\underline{MC}(\underline{G}), L) ≅ H_{i−1}(Tot^{Π}σ^{≥1}N^{s}N_{c}C_{ω}^{•}(\underline{G}, L)).

Thus we have a spectral sequence

E_{2}^{i,−j} = H^{i}(σ^{≥1}π_{j}C_{ω}^{•}(\underline{G}, L)) ⟹ D_{ω}^{i−j−1}(\underline{MC}(\underline{G}), L);

in the terminology of [Wei] p.142, this is a second quadrant spectral sequence, so is weakly convergent.

The simplicial abelian group C_{ω}^{n}(\underline{G}, L) is given in level i by C_{ω}^{n}(\underline{G}, (L^{Δ^{i}})_{0}). Moreover, C^{ω}(G, −) is an exact functor; left exactness follows from homogeneity, and right exactness from formal smoothness. Thus π_{j}C_{ω}^{n}(\underline{G}, L) ≅ C_{ω}^{n}(\underline{G}, π_{j}(L^{Δ^{•}})_{0}).

Now, when dN_{R}^{♭} = sN_{R}^{♭}, we have (L^{Δ^{n}})_{0} = L_{n}, so (L^{Δ^{•}})_{0} = L. When dN_{R}^{♭} = dg_{+}N_{R}^{♭}, N^{s}(L^{Δ^{•}})_{0} is weakly equivalent to L. In either case, π_{j}(L^{Δ^{•}})_{0} ≅ H_{j}L, so our spectral sequence is

H^{i}(σ^{≥1}C_{ω}^{•}(G, H_{j}L)) ⟹ D_{ω}^{i−j−1}(\underline{MC}(\underline{G}), L).

Taking L = M[−n], the spectral sequence degenerates, giving

H^{i}(σ^{≥1}C_{ω}^{•}(G, M)) ≅ D_{ω}^{i−n−1}(\underline{MC}(\underline{G}), M[−n]) = D_{ω}^{i−1}(\underline{MC}(\underline{G}), M),

completing the proof for \underline{MC}.

Now, T_{ω}(\underline{MC}, −) : dMod_{A} → 𝕊 preserves fibrations and trivial fibrations. Since M[−j]⊕ cone(M)[1 − j] is a path object for M[−j] when j ≥ 1 (recalling that M is a discrete Amodule), this means that T_{ω}(\underline{MC}, M[−j] ⊕ cone(M)[1 − j]) must be a path object for T_{ω}(\underline{MC}, M[−j]). Therefore for j ≥ 1,

π_{0}T_{ω}(\underline{MC}, M[−j]) = T_{ω}(MC, M[−j])/T_{ω}(MC, cone(M)[1 − j])
D_{ω}^{j}(\underline{MC}, M) = D_{ω}^{j}(MC, M).

The proof for j = 0 is even simpler, as M^{Δ^{n}} = M, so T_{ω}(\underline{MC}, M) = T_{ω}(MC, M). □

Lemma 4.20. If G : Alg_{R} → cGp is a homogeneous, levelwise formally smooth functor, with A ∈ Alg_{R} and M an A-module, then

D_{ω}^{i}(\underline{Del}(\underline{G})) ≅ D_{ω}^{i}(Del(\underline{G})) ≅ H_{ω}^{i+1}(G, M).

Proof. The description of D_{ω}^{i}(\underline{Del}(\underline{G})) follows with the same reasoning as Lemma 4.19, substituting Lemma 4.13 for Lemma 4.10.

Now, there is a morphism

\xymatrix{MC(\underline{G}) \ar[r] \ar[d] & Del(\underline{G}) \ar[r] \ar[d] & BG^{0} \ar[d] \\ \underline{MC}(\underline{G}) \ar[r] & \underline{Del}(\underline{G}) \ar[r] & _{W̄ \underline{G}_{0}}}

of fibration sequences, with the outer maps inducing isomorphisms on D^{i}, so Proposition 1.12 gives the required isomorphisms D_{ω}^{i}(Del(\underline{G})) ≅ D_{ω}^{i}(\underline{Del}(\underline{G})). □

4.3. Denormalisation.

Definition 4.21. Given a DGLA L in non-negative degrees, let DL be its cosimplicial denormalisation. Explicitly,

D^{n}L := ⨁_{\begin{matrix}m+s=n \\ 1≤j_{1}<...<j_{s}≤n \end{matrix}} ∂^{j_{s}} … ∂^{j_{1}}L^{m},

for formal symbols ∂^{j}. We then define operations ∂^{j} and σ^{i} using the cosimplicial identities, subject to the conditions that σ^{i}L = 0 and ∂^{0}v = dv − ∑_{i=1}^{n+1}(−1)^{i}∂^{i}v for all v ∈ L^{n}.

We now have to define the Lie bracket ⟦−, −⟧ from D^{n}L ⊗ D^{n}L to D^{n}L. Given a finite set I of distinct strictly positive integers, write ∂^{I} = ∂^{i_{s}} … ∂^{i_{1}}, for I = {i_{1}, … i_{s}}, with i_{1} < … < i_{s}. The Lie bracket is then defined on the basis by

⟦∂^{I}v, ∂^{J}w⟧ := \begin{cases}∂^{I∩J}(−1)^{(J\I,I\J)}[v, w] & v ∈ L^{|J\I|}, w ∈ L^{|I\J|}, \\ 0 & otherwise, \end{cases}

where for disjoint sets S, T of integers, (−1)^{(S,T)} is the sign of the shuffle permutation of S ⊔ T which sends the first |S| elements to S (in order), and the remaining |T | elements to T (in order). Beware that this formula cannot be used to calculate ⟦∂^{I}v, ∂^{J}w⟧ when 0 ∈ I ∪ J (for the obvious generalisation of ∂^{I} to finite sets I of distinct non-negative integers).

Of course, the denormalisation functor above extends a denormalisation functor D from non-negatively graded cochain complexes to cosimplicial complexes. The latter D is quasiinverse to the normalisation functor N_{c} of Definition 4.6.

Definition 4.22. Given a pro-nilpotent Lie algebra 𝔤, define Û(𝔤) to be the pro-unipotent completion of the universal enveloping algebra of 𝔤, regarded as a pro-object in the category of algebras. As in [Qui1] Appendix A, this is a pro-Hopf algebra, and we define exp(𝔤) ⊂ Û(𝔤) to consist of elements g with ε(g) = 1 and Δ(g) = g ⊗ g, for ε : Û(𝔤) → k the augmentation (sending 𝔤 to 0), and Δ : Û(𝔤) → Û(𝔤) ⊗ Û(𝔤) the comultiplication.

Since k is assumed to have characteristic 0, exponentiation gives an isomorphism from 𝔤 to exp(𝔤), so we may regard exp(𝔤) as having the same elements as 𝔤, but with multiplication given by the Campbell–Baker–Hausdorff formula.

If L is a DGLA in strictly positive degrees, observe that we can write L as the inverse limit L = \underleftarrow{lim}_{n} σ^{≤n}L of nilpotent DGLAs, where σ^{≤n} denotes brutal truncation. We may thus regard DL as the pro-nilpotent cosimplicial Lie algebra \underleftarrow{lim}_{n} D(σ^{≤n}L), so we can exponentiate to obtain exp(DL) := \underleftarrow{lim}_{n} exp(D(σ^{≤n}L)).

Lemma 4.23. Given a simplicial DGLA L_{•}^{•} in strictly positive cochain degrees, there is a canonical isomorphism

MC(exp(DL)) ≅ MC(Tot ΠN^{s}L),

Here, N^{s} is simplicial normalisation (as in Definition 1.1).

Proof. This is [Pri4] Theorem 6.23.

□

Definition 4.24. Given a DGLA L with gauge G_{L}, define the cosimplicial group D(exp(L), G_{L}) as follows.

D^{n}(exp(L), G_{L}) := exp(D^{n}L^{>0}) ⋊ G_{L},

(with G_{L} acting on exp(D^{n}L^{>0}) via the adjoint action ad), with operations

σ^{i}(a, g) = (σ^{i}a, g)
∂^{i}(a, g) = \begin{cases}(∂^{i}a, g) & i > 0 \\ (∂^{0}a · exp((∂^{2})^{n}Dg), g) & i = 0, \end{cases}

for (a, g) ∈ D^{n}(exp(L), G_{L}), and Dg ∈ L^{1}.

Remark 4.25. If L is a nilpotent DGLA in non-negative degrees and G_{L} = exp(L^{0}) as in Example 3.5, then observe that D(exp(L), G_{L}) ≅ exp(DL), with the isomorphism given in level n by

(a, g) ↦ a · (∂^{1})^{n}g.

The only difficult part of the comparison is checking that the isomorphism preserves ∂^{0}. This follows because for v ∈ L^{0}, we have ∂^{0}v = ∂^{1}v + dv. Since ⟦dv, ∂^{1}v⟧ = 0, this gives

exp(∂^{0}v) = exp(∂^{1}v) + d exp(v)
= (1 + (d exp(v)) exp(−v)) · exp(∂^{1}v)
= (1 + D exp(v)) · exp(∂^{1}v)
= exp(D exp(v)) · exp(∂^{1}v).

Lemma 4.26. Given a simplicial DGLA L with gauge G_{L} for L_{0}, the isomorphism MC(D(exp(L), G_{L})) ≅ MC(Tot ΠN^{s}L) of Lemma 4.23 is G_{L}-equivariant for the respective gauge actions.

Proof. This is a consequence of the proof of [Pri2] Theorem 4.44, which deals with the case when G_{L} = exp(L^{0}) is defined as in Example 3.5. □

Corollary 4.27. Given a DGLA L over R, with gauges G_{L}(A) for L ⊗_{R} A, functorial in A ∈ Alg_{R}, there are canonical isomorphisms

MC(L ⊗_{R} −) ≅ MC(D(exp(L ⊗_{R} −), G_{L}(−))
[MC(L ⊗_{R} −)/G_{L}(−)] ≅ 𝔇𝔢𝔩(D(exp(L ⊗_{R} −), G_{L}(−))

of functors on dg_{+}Alg_{R}.

Thus cosimplicial groups generalise DGLAs, with the added advantage that they can also give functors on sAlg_{R}, and hence work in all characteristics.

4.4. Sheafification. Another advantage of cosimplicial groups over DGLAs is that they have a good notion of sheafification.

Definition 4.28. Given a levelwise formally smooth, homogeneous functor G : Alg_{R} → cGp preserving finite products, and some class P of covering morphisms in Alg_{R}, define the sheafification G^{♯} of G with respect to P by first defining a cosimplicial commutative

A-algebra (B/A)^{•} for every P-covering A → B, as (B/A)^{n} := \overbrace{B ⊗_{A} B ⊗_{A} … ⊗_{A} B}^{n+1}, then setting

G^{♯}(A) = diag \underrightarrow{lim}_{B} G((B/A)^{•}),

where diag is the diagonal functor (diag X)^{n} = X^{nn} from bicosimplicial groups to cosimplicial groups.

Definition 4.29. Given a groupoid-valued functor Γ : dN_{R}^{♭} → Gpd, and a class P of covering morphisms in Alg_{R}, define Γ^{♯} : dN_{R}^{♭} → Gpd to be the stackification of the groupoid presheaf Γ in the strict P-topology of Definition 2.17.

Lemma 4.30. For G and P as above, there is a canonical morphism

𝔇𝔢𝔩(\underline{G})^{♯} → 𝔇𝔢𝔩(\underline{G}^{♯})

of groupoid-valued functors on dN_{R}^{♭}, inducing an equivalence

π^{0}𝔇𝔢𝔩(\underline{G})^{♯} → π^{0}𝔇𝔢𝔩(\underline{G}^{♯}).

Proof. An object of 𝔇𝔢𝔩(\underline{G})^{♯} is a pair (ω, g) ∈ MC(\underline{G}(B⊗_{A_{0}} A))×G^{0}(B⊗_{A_{0}} B), for A_{0} → B a P-covering, satisfying the following conditions:

(1) g ⋆ (pr_{1}^{∗}ω) = (pr_{0}^{∗}ω) ∈ MC(\underline{G}(A ⊗_{A_{0}} B ⊗_{A_{0}} B)),

(2) pr_{02}^{∗}g = (pr_{01}^{∗}g) · (pr_{12}^{∗}g) ∈ G^{0}(B ⊗_{A_{0}} B ⊗_{A_{0}} B), and we now describe the image of (ω, g).

First, form the cosimplicial A_{0}-algebra (B/A_{0})^{•} as in Definition 4.28. We now map (B, ω, g) to the object ω′ of MC(diag \underline{G}(A ⊗_{A_{0}} (B/A_{0})^{•})) given by

ω′_{n} := (pr_{01}^{∗}(∂^{1})_{G}^{n+1}g) · pr_{1}^{∗}ω_{n} ∈ G^{n+1}((A^{Δ^{n}})_{0} ⊗_{A_{0}} (B/A_{0})^{n+1}).

An isomorphism from (B, ω, g) to (C, ν, h) is a P-covering B ⊗_{A_{0}} C → D with A → D a P-covering, together with an element α ∈ G^{0}(D) such that α ⋆ ω = ν ∈ MC(\underline{G}(A ⊗_{A_{0}} D)), with (pr_{0}^{∗}α) · g = h · (pr_{1}^{∗}α) ∈ G^{0}(D ⊗_{A_{0}} D). We just map this to α ∈ G^{0}(D).

To see that this induces an equivalence 𝔇𝔢𝔩(\underline{G})^{♯}(A) → 𝔇𝔢𝔩(\underline{G}^{♯}(A)) for A ∈ Alg_{R}, we appeal to [Pri4] Lemma 1.21, which shows that objects of

MC(diag G((B/A)^{•}))

correspond to pairs (ω, g) ∈ MC(G(B))×MC(G^{0}((B/A)^{•})) satisfying ∂_{B}^{1}ω·∂_{G}^{0}g = ∂^{1}g·∂_{B}^{0}ω. Since ∂_{B}^{1}ω = pr_{0}^{∗}ω and ∂_{B}^{0}ω = pr_{1}^{∗}ω, this amounts to saying that g and ω satisfy condition (1) above, while condition (2) is equivalent to saying that g ∈ MC(G^{0}((B/A)^{•}). Morphisms in MC(diag G((B/A)^{•})) are given by the gauge actions of G^{0}(B^{0}), so the equivalence of groupoids follows. □

Now recall that P-covering morphisms are all assumed faithfully flat.

Lemma 4.31. Take G : Alg_{R} → cGp satisfying the conditions of Definition 4.28, a ring A ∈ Alg_{R}, an A-module M, and an object of 𝔇𝔢𝔩(G)^{♯}(A) represented by (ω, g) for ω ∈ MC(G(B)). If the maps

H_{ω}^{∗}(G, M ⊗_{A} B) ⊗_{B} B′ → H_{ω}^{∗}(G, M ⊗_{A} B′)

are isomorphisms for all P-coverings B → B′, then the morphism α of Lemma 4.30 induces an isomorphism

D_{(ω,g)}^{∗}(B𝔇𝔢𝔩(\underline{G})^{♯}, M) → D_{α(ω,g)}^{∗}(B𝔇𝔢𝔩(\underline{G}^{♯}), M).

Proof. We begin by calculating the cohomology groups D_{(ω,g)}^{∗}(B𝔇𝔢𝔩(\underline{G})^{♯}, M). It follows from Lemma 4.20 that these are given by first taking a cosimplicial A-module K(B′) given by the equaliser of

\xymatrix{C_{ω}^{•}(G, M ⊗_{A} B′) \ar[r]^{^{pr∗ 1}} \ar[r]_{_{ad_{g}pr∗ 0}} & C_{pr_{1}^{∗}ω}^{•}(G, M ⊗_{A} B′ ⊗_{A} B′),}

then getting

D_{(ω,g)}^{i}(B𝔇𝔢𝔩(\underline{G})^{♯}, M) ≅ \underrightarrow{lim}_{B′} H^{i+1}K(B′),

where B′ ranges over all P-hypercoverings B → B′.

Now, the requirement that H_{ω}^{∗} commute with base change ensures that ad_{g} gives an effective descent datum on cohomology, giving an isomorphism

H^{∗}K(B′) ⊗_{A} B′ ≅ H_{ω}^{∗}(G, M ⊗_{A} B′).

Thus taking the colimit over B′ does not affect the calculation, so

D_{(ω,g)}^{i}(B𝔇𝔢𝔩(\underline{G})^{♯}, M) ≅ H^{i+1}K(B).

Meanwhile, g allows us to extend the fork above to form a bicosimplicial complex Č^{•}(B′/A, C_{ω}^{•}(G, M)), with

Č^{i}(B′/A, C_{ω}^{•}(G, M)) = C_{pr_{i}^{∗}ω}^{•}(G, M ⊗_{A} (B′/A)^{i})

and horizontal cohomology Ȟ^{0}(B′/A, C_{ω}^{•}(G, M)) = K(B′). Lemma 4.20 then shows that

D_{α(ω,g)}^{i}(B𝔇𝔢𝔩(\underline{G}^{♯}), M) ≅ \underrightarrow{lim}_{B′} H^{i+1}(diag Č^{•}(B′/A, C_{ω}^{•}(G, M))),

and the Eilenberg–Zilber Theorem allows us to replace diag with the total complex functor Tot .

Now, there are canonical maps K(B)⊗_{A} (B′/A)^{n} → Č^{n}(B′/A, C_{ω}^{•}(G, M)), and we know that these give isomorphisms on cohomology, so

D_{α(ω,g)}^{i}(B𝔇𝔢𝔩(\underline{G}^{♯}), M) ≅ \underrightarrow{lim}_{B′} H^{i+1}(Tot K(B) ⊗_{A} (B′/A)^{•}).

Since H^{j}((B′/A)^{•}) = 0 for all j > 0, this becomes

D_{α(ω,g)}^{i}(B𝔇𝔢𝔩(\underline{G}^{♯}), M) ≅ \underrightarrow{lim}_{B′} H^{i+1}K(B),

as required. □

Remark 4.32. In particular, this means that the groupoid-valued functor G of Definition 3.29 can be replaced by a functor coming straight from a cosimplicial group-valued functor. Explicitly, set G_{[p,q]} := D(exp(L_{[p,q]}), G_{L_{[p,q]}}); then G := \underrightarrow{lim}_{p} \underleftarrow{lim}_{q} G_{[p,q]} satisfies the condi-

tions of Definition 4.28, so Lemma 4.30 gives a map G → 𝔇𝔢𝔩(G^{♯}), inducing an equivalence on π^{0}, and isomorphisms on D^{i} for all points of M ⊂ π^{0}G.

Remark 4.33. In Definition 4.28, instead of just taking hypercovers A_{0} → B^{•} coming from P-covering morphisms A_{0} → B, we could have taken the filtered colimit over the category of all simplicial P-hypercovers Spec Ã^{•} → Spec A. This corresponds to a kind of hypersheafification, rather than just sheafification, and all the results above still carry over, by faithfully flat descent.

4.5. Derived moduli of G-torsors. We now show how cosimplicial groups can govern derived moduli of torsors. Fix a smooth algebraic group space G over R, and a Deligne– Mumford stack X over R.

Definition 4.34. Define C_{ét}^{•}(X, G) : Alg_{R} → cGp as follows. Given A ∈ Alg_{R}, let hyp_{ét}(X, A) be the inverse category whose objects are simplicial étale hypercovers Y_{•} of X × Spec A, and whose morphisms are simplicial étale hypercovers. Then set

C_{ét}^{n}(X, G)(A) := \underrightarrow{lim}_{Y_{•}∈hyp_{ét}(X,A)} Hom(Y_{n}, G).

Remark 4.35. Since G is finitely presented, we can work just as well with pro-étale hypercovers. This has the advantage that Lemma 2.13 can be applied to provide a weakly initial object among simplicial hypercovers, giving a smaller, non-functorial, model for C_{ét}^{•}(X, G)(A).

Lemma 4.36. For A ∈ Alg_{R}, the groupoid 𝔇𝔢𝔩(C_{ét}^{n}(X, G)(A)) is equivalent to the groupoid of G-torsors on X × Spec A.

Proof. An object of MC(C_{ét}^{•}(X, G)(A)) is a descent datum ω ∈ Hom(Y_{1}, G), with the Maurer–Cartan relations giving the gluing conditions. Thus ω gives rise to an étale Gtorsor 𝔹_{ω} on X × Spec A, and it is straightforward to check that the gauge action of Hom(Y_{0}, G) corresponds to isomorphisms of torsors. Every torsor is trivialised by some étale cover, so this functor is an equivalence. □

Lemma 4.37. Given A ∈ Alg_{R}, an A-module M, and ω ∈ MC(C_{ét}^{•}(X, G)(A)) corresponding to a G-torsor 𝔹_{ω} on X × Spec A, there are canonical isomorphisms

H_{ω}^{i}(C_{ét}^{•}(X, G), M) ≅ H_{ét}^{i}(X × Spec A, M ⊗_{A} ad𝔹_{ω}),

where ad𝔹_{ω} is the adjoint bundle

(𝔤 ⊗_{R} 𝒪_{X} ⊗_{R} A) ×_{G(𝒪_{X}⊗_{R}A)} 𝔹_{ω},

for 𝔤 the Lie algebra of G, equipped with its adjoint G-action.

Proof. First, observe that

G(𝒪_{X} ⊗_{R} (A ⊕ M)) ≅ G(𝒪_{X} ⊗_{R} A) ⋉ (𝔤 ⊗_{R} 𝒪_{X} ⊗_{R} M),

which gives functorial isomorphisms

C_{ω}^{i}(C_{ét}^{•}(X, G), M) ≅ \underrightarrow{lim}_{Y_{•}} Γ(Y_{i}, M ⊗_{A} ad𝔹_{ω}).

Since étale hypercovers compute cohomology, this gives the required isomorphism. □

Proposition 4.38. The functor \underline{Del}(\underline{C_{ét}^{•}(X, G)}) is canonically weakly equivalent to the derived stack of étale derived G-torsors on X from [Pri6] Example 3.38.

Proof. Combining Lemma 4.20 and Proposition 4.15 with Remark 1.30, it suffices to construct a functorial natural transformation from 𝔇𝔢𝔩(\underline{C_{ét}^{•}(X, G)})(A) to the ∞-groupoid Tors(X, G)(A) of G-torsors on X × Spec A, and to show that this is an equivalence for A ∈ Alg_{R}, inducing isomorphisms on cohomology groups D^{i}.

Our first key observation is that for A ∈ dN_{R}^{♭}, the ring (A^{Δ^{n}})_{0} is a nilpotent extension of H_{0}A, so its étale site is isomorphic to that of A_{0}. In particular, every simplicial étale hypercover of X × Spec (A^{Δ^{n}})_{0} is of the form Y_{•} ⊗_{A_{0}} (A^{Δ^{n}})_{0}, for Y_{•} a simplicial étale hypercover of X × Spec A_{0}.

Thus an element ω ∈ MC(\underline{C_{ét}^{•}(X, G)}(A)) lies in MC applied to the simplicial cosimplicial

group Γ(Y_{•} ⊗_{A_{0}} (A^{Δ^{•}}), G) given by (i, j) ↦ Hom(Y_{i} ⊗_{A_{0}} (A^{Δ^{j}})_{0}, G), for some simplicial étale hypercover Y_{•} of X × Spec A_{0}. Now,

MC(Γ(Y_{•} ⊗_{A_{0}} (A^{Δ^{•}}), G)) = Hom_{c𝕊}(Δ, W̄ Γ(Y_{•} ⊗_{A_{0}} (A^{Δ^{•}}), G))
≅ Hom_{s Pr(dN_{R}^{♭})}(Y_{•} ×_{Spec A_{0}} Spec A, W̄ \underline{G}),

where s Pr(dN_{R}^{♭}) denotes the category of functors dN_{R}^{♭} → 𝕊.

For a simplicial group Γ, and W̄ as in Definition 3.15, there is a universal principal Γ-space W Γ over W̄ Γ, as in [GJ] §V.4, given by W Γ = W̄ [Γ/Γ], (whereas W̄ Γ = W̄ [•/Γ], regarding a group as a groupoid on one object). Thus W Γ has a group structure inherited from Γ, and Γ = W̄ [Γ/{1}] is a subgroup; the Γ-action on W Γ is then given by left multiplication, with W̄ Γ = Γ\W Γ.

Thus we may associate the \underline{G}-space P_{ω} := (Y_{•} ×_{Spec A_{0}} Spec A) ×_{W̄ \underline{G}} W \underline{G} to ω. Since \underline{G} is the derived stack RG associated to G, the derived stack RP_{ω} associated to P_{ω} is a derived RG-torsor on R(Y_{•} ×_{Spec A_{0}} Spec A) ≃ R(X ×_{Spec A_{0}} Spec A), so we have defined our functor on objects.

Now, the constant group Γ_{0} is a simplicial subgroup of Γ, giving a simplicial group homomorphism W Γ_{0} → W Γ. Moreover, W Γ_{0} is the 0-coskeleton cosk_{0}Γ_{0} of Γ_{0}, so Hom_{𝕊}(Y, W Γ_{0}) ≅ Hom_{Set}(Y_{0}, Γ_{0}). From the proof of [Pri2] Proposition 3.9, the gauge action (Definition 4.11) of Hom(Y_{0}, Γ_{0}) on MC(C^{•}(Y, Γ)) = Hom(Y, W̄ Γ) corresponds to the right multiplication by W Γ_{0} on W̄ Γ = Γ\W Γ.

Hence, given ω, ω′ ∈ Hom_{s Pr(dN_{R}^{♭})}(Y_{•} ×_{Spec A_{0}} Spec A, W̄ \underline{G}) and g ∈ Hom_{Pr(dN_{R}^{♭})}(Y_{0}, G) with g ⋆ ω = ω′, this means that ω′(y) = ω(y) · g(y)^{−1}. We therefore construct an isomorphism P_{ω} → P_{ω′} by (y, w) ↦ (y, w·g(y)^{−1}), for y ∈ Y_{•} ×_{Spec A_{0}} Spec A and w ∈ W \underline{G}.

We have thus constructed a morphism 𝔇𝔢𝔩(\underline{C_{ét}^{•}(X, G)}) → Tors(X, G). The nerve of Tors(X, G) is weakly equivalent to the derived stack Hom(X, BG) from [Pri6] Example 3.38. Since 𝔤 = Hom_{R}(e^{∗}Ω_{G/R}, R), the calculation of [Pri6] Example 3.38 gives

D_{𝔹}^{i}(Hom(X, BG), M) ≅ H_{ét}^{i+1}(X × Spec A, M ⊗_{A} ad𝔹_{ω}).

On taking nerves, we thus get maps

\underline{Del}(\underline{C_{ét}^{•}(X, G)}) ← Del(\underline{C_{ét}^{•}(X, G)}) → W̄ Tors(X, G) ≃ Hom(X, BG),

all of which give isomorphisms on D^{i} and equivalences on π^{0}. Remark 1.30 thus shows that

\underline{Del}(\underline{C_{ét}^{•}(X, G)}) ≃ W̄ \underline{Del(\underline{C_{ét}^{•}(X, G)})} ≃ Hom(X, BG).
□

Remark 4.39. If G = GL_{r}, this gives us a construction for derived moduli of rank r vector bundles on X. If instead G = SL_{r}, we get derived moduli of determinant 1, rank r vector bundles.

5. Moduli from quasi-comonoids

Although cosimplicial groups can be used to construct derived moduli in all characteristics for many problems, they are insufficiently flexible to arise in the generality we need. Instead, we use the quasi-comonoids introduced in [Pri4]

5.1. Quasi-comonoids. The following is a special case of [Pri4] Lemma 1.5:

Definition 5.1. Define a quasi-comonoid E to consist of sets E^{n} for n ∈ ℕ_{0}, together with maps

\begin{matrix}∂^{i} : E^{n} → E^{n+1} & 1 ≤ i ≤ n \\ σ^{i} : E^{n} → E^{n−1} & 0 ≤ i < n, \end{matrix}

an associative product ∗ : E^{m} × E^{n} → E^{m+n}, with identity 1 ∈ E^{0}, such that: (1) ∂^{j}∂^{i} = ∂^{i}∂^{j−1} i < j.

(2) σ^{j}σ^{i} = σ^{i}σ^{j+1} i ≤ j.

(3) σ^{j}∂^{i} = \begin{cases}∂^{i}σ^{j−1} & i < j \\ id & i = j, i = j + 1 \\ ∂^{i−1}σ^{j} & i > j + 1 \end{cases} .

(4) ∂^{i}(e) ∗ f = ∂^{i}(e ∗ f).

(5) e ∗ ∂^{i}(f) = ∂^{i+m}(e ∗ f), for e ∈ E^{m}.

(6) σ^{i}(e) ∗ f = σ^{i}(e ∗ f).

(7) e ∗ σ^{i}(f) = σ^{i+m}(e ∗ f), for e ∈ E^{m}.

Denote the category of quasi-comonoids by QM^{∗}.

Example 5.2. Given a cosimplicial group G (or even a cosimplicial monoid G), there is an associated quasi-comonoid E(G) given by E(G)^{n} = G^{n}, with identity 1 ∈ G^{0}, operations ∂_{E(G)}^{i} = ∂_{G}^{i}, σ_{E(G)}^{i} = σ_{G}^{i}, and Alexander-Whitney product

g ∗ h = ((∂_{G}^{m+1})^{n}g) · ((∂_{G}^{0})^{m}h),

for g ∈ G^{m}, h ∈ G^{n}.

5.1.1. Maurer–Cartan. We now construct a Maurer–Cartan functor analogous to the one for cosimplicial groups.

Definition 5.3. Define MC : QM^{∗}(Set) → Set by

MC(E) = {ω ∈ E^{1} : σ^{0}ω = 1, ∂^{1}ω = ω ∗ ω}.

Now let QM^{∗}(𝕊) be the category of simplicial objects in QM^{∗}. The following is [Pri4] Definition 3.5:

Definition 5.4. Define \underline{MC} : QM^{∗}(𝕊) → 𝕊 by

\underline{MC}(E) ⊂ ∏_{n≥0}(E^{n+1})^{I^{n}}

(where I = Δ^{1} ∈ 𝕊), consisting of those \underline{ω} satisfying:

ω_{m}(s_{1}, …, s_{m}) ∗ ω_{n}(t_{1}, …, t_{n}) = ω_{m+n+1}(s_{1}, …, s_{m}, 0, t_{1}, …, t_{n});
∂^{i}ω_{n}(t_{1}, …, t_{n}) = ω_{n+1}(t_{1}, …, t_{i−1}, 1, t_{i}, …, t_{n});
σ^{i}ω_{n}(t_{1}, …, t_{n}) = ω_{n−1}(t_{1}, …, t_{i−1}, min{t_{i}, t_{i+1}}, t_{i+2}, …, t_{n});
σ^{0}ω_{n}(t_{1}, …, t_{n}) = ω_{n−1}(t_{2}, …, t_{n});
σ^{n}ω_{n}(t_{1}, …, t_{n}) = ω_{n−1}(t_{1}, …, t_{n−1}),
σ^{0}ω_{0} = 1.

Define MC : QM^{∗}(𝕊) → Set by MC(E) = \underline{MC}(E)_{0}, noting that this agrees with Definition 5.3 when E ∈ QM^{∗}(Set).

Definition 5.5. Given a quasi-comonoid E, define the nth matching object M^{n}E to be the set

M^{n}E = {(e_{0}, e_{1}, …, e_{n−1}) ∈ (E^{n−1})^{n} | σ^{i}e_{j} = σ^{j−1}e_{i} ∀i < j}.

The Reedy matching map E^{n} → M^{n}W sends e to (σ^{0}e, σ^{1}e, …, σ^{n−1}e).

Then [Pri4] Lemma 3.2 gives a model structure on QM^{∗}(𝕊) in which a morphism E → F is a (trivial) fibration whenever the canonical maps

E^{n} → F^{n} ×_{M^{n}F} M^{n}E

are (trivial) fibrations in 𝕊 for all n ≥ 0.

Lemma 5.6. If f : E → F is a (trivial) fibration in QM^{∗}(𝕊), then the map

\underline{MC}(f) : \underline{MC}(E) → \underline{MC}(F)

is a (trivial) fibration in 𝕊. In particular, if f : E → F is a trivial fibration, then MC(f) is surjective.

Proof. This is a direct consequence of [Pri4] Corollary 3.12, which shows that \underline{MC} is a right Quillen functor. □

Lemma 5.7. There is an equivalence between the category of abelian group objects in QM^{∗}, and the category cAb of cosimplicial complexes of abelian groups.

Proof. This is [Pri4] Lemma 4.1. The equivalence is given by the functor E of Example 5.2. □

Lemma 5.8. For a cosimplicial simplicial abelian group A, there are canonical isomorphisms

π_{n}\underline{MC}(E(A)) ≅ H_{n−1}(Tot^{Π}σ^{≥1}N^{s}N_{c}A),

where σ^{≥1} denotes brutal truncation in cochain degrees ≥ 1, Tot^{Π} is the product total functor of Definition 4.9, and N^{s} and N_{c} are the normalisation functors of Definitions 1.1 and 4.6.

Proof. This is [Pri4] Proposition 4.12. □

5.1.2. The gauge action.

Definition 5.9. For E ∈ QM^{∗}(𝕊), let (E^{0})_{n}^{×} ⊂ E_{n}^{0} be the submonoid of invertible elements. There is then an action of the simplicial group (E^{0})^{×} on the simplicial set \underline{MC}(E), called the gauge action, and given by setting

(g ⋆ ω)_{n} = (σ_{0})^{n}g ∗ ω_{n} ∗ (σ_{0})^{n}g^{−1},

as in [Pri2] Definition 3.8, with (σ_{0})^{n} denoting the canonical map (E^{0}) → (E^{0})^{Δ^{n}}.

Definition 5.10. Given E ∈ QM^{∗}(𝕊), define the Deligne groupoid by 𝔇𝔢𝔩(E) := [MC(E)/(E_{0}^{0})^{×}] In other words, 𝔇𝔢𝔩(E) has objects MC(E), and morphisms from ω to ω′ consist of {g ∈ (E_{0}^{0})^{×} : g ∗ ω = ω′ ∗ g}.

Define the derived Deligne groupoid to be the simplicial object in groupoids given by \underline{𝔇𝔢𝔩}(E) := [\underline{MC}(E)/(E^{0})^{×}], so 𝔇𝔢𝔩(E) = \underline{𝔇𝔢𝔩}(E)_{0}.

Define the simplicial sets Del(E), \underline{Del}(E) ∈ 𝕊 to be the nerves B𝔇𝔢𝔩(E) and W̄ \underline{𝔇𝔢𝔩}(E), respectively.

Lemma 5.11. If A is a simplicial cosimplicial abelian group, then

π_{n}\underline{Del}(E(A)) ≅ H_{n−1}(Tot^{Π}N^{s}N_{c}A),

whereas π_{1}Del(E(A)) ≅ H^{0}(A_{0}), with

π_{0}Del(E(A)) ≅ Z_{−1}(Tot^{Π}N^{s}N_{c}A)/d_{c}(A_{0}^{0}).

Proof. This is [Pri4] Proposition 4.13. □

5.1.3. Comparison with cosimplicial groups.

Lemma 5.12. Given a simplicial cosimplicial group G, and associated simplicial quasicomonoid E(G) as in Example 5.2, there are G^{0}-equivariant weak equivalences

\underline{MC}(E(G)) ≃ \underline{MC}(G)

and hence weak equivalences

Del(E(G)) ≃ Del(G)

in 𝕊, functorial in objects G ∈ csGp. Here, the functors \underline{MC} on the left and right are those from Definitions 5.4 and 4.3 respectively, while the functors Del are those from Definitions 5.10 and 4.12.

Proof. This is [Pri4] Propositions 6.8 and 6.11. □

5.2. Constructing quasi-comonoids.

5.2.1. Monads.

Definition 5.13. A monad (or triple) on a category B is a monoid in the category of endofunctors of B (with the monoidal structure given by composition of functors).

Example 5.14. Given an adjunction

\xymatrix{D \ar[r]^{U}_{⊤} & B \ar[l]^{F}}

with unit η : id → UF and co-unit ε : F U → id, the associated monad on B is given by ⊤ = UF, with unit η : id → ⊤ and multiplication μ := UεF : ⊤^{2} → ⊤.

Definition 5.15. Given a monad (⊤, η, μ) on a category B, define the category of ⊤- algebras, B^{⊤}, to have objects

⊤B ⟶^{θ} B,

such that θ ∘ η_{B} = id and θ ∘ ⊤θ = θ ∘ μ_{B} : ⊤^{2}B → B.

A morphism

g : (⊤B_{1} ⟶^{θ} B_{1}) → (⊤B_{2} ⟶^{ϕ} B_{2})

of ⊤-algebras is a morphism g : B_{1} → B_{2} in B such that ϕ ∘ ⊤g = g ∘ θ.

Example 5.16. Let ⊤ := Symm_{R} be the symmetric functor on Mod_{R}, with unit η_{M} : M → ⊤M given by the inclusion of degree 1 monomials, and μ_{M} : ⊤^{2}M → ⊤M given by expanding out polynomials of polynomials. Then (Mod_{R})^{⊤} is equivalent to the category of unital commutative R-algebras.

Given a monad (⊤, μ, η) on a category B, and an object B ∈ B, there is a quasi-comonoid E(B) given by

E^{n}(B) = Hom_{B}(⊤^{n}B, B)

in (Set, ×), with product g ∗ h = g ∘ ⊤^{n}h, and for g ∈ E^{n}(B),

∂^{i}(g) = g ∘ ⊤^{i−1}μ_{⊤^{n−i}B}
σ^{i}(g) = g ∘ ⊤^{i}η_{⊤^{n−i−1}B}.

Note that these constructions also all work for a comonad (⊥, Δ, ε), by contravariance. There is even a generalisation to bialgebras for a distributive monad-comonad pair: see [Pri4] Proposition 2.12.

Lemma 5.17. Given an object B ∈ B, the set of ⊤-algebra structures on B is MC(E(B)), while 𝔇𝔢𝔩(E(B)) is equivalent to the groupoid of ⊤-algebras overlying B.

Proof. This follows immediately from the explicit description in Definition 5.3. □

5.2.2. Diagrams.

Definition 5.18. Given a category B equipped with a monad ⊤, together with K ∈ 𝕊 and a map B : K_{0} → Ob B, define the quasi-comonoid E_{K}(B) by

E^{n}(B/K) = ∏_{x∈K_{n}} Hom_{B}(⊤^{n}B((∂_{0})^{n}x), B((∂_{1})^{n}x)),

with operations

∂^{i}(e)(x) := e(∂_{i}x) ∘ ⊤^{i−1}μ_{⊤^{n−i}B((∂_{0})^{n+1}x)}
σ^{j}(e)(y) := e(σ_{j}y) ∘ ⊤^{i}η_{⊤^{n−i−1}B(∂_{0})^{n−1}x)},
(f ∗ e)(z) := f((∂_{m+1})^{n}z) ∘ ⊤^{m}e((∂_{0})^{m}z),

for f ∈ E^{m}(B/K), e ∈ E^{n}(B/K).

Definition 5.19. Given a category B equipped with a monad ⊤, together with a small category 𝕀 and a map B : Ob 𝕀 → Ob B, define the quasi-comonoid E(B/𝕀) by

E(B/𝕀) := E(B/B𝕀),

where B𝕀 is the nerve of 𝕀.

Lemma 5.20. Given B, ⊤, 𝕀 and B : Ob 𝕀 → Ob B as above,

MC(E(B/𝕀))

is isomorphic to the set of functors 𝔻 : 𝕀 → B^{⊤} with U𝔻(i) = B(i) for all i ∈ 𝕀, where U : B^{⊤} → B is the forgetful functor.

Meanwhile 𝔇𝔢𝔩(E(B)) is equivalent to the groupoid of diagrams 𝔻 : 𝕀 → B^{⊤} with 𝔻(i) overlying B(i) for all i ∈ 𝕀.

Proof. This is [Pri4] Lemma 1.36. Given ω ∈ MC(E(B/𝕀)), the algebra structure ⊤B(i) → B(i) is given by ω(i ⟶^{id} i) ∈ Hom_{B}(⊤^{n}B(i), B(i)), while the morphism 𝔻(f) : 𝔻(i) → 𝔻(j)

is given by ω(i ⟶^{f} j) ∘ η_{B(i)} ∈ Hom_{B}(B(i), B(j)). □

Corollary 5.21. Take a category B equipped with a monad ⊤, together with a small category 𝕀 and a subcategory 𝕁. Assume that we have a functor 𝔽 : 𝕁 → B^{⊤}, and a map B : Ob 𝕀 → Ob B extending UOb 𝔽 : Ob J → Ob B.

For ω_{𝔽} ∈ MC(E(B|_{𝕁}/𝕁)) corresponding to 𝔽 in Lemma 5.20, we can form a quasicomonoid E by

E^{n} := E^{n}(B/𝕀) ×_{E^{n}(B|_{𝕁}/𝕁)} \{\overbrace{ω_{𝔽} ∗ ω_{𝔽} ∗ … ∗ ω_{𝔽}}^{n}\}.

Then MC(E) is isomorphic to the set of functors 𝔻 : 𝕀 → B^{⊤} with U𝔻(i) = B(i) for all i ∈ 𝕀 and 𝔻|_{𝕁} = 𝔽, while 𝔇𝔢𝔩(E) is the groupoid of such functors.

Proof. This follows immediately from the observation that MC preserves limits. □

Example 5.22. The main applications of these results are to moduli of morphisms. In that case, 𝕀 is the category 0 → 1, B𝕀 ≅ Δ^{1}, and we define E(B_{0}, B_{1}) := E(B/𝕀), where B : Ob 𝕀 → B is given by B(i) := B_{i}.

The category 𝕁 will be ∅, {0}, {1} or {0, 1} depending on which endpoints we wish to fix (if any), so the quasi-comonoid E of the corollary is the fibre of E(B_{0}, B_{1}) → ∏_{j∈𝕁} E(B_{j}) over powers of ω_{j}.

Remark 5.23. In [Pri4] Definition 3.25 the construction E(B/𝕀) (for a simplicial category B equipped with a monad ⊤) is used to extend the simplicial set \underline{MC}(E) to a bisimplicial set MC(E). Explicitly, MC(E)_{n} is given by taking 𝕀 to be the category associated to the poset [0, n], and setting

MC(E)_{n} := ∐_{B:[0,n]→Ob C} \underline{MC}(E(B/n)).

By [Pri4] Proposition 5.7, MC(E) is a Segal space whenever C satisfies suitable fibrancy conditions. Segal spaces are a model for ∞-categories (whereas simplicial sets are a model for ∞-groupoids), and [Pri4] Propositions 5.15 and 5.24 show that 𝔇𝔢𝔩(E) is effectively the core of MC(E). This means that for all of our moduli constructions based on quasicomonoids in §6, we could construct derived moduli as ∞-categories, rather than just ∞-groupoids.

Definition 5.24. Say that an ordered pair B, B′ of objects in B induces fibrant quasidescent data if E(B) and E(B′) are fibrant simplicial quasi-comonoids, and the matching maps \underline{Hom}_{B}(⊤^{n}B, B′) → M^{n}\underline{Hom}B(⊤^{•}B, B′) are also Kan fibrations for all n ≥ 0.

Note that this is the same as regarding B as a simplicial quasi-descent datum (in the sense of [Pri4] Proposition 2.9), then restricting to objects B, B′, discarding morphisms B′ → B, and requiring that the resulting simplicial quasi-descent datum D be fibrant.

Lemma 5.25. Take objects D_{0}, D_{1} ∈ B^{⊤}, with UD_{0}, UD_{1} ∈ B inducing fibrant quasidescent data. Then there is a natural weak equivalence

\underline{MC}(UD_{0}, UD_{1}) ×_{\underline{MC}(E(UD_{0}))×\underline{MC}(E(UD_{1}))} \{(D_{0}, D_{1})\} ≃ Tot \underline{Hom}_{B}(⊤^{•}UD_{0}, UD_{1}),

where Tot : c𝕊 → 𝕊 the total space functor of [GJ] Ch. VIII, and the cosimplicial structure on \underline{Hom}_{B}(⊤^{•}UD_{0}, UD_{1}) is the usual cotriple resolution ([Wei] §8.7) defined via the isomorphisms

\underline{Hom}_{B}(⊤^{n}UD_{0}, UD_{1}) ≅ \underline{Hom}_{B^{⊤}}((F U)^{n+1}D_{0}, D_{1}).

Proof. This is [Pri4] Proposition 5.10. □

5.2.3. A bar construction. Now fix a simplicial category B equipped with a monad (⊤, μ, η) respecting the simplicial structure. Assume that B^{⊤} is a cocomplete simplicial category, equipped with a functor 𝕊 × B^{⊤} ⟶^{⊗} B^{⊤} for which

Hom_{B^{⊤}}(K ⊗ D, D′) ≅ Hom_{𝕊}(K, \underline{Hom}_{B^{⊤}}(D, D′)).

Write F : B → B^{⊤} for the free algebra functor sending M to the map (μ_{M} : ⊤^{2}M → ⊤M), and U : B^{⊤} → B for the forgetful functor sending (θ : ⊤M → M) to M. In particular, UF = ⊤.

Definition 5.26. Given an object M ∈ B and an element ω ∈ MC(E(B)) (for MC as in Definition 5.4), define β_{F}^{∗} (M) ∈ B^{⊤} by the property that

Hom(β_{F}^{∗} (M), D) ⊂ ∏_{q≥0} Hom_{B^{⊤}}((Δ^{1})^{q} ⊗ F ⊤^{q}M, D)

consists of \underline{ϕ} satisfying

ϕ_{q}(t_{1}, …, t_{i−1}, 1, t_{i}, … t_{q−1}) = ∂^{i}ϕ_{q−1}(t_{1}, …, t_{q−1})
ϕ_{q}(t_{1}, …, t_{i−1}, 0, t_{i}, … t_{q−1}) = ϕ_{i−1}(t_{1}, …, t_{i−1}) ∘ F ⊤^{i−1}ω_{q−i}(t_{i}, … t_{q−1})
σ^{i}ϕ_{q}(t_{1}, …, t_{q}) = ϕ_{q−1}(t_{1}, …, min(t_{i}, t_{i+1}), …, t_{q}) 1 ≤ i ≤ q
σ^{q}ϕ_{q}(t_{1}, …, t_{q}) = ϕ_{q−1}(t_{1}, …, t_{q−1}),

where ∂^{0} = λ_{⊤^{q−1}M}^{∗}, ∂^{i} = F ⊤^{i−1}μ_{⊤^{q−i}M}^{∗}, and σ^{i} = F ⊤^{i−1}η_{⊤^{q−i}M}^{∗}.

Beware that, unlike Definition 5.4, there is no relation for σ^{0}.

Remarks 5.27. Note that β_{F}^{∗} (M, ω) is a ⊤-algebra R generated by ϕ_{q}(⊤^{q}M ⊗ I^{q}), subject to various conditions, all of which are linear on generators except for

ϕ_{q}(1, t_{1}, …, t_{q−1}) = Uε_{R} ∘ ⊤ϕ_{q−1}(t_{1}, …, t_{q−1}).

Also observe that β_{F}^{∗} defines a functor 𝔇𝔢𝔩(E(M)) → B^{⊤}.

This construction is inspired by Lada’s bar construction in [CLM], which uses similar data to define a bar construction as an object of B, then shows that it carries a canonical ⊤-algebra structure. However, Lada’s construction only applies when ⊤ is an operad and B is the category of topological spaces, since it uses special properties of both. Beware that although similar expressions arise in both β_{F}^{∗} and in Lada’s bar construction, they are not directly comparable.

Proposition 5.28. Fix B, ⊤, M, ω as above, and take any D ∈ B^{⊤} for which the pair M, UD induces fibrant quasi-descent data (Definition 5.24). Then there is a functorial weak equivalence between \underline{Hom}_{B^{⊤}}(β_{F}^{∗} (M, ω), D) and the fibre of

\underline{MC}(E(M, UD)) → \underline{MC}(E(M)) × \underline{MC}(E(UD))

over (ω, μ_{D}), where D = (μ_{D} : ⊤UD → UD), and E(M, UD) is the quasi-comonoid defined in Example 5.22.

Proof. We adapt the proof of [Pri4] Proposition 5.10. In the notation of [Pri4], we have D ∈ sQDat_{1} given by D(0, 0) = E(M), D(1, 1) = E(UD), D(1, 0) = ∅ and D(0, 1)^{n} = \underline{Hom}_{B}(⊤^{n}M, UD). Using further notation from [Pri4], we can define A ∈ sQDat_{1} by

A := (Ξ × alg^{∗}1) ∪_{Ξ×{1}} (alg^{∗}0 × {1}),

and we then have

\underline{MC}(E(M, UD)) ×_{\underline{MC}(E(UD))} MC(E(D)_{0}) ≅ \underline{Hom}_{sQDat_{1}}(A, D).

Meanwhile, we can construct another object C ∈ sQDat_{1} with C(0, 0) = Ξ (so C(0, 0)^{n} = I^{n−1}), C(1, 1) = •, C(1, 0) = ∅ and C(0, 1)^{n} = I^{n}, where I = Δ^{1}. The multiplication operation C(0, 0)^{m} × C(0, 1)^{n} → C(0, 1)^{mn} is given by I^{m−1} × I^{n} → I^{m−1} × {0} × I^{n}. The operations on C(0, 1) are

∂^{i}(t_{1}, …, t_{n}) = (t_{1}, …, t_{i−1}, 1, t_{i}, … t_{n}) 1 ≤ i ≤ n
σ^{i}(t_{1}, …, t_{n}) = (t_{1}, …, min(t_{i}, t_{i+1}), …, t_{n}) 0 ≤ i < n.

The multiplication operation C(0, 1)^{m} × C(1, 1)^{n} → C(0, 1)^{m+n} is given by

(t_{1}, …, t_{m}, •) ↦ (t_{1}, …, t_{m}, 1, 1, …, 1).

Now, the inclusion Ξ = C(0, 0) → C gives a Kan fibration

\underline{Hom}_{sQDat_{1}}(C, D) →  \underline{Hom}_{sQM^{∗}(𝕊)}(Ξ, E(M)) × \underline{Hom}(alg^{∗}0, E(UD))
= \underline{MC}(E(M)) × MC(E(D)_{0})

whose fibre over (ω, D) is \underline{Hom}_{B^{⊤}}(β_{F}^{∗} (M, ω), D). It therefore suffices to show that

\underline{Hom}_{sQDat_{1}}(C, D) ×_{MC(E(UD)_{0})} \{D\} ≃ \underline{MC}(E(M, UD)) ×_{MC(E(UD)_{0})} \{D\}

as fibrant objects over \underline{MC}(E(M)), since taking the fibre over ω yields the required result.

Now, alg^{∗}1 ∈ sQDat_{1} is given by (alg^{∗}1)(i, j)^{n} = • for all n and for all 0 ≤ i ≤ j ≤ 1, while (alg^{∗}1)(1, 0) = ∅. Thus the unique maps C → alg^{∗}1 and A → alg^{∗}1 are both weak equivalences, and hence both are cofibrant replacements for alg^{∗}1 in the comma category (alg^{∗}0 × {1})↓sQDat_{1}.

If we regard D as an object of (alg^{∗}0 × {1}) ↓ sQDat_{1} via the morphism D : (alg^{∗}0 × {1}) → D, this means that

\underline{Hom}_{sQDat_{1}}(C, D) ×_{MC(E(UD)_{0})} \{D\} ≃ \underline{Hom}_{(alg^{∗}0×\{1\})↓sQDat_{1}}(C, D)
≃ R\underline{Hom}_{(alg^{∗}0×\{1\})↓sQDat_{1}}(alg^{∗}1, D)
≃ \underline{Hom}_{(alg^{∗}0×\{1\})↓sQDat_{1}}(A, D)
≃ \underline{MC}(E(M, UD)) ×_{\underline{MC}(E(UD))} \{D\},

which completes the proof. □

Definition 5.29. Define ⊥_{•} : B^{⊤} → B^{⊤} by the property that

\underline{Hom}_{B^{⊤}}(⊥_{•}A, D) ≅ Tot \underline{Hom}_{B^{⊤}}((F U)^{•+1}A, D).

Explicitly, we form the simplicial diagram n ↦ (F U)^{n+1}A in B^{⊤} (the cotriple resolution), then let ⊥_{•}A be the coend

∫ ^{n∈Δ} Δ^{n} ⊗ (F U)^{n+1}.

Corollary 5.30. If A, D ∈ B^{⊤}, with UA, UD inducing fibrant quasi-descent data, then there are functorial weak equivalences

\underline{Hom}_{B^{⊤}}(β_{F}^{∗} (UA, μ_{A}), D) ≃ \underline{Hom}_{B^{⊤}}(⊥_{•}A, D).

Proof. This just combines Proposition 5.25 with Proposition 5.28. □

5.3. Linear quasi-comonoids.

Definition 5.31. Say that a quasi-comonoid A is linear if each A^{n} is an abelian group, with the operations ∂^{i}, σ^{i} being linear, and ∗ : A^{m} × A^{n} → A^{m+n} bilinear. As explained in [Pri4] §4.4, this corresponds to working with the monoidal structure ⊗ rather than ×.

Denote the category of linear quasi-comonoids by QM^{∗}(Ab, ⊗), and the category of simplicial objects in QM^{∗}(Ab, ⊗) by QM^{∗}(sAb, ⊗)

Example 5.32. The quasi-comonoid E(B) constructed in §5.2.1 is a linear quasi-comonoid whenever B is a pre-additive category and ⊤ is an additive functor.

Lemma 5.33. Given A ∈ QM^{∗}(Ab, ⊗), the normalisation NA has the natural structure of a (not necessarily commutative) DG ring.

Proof. The normalisation is given by (NA)^{n} = A^{n} ∩ ⋂_{i=0}^{n−1} ker σ^{i}. If we write 0_{m} for the additive identity in A^{m}, then define ∂^{0}, ∂^{n+1} : A^{n} → A^{n+1} by ∂^{0}a := 0_{1}∗a, ∂^{n+1}a := a∗0_{1}. This makes A into a cosimplicial complex, so NA is a chain complex, with d : N^{n}A → N^{n+1}A given by

da := ∑_{i=0}^{n+1}(−1)^{i}∂^{i}.

For a ∈ N^{m}A and b ∈ N^{n}A, the product a ∗ b lies in N^{m+n}B, and d(a ∗ b) = (da) ∗ b + (−1)^{m}a ∗ (db), so NA is indeed a DG ring, with unit 1 ∈ N^{0}A. □

Definition 5.34. Given a DG ring B in non-negative cochain degrees with multiplication denoted by ∧, we now define the cosimplicial ring DB. As a cosimplicial complex, DB is given by the formula of Definition 4.21, with multiplication

(∂^{I}a) · (∂^{J}b) := \begin{cases}∂^{I∩J}(−1)^{(J\I,I\J)}a ∧ b & a ∈ L^{|J\I|}, b ∈ L^{|I\J|}, \\ 0 & otherwise, \end{cases}

for (−1)^{(S,T)} defined as in Definition 4.21.

Lemma 5.35. Given A ∈ QM^{∗}(Ab, ⊗), the quasi-comonoid E(DNA) is isomorphic to A (for E as in Example 5.2, regarding DNA as a cosimplicial multiplicative monoid).

Proof. Since DNA ≅ A as a cosimplicial complex, it is automatic that we have isomorphisms E(DNA)^{n} ≅ A^{n}, compatible with the structural operations σ^{i}, ∂^{i}, as well as with the operations a ↦ 0_{1} ∗ a and a ↦ a ∗ 0_{1} (corresponding to the additional operations ∂^{0}, ∂^{n+1}). Now, since Eilenberg–Zilber shuffles are left inverse to Alexander–Whitney, it follows that for a, b ∈ NA, a ∗_{E} b = a ∗ b. Since A is spanned by elements of the form ∂^{I}a for a ∈ NA, the defining equations of a quasi-comonoid ensure that a ∗_{E} b = a ∗ b for all a, b ∈ A. □

Now, any DG R-algebra B has an underlying DGLA over R, with bracket [a, b] = ab − (−1)^{deg deg b}ba. If a non-unital DG ℚ-algebra B is pro-nilpotent (i.e. B ≅ \underleftarrow{lim}_{n} B/(B)^{n}), we can thus define a group exp(DN^{>0}B) as in Lemma 4.23.

Lemma 5.36. For any non-unital pro-nilpotent DG ℚ-algebra B, there is a canonical isomorphism

exp(DB) ≅ 1 + DB

of groups.

Proof. Since B is pro-nilpotent, DB is as well. The isomorphism is given by evaluating the exponential in the ring DB, with inverse given by log. □

Definition 5.37. Given A ∈ QM^{∗}(Ab, ⊗), make (A^{0})^{×} into a gauge on the DGLA underlying NA by giving it the obvious adjoint action, and with D : (A^{0})^{×} → N^{1}A given by Da = da · a^{−1}.

Lemma 5.38. For A ∈ QM^{∗}(Ab, ⊗) with A^{0} a ℚ-algebra, there is a canonical isomorphism

D(exp(NA), (A^{0})^{×}) ≅ (DNA) ×_{A^{0}} (A^{0})^{×}

of cosimplicial groups.

Proof. By applying Lemma 5.36 to N^{>0}A,

D(exp(NA), (A^{0})^{×})^{n} ≅ (A^{0})^{×}⋉(1+ker(A^{n} → A^{0})) = (A^{0})^{×}⋉(A^{n}×_{A^{0}} 1) ≅ A^{n}×_{A^{0}} (A^{0})^{×},

and this is automatically compatible with the cosimplicial operations in higher degrees. A short calculation show that it is also compatible with ∂^{0}, ∂^{1} : A^{0} → A^{1}. □

Definition 5.39. Given a simplicial DGLA L, define the simplicial set \underline{MC}(L) as follows. first, for any simplicial set K, define the simplicial DGLA L^{K} by (L^{K})^{n} := (L^{n})^{K}, defined with the formula of Definition 1.22. Then \underline{MC}(L) is given by

\underline{MC}(L)_{n} := MC(Tot^{Π}N^{s}(L^{Δ^{n}})),

where the normalisation N^{s}(L) has a bracket N^{s}(L)_{m}^{i} × N^{s}(L)_{n}^{j} → N^{s}(L)_{m+n}^{i+j} given by the Eilenberg–Zilber shuffle product ([Wei] 8.5.4).

Proposition 5.40. For A ∈ QM^{∗}(sAb, ⊗), there is a canonical (A^{0})^{×}-equivariant weak equivalence

\underline{MC}(A) ≃ \underline{MC}(NA),

of simplicial sets, and hence a canonical weak equivalence

Del(A) ≃ W̄ [\underline{MC}(NA)/(A^{0})^{×}].

Proof. By combining Corollary 4.23 with Lemma 5.38, we have an (A^{0})^{×}-equivariant isomorphism

\underline{MC}(NA) ≅ \underline{MC}((DNA) ×_{A^{0}} (A^{0})^{×}).

Now, Lemma 5.12 combines with Lemma 5.35 to give an (A^{0})^{×}-equivariant weak equivalence

\underline{MC}((DNA) ×_{A^{0}} (A^{0})^{×}) ≃ \underline{MC}(A ×_{A^{0}} (A^{0})^{×}).

Since \underline{MC}(A ×_{A^{0}} (A^{0})^{×}) = \underline{MC}(A) and Del(A) = W̄ [\underline{MC}(A)/(A^{0})^{×}], this completes the proof. □

5.4. Moduli functors from quasi-comonoids.

Definition 5.41. Given a homogeneous, levelwise formally smooth functor E : Alg_{R} → QM^{∗}, a ring A ∈ Alg_{R}, an A-module M and ω ∈ MC(E(A)), define the cosimplicial A-module C_{ω}^{•}(E, M) by

C_{ω}^{n}(E, M) := T_{ω^{n}}(E^{n}, M)

with operations on a ∈ C_{ω}^{n}(E, M) given by

σ^{i}a = σ_{E}^{i}a
∂^{i}a = \begin{cases}ω ∗ a & i = 0 \\ ∂_{E}^{i}a & 1 ≤ i ≤ n \\ a ∗ ω & i = n + 1. \end{cases}

Proposition 5.42. If E : Alg_{R} → QM^{∗} is a homogeneous functor, with each E^{n} formally smooth, then the functor

\underline{MC}(\underline{E}) : dN_{R}^{♭} → 𝕊

is homogeneous and formally quasi-smooth, so

MC(\underline{E}) : dN_{R}^{♭} → Set

is homogeneous and pre-homotopic.

Proof. This proceeds along the same lines as Proposition 4.14. Homogeneity is automatic, as \underline{MC} preserves arbitrary limits.

We can extend E^{n} to a functor E^{n} : dN_{R}^{♭} → Set, given by E^{n}(A) := E^{n}(A_{0}). Then formal smoothness of E^{n} implies that the extended E^{n} is pre-homotopic. It is automatically formally quasi-presmooth, as all discrete morphisms are fibrations. Thus Proposition 1.27 implies that \underline{E}^{n} is formally smooth, and hence formally quasi-smooth.

For our next step, there is no analogue of Lemma 4.7 for quasi-comonoids, so we have to work a little harder. We wish to show that \underline{MC}(\underline{E})(A′) → \underline{MC}(\underline{E})(A) is a (trivial) fibration for all (acyclic) square-zero extensions A′ → A. Write C := H_{0}A, and note that \underline{E}(C) = E(C), so \underline{MC}(\underline{E}(C)) = MC(E(C)) is a set. For any ω ∈ MC(E(C)), it thus suffices to show that the morphism \underline{MC}(\underline{E})(A′)_{ω} → \underline{MC}(\underline{E})(A)_{ω} of fibres over ω is a (trivial) fibration.

Now, on the subcategory of dN^{♭} ↓C consisting of nilpotent extensions B → C, define a functor E_{ω} by E_{ω}^{n}(B) = E^{n}(B) ×_{E^{n}(C)} ω^{n}. Thus E_{ω}(B) ∈ QM^{∗}; since C^{Δ^{n}} = C, we also have \underline{E}_{ω}(B) ∈ QM^{∗}(𝕊). Now, the crucial observation is that

\underline{MC}(\underline{E})(B)_{ω} = \underline{MC}(\underline{E}_{ω}(B)),

so by Lemma 5.6, it suffices to show that \underline{E}_{ω}(A′) → \underline{E}_{ω}(A) is a (trivial) fibration in QM^{∗}(𝕊).

This amounts to saying that μ_{n} : \underline{E}_{ω}^{n} → M^{n}\underline{E}_{ω} is formally quasi-smooth for all n ≥ 0. In fact, it is formally smooth; we prove this inductively on n. For n = 0, this follows immediately from the observation above that \underline{E}^{0} is formally smooth. If this holds up to level n − 1, then M^{n}\underline{E}_{ω} is also formally quasi-smooth, being a pullback of formally quasismooth maps. Since \underline{E}_{ω}^{n} is formally smooth, it follows from Corollary 1.16 that μ_{n} will be formally smooth provided

D_{ω}^{i}(\underline{E}^{n}, M) → D_{ω}^{i}(M^{n}\underline{E}, M)

is surjective for i = 0, and an isomorphism for i > 0, for all C-modules M. Now, Lemma 1.28 shows that D_{ω}^{i}(\underline{E}^{n}, M) = D_{ω}^{i}(E^{n}, M), and similarly for M^{n}E. These are 0 for i ≠ 0, and D_{ω}^{0}(E^{n}, M) = C_{ω}^{n}(E, M). Hence we need only show that

C_{ω}^{n}(E, M) → M^{n}C_{ω}^{•}(E, M)

is surjective, which follows from Lemma 4.7. □

Corollary 5.43. If E : Alg_{R} → QM^{∗} is a homogeneous functor, with each E^{n} formally smooth, then the functor

\underline{Del}(\underline{E}) : dN_{R}^{♭} → 𝕊

is homogeneous and formally quasi-smooth, while

Del(\underline{E}) : sN_{R}^{♭} → 𝕊

is homogeneous, pre-homotopic and formally quasi-presmooth.

Proof. The proof of Proposition 4.15 adapts, substituting Proposition 5.42 for Proposition 4.14. □

5.4.1. Cohomology.

Definition 5.44. Given a homogeneous, levelwise formally smooth functor E : Alg_{R} → QM^{∗}, a ring A ∈ Alg_{R}, an A-module M and ω ∈ MC(E(A)), define

H_{ω}^{i}(E, M) := H^{i}C_{ω}^{•}(E, M),

for C_{ω}^{•}(E, M) as in Definition 5.41.

The following is immediate.

Lemma 5.45. Given a homogeneous, levelwise formally smooth functor E : Alg_{R} → QM^{∗}(𝕊), a ring A ∈ Alg_{R}, an A-module M and ω ∈ MC(E_{0}(A)), the fibre of MC(E(A ⊕ M)) → MC(E(A)) over ω is canonically isomorphic to MC(C_{ω}^{n}(G, M)).

Lemma 5.46. If E : Alg_{R} → QM^{∗} is a homogeneous, levelwise formally smooth functor, with A ∈ Alg_{R} and M an A-module, then

D_{ω}^{i}(\underline{MC}(\underline{E}), M) ≅ D_{ω}^{i}(MC(\underline{E}), M) ≅ \begin{cases}H_{ω}^{i+1}(E, M) & i > 0 \\ Z^{1}C_{ω}^{•}(E, M) & i = 1. \end{cases}

Proof. First, observe that there is a constant quasi-comonoid ι•, given by the one-point set in every level. The element ω thus defines a morphism ω^{•} : ι• → E(A) of quasi-comonoids, given by ω^{n} in level n, and the fibre product

T_{ω^{•}}(\underline{E}, L) := \underline{E}(A ⊕ L) ×_{E(A),ω^{•}} ι•

is thus an abelian quasi-comonoid, for all L ∈ dMod_{A}. By Lemma 5.7, this corresponds to a simplicial cosimplicial abelian group, which is just given in simplicial level n by

T_{ω^{•}}(\underline{E}, L)_{n} = C_{ω}^{•}(E, (L^{Δ^{n}})_{0}).

Therefore

T_{ω}(\underline{MC}(\underline{E}), L) ≅ \underline{MC}(T_{ω^{•}}(\underline{E}, L)),

and the result is an immediate consequence of Lemma 5.8. □

Lemma 5.47. If E : Alg_{R} → QM^{∗} is a homogeneous, levelwise formally smooth functor, with A ∈ Alg_{R} and M an A-module, then

D_{ω}^{i}(\underline{Del}(\underline{E}), M) ≅ D_{ω}^{i}(Del(\underline{E}), M) ≅ H_{ω}^{i+1}(E, M).

Proof. The proof of Lemma 4.20 carries over. □

5.5. Sheafification.

Definition 5.48. Given a cosimplicial diagram C^{•}(E) in QM^{∗}, define the quasi-comonoid diag C^{•}(E) by

diag C^{•}(E)^{n} := C^{n}(E^{n}),

with operations ∂^{i} = ∂_{C}^{i}∂_{E}^{i}, σ^{i} = σ_{C}^{i}σ_{E}^{i}, identity 1 ∈ C^{0}(E), and multiplication

a ∗ b := (∂_{C}^{m+1})^{n}a ∗_{E} (∂_{C}^{0})^{m}b,

for a ∈ C^{m}(E^{m}), b ∈ C^{n}(E^{n}).

Definition 5.49. Given a levelwise formally smooth, homogeneous functor E : Alg_{R} → QM^{∗} preserving finite products, and some class P of covering morphisms in Alg_{R}, define the sheafification E^{♯} of E with respect to P by first defining a cosimplicial commutative

A-algebra (B/A)^{•} for every P-covering A → B, as (B/A)^{n} := \overbrace{B ⊗_{A} B ⊗_{A} … ⊗_{A} B}^{n+1}, then setting

E^{♯}(A) = diag \underrightarrow{lim}_{B} E((B/A)^{•}).

Lemma 5.50. For E and P as above, there is a canonical morphism

𝔇𝔢𝔩(\underline{E})^{♯} → 𝔇𝔢𝔩(\underline{E}^{♯})

of groupoid-valued functors on dN_{R}^{♭}, for sheafification 𝔇𝔢𝔩(\underline{E})^{♯} defined as in Definition 4.29. This induces an equivalence

π^{0}𝔇𝔢𝔩(\underline{E})^{♯} → π^{0}𝔇𝔢𝔩(\underline{E}^{♯}).

Proof. An object of 𝔇𝔢𝔩(\underline{E})^{♯} is a pair (ω, g) ∈ MC(\underline{E}(A ⊗_{A_{0}} B)) × E^{0}(B ⊗_{A_{0}} B)^{×}, for A_{0} → B a P-covering, satisfying the following conditions:

(1) g ∗ (pr_{1}^{∗}ω) = (pr_{0}^{∗}ω) ∗ g ∈ MC(\underline{E}(A ⊗_{A_{0}} B ⊗_{A_{0}} B)),

(2) pr_{02}^{∗}g = (pr_{01}^{∗}g) · (pr_{12}^{∗}g) ∈ E^{0}(B ⊗_{A_{0}} B ⊗_{A_{0}} B)^{×}, and we now describe the image of (ω, g).

First, form the cosimplicial A_{0}-algebra (B/A_{0})^{•} as in Definition 5.49. We now map (B, ω, g) to the object ω′ of MC(diag \underline{E}(A ⊗_{A_{0}} (B/A_{0})^{•})) given by

ω′_{n} := (pr_{0,n+1}^{∗}g) ∗ pr_{1}^{∗}ω_{n} ∈ E^{n+1}((A^{Δ^{n}})_{0} ⊗_{A_{0}} (B/A_{0})^{n+1}).

The remainder of the proof now follows exactly as for that of Lemma 4.30. □

Now recall that P-covering morphisms are all assumed faithfully flat.

Lemma 5.51. Take E : Alg_{R} → QM^{∗} satisfying the conditions of Definition 5.49, a ring A ∈ Alg_{R}, an A-module M, and an object of 𝔇𝔢𝔩(E)^{♯}(A) represented by (ω, g) for ω ∈ MC(E(B)). If the maps

H_{ω}^{∗}(E, M ⊗_{A} B) ⊗_{B} B′ → H_{ω}^{∗}(E, M ⊗_{A} B′)

are isomorphisms for all P-coverings B → B′, then the morphism α of Lemma 5.50 induces an isomorphism

D_{(ω,g)}^{∗}(B𝔇𝔢𝔩(\underline{E})^{♯}, M) → D_{α(ω,g)}^{∗}(B𝔇𝔢𝔩(\underline{E}^{♯}), M).

Proof. The proof of Lemma 4.31 carries over verbatim. □

6. Examples of derived moduli via quasi-comonoids

We now show how to apply the machinery of the previous section to construct derived moduli stacks for several specific examples. The approach is the same as that used to construct derived deformations in [Pri1] and [Pri5], by finding suitable monads to construct a quasi-comonoid, and then taking the Deligne groupoid.

Throughout this section, R will be a G-ring admitting a dualising complex in the sense of [Har1] Ch. V. Examples are ℤ, any field, or any Gorenstein local ring.

6.1. Finite schemes. For a fixed r ∈ ℕ, we now study a quasi-comonoid governing finite schemes of rank r. For any commutative R-algebra A, our moduli groupoid consists of algebra homomorphisms A → B, with B a locally free A-module of rank r.

In order to construct a quasi-comonoid, we take the approach of §5.2.1. Let B(A) be the category of A-modules, and define the monad ⊤_{A} on B(A) to be Symm_{A}, so D(A) := B(A)^{⊤_{A}} is the category of commutative A-algebras.

Definition 6.1. Working in B(A), define E_{r} : Alg_{R} → QM^{∗} by setting E_{r}(A) to be the quasi-comonoid E_{r}(A) := E(A^{r}), given by

E_{r}^{n}(A) = Hom_{A}(⊤_{A}^{n}(A^{r}), A^{r}).

The following is then just Lemma 5.17 in this context:

Lemma 6.2. For any commutative R-algebra A, 𝔇𝔢𝔩(E_{r}(A)) is canonically isomorphic to the groupoid of commutative A-algebra structures on the A-module A^{r}.

For our next step, note that 𝔇𝔢𝔩(E_{r}) is not a stack, although the core of D is. However, the stackification 𝔇𝔢𝔩(E_{r})^{♯} in the Zariski topology is equivalent to the subgroupoid of D(A) consisting of commutative A-algebras B which are locally free A-modules of rank r.

Definition 6.3. Define 𝔇𝔢𝔩(\underline{E_{r}})^{♯} : dN_{R}^{♭} → Gpd to be the stackification of 𝔇𝔢𝔩(\underline{E_{r}}) in the strict Zariski topology of Definition 2.17. Likewise, define the simplicial groupoid-valued functor \underline{𝔇𝔢𝔩}(\underline{E_{r}})^{♯} on dN_{R}^{♭} by stackifying levelwise, so (\underline{𝔇𝔢𝔩}(\underline{E_{r}})^{♯})_{n} = (\underline{𝔇𝔢𝔩}(\underline{E_{r}})_{n})^{♯}.

Proposition 6.4. The functor W̄ \underline{𝔇𝔢𝔩(\underline{E_{r}})^{♯}} → 𝕊 is representable by an almost finitely presented derived geometric 1-stack. Moreover,

W̄ \underline{𝔇𝔢𝔩(\underline{E_{r}})^{♯}} ≃ W̄ \underline{𝔇𝔢𝔩}(\underline{E_{r}})^{♯} ≃ \underline{Del}(\underline{E_{r}}^{♯}),

where the last is defined using the quasi-comonoid sheafification of Definition 5.49.

Proof. For the first statement, we just show that B𝔇𝔢𝔩(\underline{E_{r}})^{♯} satisfies the conditions of Theorem 1.31. Corollary 5.43 ensures that B𝔇𝔢𝔩(\underline{E_{r}})^{♯} is homogeneous and pre-homotopic. It follows from Lemma 5.47 and the cotriple characterisation of André-Quillen cohomology ([Wei] §8.8) that

D_{C}^{i}(B𝔇𝔢𝔩(\underline{E_{r}})^{♯}, M) ≅ Ext^{i+1}(𝕃_{•}^{C/A}, C ⊗_{A} M),

so the finiteness conditions of Theorem 1.31 all hold, making W̄ \underline{𝔇𝔢𝔩(\underline{E_{r}})^{♯}} representable.

Similar arguments show that W̄ \underline{𝔇𝔢𝔩}(\underline{E_{r}})^{♯} satisfies the conditions of Theorem 1.19, so is representable by a derived geometric 1-stack. For the first equivalence, we thus apply Corollary 1.29 to the morphism

B𝔇𝔢𝔩(\underline{E_{r}})^{♯} → W̄ \underline{𝔇𝔢𝔩}(\underline{E_{r}})^{♯}

coming from the map 𝔇𝔢𝔩(\underline{E_{r}})^{♯} → \underline{𝔇𝔢𝔩}(\underline{E_{r}})^{♯} of simplicial diagrams of groupoids.

For the second equivalence, we just use Lemmas 5.50 and 5.51 to show that the composite map

B𝔇𝔢𝔩(\underline{E_{r}})^{♯} → B𝔇𝔢𝔩(\underline{E_{r}}^{♯}) → W̄ \underline{𝔇𝔢𝔩}(\underline{E_{r}})^{♯}

satisfies the conditions of Corollary 1.29. □

Remark 6.5. Alternatively, we can describe the associated derived geometric 1-stack explicitly. The functor A ↦ MC(E_{r}(A)) is an affine dg scheme, (E_{r}^{0})^{×} = GL_{r}, and W̄ \underline{𝔇𝔢𝔩(\underline{E_{r}})^{♯}} is just the hypersheafification of the quotient B[MC(E_{r})/(E_{r}^{0})^{×}] in the homotopy-Zariski (and indeed homotopy-étale) topologies. In the terminology of [Pri8], the simplicial affine dg scheme B[MC(E_{r})/(E_{r}^{0})^{×}] is a homotopy derived Artin 1-hypergroupoid representing W̄ \underline{𝔇𝔢𝔩(\underline{E_{r}})^{♯}}.

Proposition 6.6. For A ∈ sN_{R}^{♭}, the space \underline{Del}(\underline{E_{r}}^{♯})(A) is functorially weakly equivalent to the nerve W̄ 𝔊(A) of the ∞-groupoid 𝔊(A) of simplicial commutative A-algebras B for which B ⊗_{A}^{L} π_{0}A is weakly equivalent to a locally free module of rank r over π_{0}A.

Proof. For F : sMod(A) → sAlg(A) the free commutative algebra functor on simplicial Amodules, the functor β_{F}^{∗} of Definition 5.26 maps from 𝔇𝔢𝔩(E_{r}^{♯})(A) to sAlg(A), functorially in A ∈ sN_{R}^{♭}, and all objects in its image are cofibrant.

When A ∈ Alg_{R}, Corollary 5.30 implies that β_{F}^{∗} (C) is homotopy equivalent to the cotriple resolution ⊥_{•}C of C. Thus π_{0}(β_{F}^{∗} (C)) ≅ C, and π_{i}(β_{F}^{∗} (C))) = 0 for all i > 0, so we indeed have a functor

β_{F}^{∗} : 𝔇𝔢𝔩(E_{r}^{♯})(A) → 𝔊(A)

for all A ∈ sN_{R}^{♭} (using compatibility with base change). Moreover, this functor is an equivalence of ∞-groupoids when A ∈ Alg_{R}.

Now, [Pri6] Corollary 3.10 and Example 3.11 give

D_{[B]}^{i}(W̄ 𝔊, N) ≅ Ext_{B}^{i+1}(𝕃^{B/A}, N ⊗_{A} B),

so β_{F}^{∗} satisfies the conditions of Remark 1.30, giving an equivalence W̄ \underline{𝔇𝔢𝔩(\underline{E_{r}})^{♯}} → W̄ 𝔊. Combining with Proposition 6.4 gives the required equivalence \underline{Del}(\underline{E_{r}}^{♯}) ≃ W̄ 𝔊. □

6.2. Coherent sheaves. Take a projective scheme X over R, and fix a numerical polynomial h. We now consider moduli of coherent sheaves on X with Hilbert polynomial h. In other words, our underived moduli functor M_{h} will set M_{h}(A) to be the groupoid of coherent sheaves ℱ on X × Spec A with Γ(X × Spec A, ℱ (n)) locally free of rank n for n ≫ 0.

Definition 6.7. For A ∈ Alg_{R}, we now define our base category B(A) as follows. First form the category C(A) of graded A-modules M = ⨁_{n≥0} M{n} in non-negative degrees, then let B(A) := pro(C(A)). Explicitly, objects of B(A) are inverse systems {M^{α}}_{α} in C(A), with

Hom_{B(A)}(\{M^{α}\}_{α}, \{N^{β}\}_{β}) = \underleftarrow{lim}_{β} \underrightarrow{lim}_{α} Hom_{C(A)}(M^{α}, N^{β}).

If we let S := ⨁_{n≥0} Γ(X, 𝒪_{X}(n)), then there is a monad ⊤ on C(A) given by ⊤M = S ⊗_{R} M, with multiplication μ : ⊤^{2} → ⊤ coming from the multiplication S ⊗_{R} A → S, and unit id → ⊤ coming from R → S.

This extends naturally to a monad on B(A), and we are now in the setting of §5.2.1, since the category D(A) := B(A)^{⊤} of ⊤-algebras is just the pro-category of graded S ⊗_{R} Amodules. We will be interested in pro-modules of the form M = {⨁_{n≥p} M{n}}_{p}; for M, M′ any two such, we get

Hom_{D(A)}(M, M′) = \underleftarrow{lim}_{q} \underrightarrow{lim}_{p} Hom_{S⊗_{R}A}^{𝔾_{m}}(M\{≥ p\}, M′\{≥ q\})
≅ \underleftarrow{lim}_{q} \underrightarrow{lim}_{p≥q} Hom_{S⊗_{R}A}^{𝔾_{m}}(M\{≥ p\}, M′\{≥ q\})
≅ \underleftarrow{lim}_{q} \underrightarrow{lim}_{p≥q} Hom_{S⊗_{R}A}^{𝔾_{m}}(M\{≥ p\}, M′)
≅ \underrightarrow{lim}_{p} Hom_{S⊗_{R}A}^{𝔾_{m}}(M\{≥ p\}, M′),

which ties in with [Ser].

Definition 6.8. Let R^{h} ∈ B(R) be the inverse system {⨁_{n≥p} R^{h(n)}(n)}_{p} of graded modules, and form the quasi-comonoid E_{h}(A) := E(R^{h} ⊗_{R} A) given by

E_{h}^{n}(A) = Hom_{B(A)}(⊤^{n}(R^{h} ⊗_{R} A), R^{h} ⊗_{R} A).

Lemma 5.17 then implies that 𝔇𝔢𝔩(E_{h}(A)) is the subgroupoid of D(A) consisting of S-module structures on R^{h} ⊗_{R} A, and all isomorphisms between them.

For our next step, note that 𝔇𝔢𝔩(E_{h}) is not a stack, although the core of D is. However, Lemma 3.25 adapts to show that the stackification 𝔇𝔢𝔩(E_{h})^{♯} in the pro-Zariski topology is equivalent to the subgroupoid of D(A) consisting of pro-(S ⊗_{R} A)-modules M = {⨁_{n≥p} M{n}}_{p}, with M{n} locally free of rank h(n) over A for n ≫ 0.

Now, there is a functor ℱ ↦ ⨁_{n≥0} Γ(X, ℱ (n)) from coherent sheaves to D(A), and the essential image just consists of finitely generated (S ⊗_{R} A)-modules.

Definition 6.9. Define the functor MC_{f}(E_{h}) : Alg_{R} → Set by letting MC_{f}(E_{h}, A) ⊂ MC(E_{h}, A) consist of (S ⊗_{R} A)-modules isomorphic to finitely generated modules. Next, define \underline{MC}_{f}(\underline{E}_{h}) : dN_{R}^{♭} → 𝕊 by

\underline{MC}_{f}(\underline{E}_{h}, A) := \underline{MC}(\underline{E}_{h}, A) ×_{MC(E_{h},H_{0}A)} MC_{f}(E_{h}, H_{0}A),

with MC_{f}(\underline{E}_{h}, A) = \underline{MC}_{f}(\underline{E}_{h}, A)_{0}.

Likewise, define 𝔇𝔢𝔩_{f}(E_{h}) := [MC_{f}(E_{h})/(E^{0})^{×}], 𝔇𝔢𝔩_{f}(\underline{E}_{h}) := [MC_{f}(\underline{E}_{h})/(E^{0})^{×}] and \underline{𝔇𝔢𝔩}_{f}(\underline{E}_{h}) := [\underline{MC}_{f}(\underline{E}_{h})/(\underline{E}^{0})^{×}].

Definition 6.10. Let E_{h}^{♯} be the quasi-comonoid sheafification (see Definition 5.49) of E_{h} in the pro-Zariski topology , and recall from Lemma 5.50 that 𝔇𝔢𝔩(E_{h}^{♯}) ≃ 𝔇𝔢𝔩(E_{h})^{♯}. Define MC_{f}(E_{h}^{♯}) ⊂ MC(E_{h}^{♯}) to be the essential image of 𝔇𝔢𝔩_{f}(E_{h})^{♯} (which consists of modules isomorphic to finitely generated modules).

Then define \underline{MC}_{f}(E_{h}^{♯}), \underline{MC}(\underline{E}_{h}^{♯}), 𝔇𝔢𝔩_{f}(E_{h}^{♯}), 𝔇𝔢𝔩_{f}(\underline{E}_{h}^{♯}) and \underline{𝔇𝔢𝔩}_{f}(\underline{E}_{h}^{♯}) by adapting the formulae of Definition 6.9.

Lemma 3.25 adapts to show that the substack 𝔇𝔢𝔩_{f}(E_{h})^{♯} of 𝔇𝔢𝔩_{f}(E_{h})^{♯} is equivalent to the stack M_{h}(A) defined at the beginning of the section. Note that MC_{f}(E_{h}) → MC(E_{h}) is formally étale, so \underline{𝔇𝔢𝔩}_{f}(\underline{E}_{h}) → \underline{𝔇𝔢𝔩}(\underline{E}_{h}) is also formally étale, as is \underline{𝔇𝔢𝔩}_{f}(\underline{E}_{h}^{♯}) → \underline{𝔇𝔢𝔩}(\underline{E}_{h}^{♯}).

Proposition 6.11. The functor W̄ \underline{𝔇𝔢𝔩_{f}(\underline{E_{h}})^{♯}} → 𝕊 is representable by an almost finitely presented derived geometric 1-stack. Moreover,

W̄ \underline{𝔇𝔢𝔩_{f}(\underline{E_{h}})^{♯}} ≃ W̄ \underline{𝔇𝔢𝔩}_{f}(\underline{E_{h}})^{♯} ≃ \underline{Del}_{f}(\underline{E_{h}}^{♯}).

Proof. By exploiting the fact that MC_{f}(E_{h}) → MC_{f}(E_{h}) is formally étale, the proof of Proposition 6.4 carries over. The only substantial differences lie in the calculation of cohomology:

D_{L}^{i}(B𝔇𝔢𝔩(\underline{E_{h}})^{♯}, M) ≅ Ext_{S⊗_{R}A}^{i+1}(L, L ⊗_{A} M),

and in establishing local finite presentation, which comes from adapting Lemma 3.28. □

Proposition 6.12. For A ∈ sN_{R}^{♭}, the space \underline{Del}_{f}(\underline{E_{h}}^{♯})(A) is functorially weakly equivalent to the nerve of the ∞-groupoid 𝔐_{h} of derived quasi-coherent sheaves ℱ on X × Spec A for which

(1) ℱ ⊗_{A}^{L} π_{0}A is weakly equivalent to a quasi-coherent sheaf ℱ̄ on X × Spec π_{0}A, and (2) for all n ≫ 0, Γ(X × Spec π_{0}A, ℱ̄(n)) is a locally free A-module of rank h(n).

Proof. Taking F : sB(A) → sD(A) to be the functor M ↦ S ⊗_{R} M from simplicial graded A-modules to simplicial graded S ⊗_{R} A-modules, Definition 5.26 gives us a functor

β_{F}^{∗} : 𝔇𝔢𝔩(E_{h}^{♯})(A) → sD(A),

preserving cofibrant objects. In particular, if A ∈ Alg_{R}, and L ∈ D(A) has L{n} locally free for all n, then β_{F}^{∗} (L) is cofibrant, and Corollary 5.30 implies that it is homotopy equivalent to the cotriple resolution S^{⊗_{R}•+1} ⊗_{R} L of L (as in [Wei] §8.7.1).

This ensures that for any A ∈ sN_{R}^{♭}, β_{F}^{∗} maps objects of 𝔇𝔢𝔩_{f}(E_{h}^{♯})(A) to modules associated to objects of 𝔐_{h}(A). Explicitly, let X̃ := (Spec S) − {0} be the canonical 𝔾_{m}-bundle over X = Proj S, with π : X̃ → X the projection and j : X̃ → Spec S the open immersion. Then our functor 𝔇𝔢𝔩_{f}(E_{h}^{♯})(A) → 𝔐_{h}(A) is

(M, ω) ↦ β_{F}^{∗} (M, ω)^{♯} = (π_{∗}j^{−1}β_{F}^{∗} (M, ω))^{𝔾_{m}}.

Now, for any A ∈ Alg_{R}, we have

D_{L}^{i}(B𝔇𝔢𝔩(E_{h}^{♯}), M) ≅ Ext_{D(A)}^{i+1}(L, L ⊗_{A} M);

adapting Serre’s Theorem ([Ser] §59) as in the proof of Proposition 3.32, this is isomorphic to Ext_{X×Spec A}^{i+1}(L^{♯}, L^{♯} ⊗_{A} M).

The remainder of the proof follows as for that of Proposition 6.6, replacing [Pri6] Example 3.11 with [Pri6] Theorem 4.12. □

Remark 6.13 (Associated DGLAs). Since the functor ⊤ = S⊗_{R} is linear, our quasicomonoid E_{h}(A) is linear in the sense of §5.3, so its normalisation NE_{h} has the natural structure of a DGLA. We could thus use Proposition 5.40 to rewrite all these results in terms of the groupoid [MC(NE_{h})/(E_{h}^{0})^{×}], thereby making them consistent with the approach of [CFK1].

Remark 6.14 (Quot schemes). If we wished to work with quotients of a fixed coherent sheaf ℳ on X, there are two equivalent approaches we could take. One is to choose M ∈ D(R) with M^{♯} = ℳ, and then to replace D(A) with the comma category (M ⊗_{R} A)↓D(A). The monad ⊤ would then be given by ⊤(L) = (M ⊗_{R} A) ⊕ (S ⊗_{A} L), with unit L → S ⊗_{A} L, and multiplication ⊤^{2}(L) → ⊤(L) being the map

(M ⊗_{R} A) ⊕ (S ⊗_{R} M ⊗_{R} A) ⊕ (S ⊗_{R} S ⊗_{R} L) → (M ⊗_{R} A) ⊕ (S ⊗_{A} L)
(m_{1}, s_{1} ⊗ m_{2}, s_{2} ⊗ s_{3} ⊗ l) ↦ (m_{1} + s_{1}m_{2}, (s_{2}s_{3}) ⊗ l).

We would also have to replace MC_{f}(E_{h}) with the subset of MC(E_{h}) consisting of finitely generated S ⊗_{R} A-modules under M ⊗_{R} A for which the map M ⊗_{R} A → L is surjective.

The alternative approach would be to work with the construction of Example 5.22, first forming the quasi-comonoid

E := E(A^{h′}, A^{h}) ×_{E(A^{h′})} {M},

where h′ is the Hilbert polynomial of M, then taking the subset of MC(E) cut out by the finiteness and surjectivity conditions above.

Beware, however, that in both of these approaches, the resulting quasi-comonoid is no longer linear, so cannot naturally be replaced with a DGLA.

6.3. Polarised projective schemes. For a fixed numerical polynomial h ∈ ℚ[t], we will now study the moduli of polarised projective schemes (X, 𝒪_{X}(1)), with 𝒪_{X}(1) ample, for which Γ(X, 𝒪_{X}(n)) is locally free of rank h(n) for n ≫ 0.

We take B(A) to be the pro-category of graded A-modules from §6.2, with monad

⊤ = Symm_{A} : B(A) → B(A).

Setting D(A) := B(A)^{⊤} gives us the pro-category of 𝔾_{m}-equivariant commutative Aalgebras in non-negative degrees.

Definition 6.15. Let R^{h} ∈ B(R) be the inverse system {⨁_{n≥p} R^{h(n)}(n)}_{p} of graded modules, and form the quasi-comonoid E_{h}(A) := E(R^{h} ⊗_{R} A) given by

E_{h}^{n}(A) = Hom_{B(A)}(⊤^{n}(R^{h} ⊗_{R} A), R^{h} ⊗_{R} A).

Lemma 5.17 then implies that 𝔇𝔢𝔩(E_{h}(A)) is the subgroupoid of D(A) consisting of commutative ring structures on R^{h}⊗_{R}A, and all isomorphisms between them. By Lemma 3.25, the stackification 𝔇𝔢𝔩(E_{h})^{♯} in the pro-Zariski topology is equivalent to the subgroupoid of D(A) consisting of commutative pro-A-algebras B = {⨁_{n≥p} B{n}}_{p}, with B{n} locally free of rank h(n) over A for n ≫ 0.

We now proceed as in Definition 6.9, letting MC_{f}(E_{h}) ⊂ MC(E_{h}) consist of finitely generated A-algebras, and so on for \underline{MC}_{f}(\underline{E}_{h}) etc. Note that MC_{f}(E_{h}) → MC(E_{h}) is formally étale.

Proposition 6.16. For A ∈ Alg_{ℚ}, 𝔇𝔢𝔩_{f}(E_{h}(A)) is equivalent to the groupoid of flat polarised schemes (X, 𝒪_{X}(1)) of finite type over A, with 𝒪_{X}(1) ample and the A-modules Γ(X, 𝒪_{X}(n)) locally free of rank h(n) for all n ≫ 0.

Proof. This is essentially just Proposition 3.30. □

Proposition 6.17. The functor W̄ \underline{𝔇𝔢𝔩_{f}(\underline{E_{h}})^{♯}} → 𝕊 is representable by an almost finitely presented derived geometric 1-stack. Moreover,

W̄ \underline{𝔇𝔢𝔩_{f}(\underline{E_{h}})^{♯}} ≃ W̄ \underline{𝔇𝔢𝔩}_{f}(\underline{E_{h}})^{♯} ≃ \underline{Del}_{f}(\underline{E_{h}}^{♯}).

Proof. The proofs of Propositions 6.4 and 6.11 carry over, substituting the relevant finiteness properties from Proposition 3.33. In particular, Proposition 3.32 adapts to show that

D_{[C]}^{i}(\underline{Del}_{f}(\underline{E_{h}}^{♯}), M) ≅ 𝔼xt_{X}^{i+1}(𝕃^{X/B𝔾_{m}⊗A}, 𝒪_{X} ⊗_{A} M),

where X = Proj (A ⊕ C). □

Proposition 6.18. For A ∈ sN_{ℚ}^{♭}, the space \underline{Del}_{f}(\underline{E_{h}}^{♯}) is functorially weakly equivalent to the nerve W̄ 𝔐(A) of the ∞-groupoid 𝔐(A) of derived geometric 0-stacks 𝔛 over B𝔾_{m} × Spec A for which X := 𝔛 ⊗_{A}^{L} H_{0}A is weakly equivalent to a flat projective scheme over H_{0}A, with the polarisation X → B𝔾_{m} ⊗ H_{0}A ample with Hilbert polynomial h.

Proof. This is essentially the same as Proposition 3.34, replacing β^{∗} with the functor β_{F}^{∗} : 𝔇𝔢𝔩(E_{h}(A)) → sD(A) on simplicial objects from Definition 5.26 (constructed similarly to those of Propositions 6.6 and 6.12). □

Remark 6.19. Note that the constructions of §5.2.2 immediately allow us to adapt E_{h} to work with moduli of diagrams of polarised projective schemes, and in particular with morphisms of such schemes. For moduli over a fixed base Proj S, an alternative approach is to replace ⊤ with the monad M ↦ S ⊗_{R} Symm_{A}M. Either of these approaches can be used to construct derived Hilbert schemes (by taking MC_{f}(E) to be the subset of MC(E) consisting of finitely generated A-algebras B for which S ⊗_{R} A → B is surjective). Propositions 6.18 and 3.34 ensure that these approaches give equivalent derived stacks, as does [CFK2].

6.4. Finite group schemes. In order to study moduli of finite group schemes, we follow the approach of [Pri6] Example 3.41, by noting that the nerve functor gives a full and faithful inclusion of the category of group schemes into the category of pointed simplicial schemes.

Given a finite group scheme G over Spec A, with O(G) := Γ(G, 𝒪_{G}) locally free of rank r, we thus look at the simplicial group scheme BG (for an explicit description, note that this is the same as W̄ G from Definition 3.15). If we write O(BG)^{n} := Γ(BG_{n}, 𝒪_{BG_{n}}), then O(BG) is a commutative cosimplicial augmented A-algebra, with O(BG)^{n} locally free of rank r^{n}.

Lemma 6.20. The functor G ↦ BG from group schemes to pointed simplicial schemes is formally étale.

Proof. A simplicial scheme X_{•} over A is of the form BG if and only if (1) X_{0} = Spec A, and

(2) for all n > 1 and all 0 ≤ k ≤ n, the maps X_{n} → Hom_{𝕊}(Λ^{n,k}, X) are isomorphisms, where Λ^{n,k} ⊂ Δ^{n} is the kth horn, obtained by removing the kth face from ∂Δ^{n}.

Since any deformation of an isomorphism is an isomorphism, the result follows. □

We could now combine §5.2.2 with §6.1 to obtain a quasi-comonoid functor governing moduli of such diagrams, but there is a far more efficient choice. If A is a local ring, then not only are the modules O(BG)^{n} independent of G: we can also describe all operations except ∂^{0}.

The following is adapted from [Pri2] Definition 3.6:

Definition 6.21. Define V̄ : sGp → 𝕊 by setting

V̄ G_{n} := G_{n−1} × G_{n−2} × … × G_{0}

with operations

∂_{0}(g_{n−1}, …, g_{0}) = ((∂_{0}g_{n−1})^{−1}g_{n−2}, …, (∂_{0}^{n−1}g_{n−1})^{−1}g_{0})
∂_{i}(g_{n−1}, …, g_{0}) = (∂_{i−1}g_{n−1}, …, ∂_{1}g_{n−i+1}, g_{n−i−1}, g_{n−i−2}, …, g_{0})
σ_{i}(g_{n−1}, …, g_{0}) = (σ_{i−1}g_{n−1}, …, σ_{0}g_{n−i}, g_{n−i}, g_{n−i−1}, …, x_{0}).

Lemma 6.22. There is a natural isomorphism ϕ̄: V̄ → W̄.

Proof. As in [Pri2] Proposition 3.9, the map ϕ̄_{G} : V̄ G → W̄ G is given by

ϕ̄(g_{n−1}, …, g_{0}) = (g_{n−1}, (∂_{0}g_{n−1})^{−1}g_{n−2} …, (∂_{0}g_{1})^{−1}g_{0}).
□

We can therefore replace B with the functor V̄, and consider the simplicial scheme V̄ G, which has the property that ∂_{0} is the only simplicial operation to depend on the group structure of G. We now proceed along the same lines as [Pri3] §5.1.

Definition 6.23. Define Δ_{∗} to be the subcategory of the ordinal number category Δ containing only those morphisms fixing 0. Given a category C, define the category c_{+}C of almost cosimplicial diagrams in C to consist of functors Δ_{∗} → C. Thus an almost cosimplicial object X^{∗} consists of objects X^{n} ∈ C, with all of the operations ∂^{i}, σ^{i} of a cosimplicial complex except ∂^{0}, satisfying the usual relations.

Definition 6.24. Define the functor F_{∂} : c_{+}Mod(A) → cMod(A) from almost cosimplicial A-modules to A-modules by

(F_{∂}M^{∗})^{n} = M^{n} ⊕ M^{n−1} ⊕ … ⊕ M^{0},

with operations

∂^{i}(v_{n}, …, v_{0}) = (∂^{i}v_{n}, ∂^{i−1}v_{n−1}, …, ∂^{1}v_{n−i+1}, 0, v_{n−i}, …, v_{1}, v_{0})
σ^{i}(v_{n}, …, v_{0}) = (σ^{i}v_{n}, …, σ^{1}v_{n−i+1}, σ^{0}v_{n−i} + v_{n−i−1}, v_{n−i−2}, …, v_{0}).

By the argument of [Pri3] Lemma 4.12, F_{∂} is left adjoint to the functor U_{∂} : cMod(A) → c_{+}Mod(A) given by forgetting ∂^{0}. Moreover, this adjunction is monadic, so for the monad ⊤_{∂} := F_{∂}U_{∂}, there is a natural equivalence

cMod(A) ≃ c_{+}Mod(A)^{⊤_{∂}}.

In fact, we can go further than this. By [Pri3] §5.1, the monad Symm distributes over ⊤_{∂}, so the composite monad Symm ∘ ⊤_{∂} is another monad. Moreover,

cAlg(A) ≃ c_{+}Mod(A)^{(Symm∘⊤_{∂})}.

We wish to modify this slightly, since we are only interested in augmented cosimplicial Aalgebras, or equivalently non-unital cosimplicial A-algebras (taking augmentation ideals). We thus replace Symm with Symm^{+} := ⨁_{n>0} S^{n}, and set ⊤ := Symm^{+} ∘ ⊤_{∂}. Then c_{+}Mod(A)^{⊤} is equivalent to the category cNAlg(A) of non-unital commutative cosimplicial A-algebras.

Definition 6.25. Define a functor E_{r} : Alg_{R} → QM^{∗} by first forming the almost cosimplicial module V̄ (A^{r}) ∈ c_{+}Mod(A) as

V̄ (A^{r})^{n} := A^{r^{n}} = \overbrace{(A^{r}) ⊗_{A} (A^{r}) ⊗_{A} … ⊗_{A} (A^{r})}^{n},

with operations dual to those of Definition 6.21 (without ∂^{0}). We then let V̄_{+}(A^{r}) := ker(V̄ (A^{r}) → A), where the A is the constant diagram V̄ (A^{0}), so

V̄_{+}(A^{r})^{n} = ker((∂^{1})^{n} : V̄ (A^{r})^{n} → V̄ (A^{r})^{0}).

We now set E_{r}(A) to be the quasi-comonoid E_{r}(A) := E(V̄_{+}(A^{r})) of §5.2.1, given by

E_{r}^{n}(A) = Hom_{c_{+}Mod(A)}(⊤_{A}^{n}V̄_{+}(A^{r}), V̄_{+}(A^{r})).

Definition 6.26. Define 𝔇𝔢𝔩(\underline{E_{r}})^{♯} : dN_{R}^{♭} → Gpd to be the stackification of 𝔇𝔢𝔩(\underline{E_{r}}) in the strict Zariski topology of Definition 2.17. Likewise, define the simplicial groupoid-valued functor \underline{𝔇𝔢𝔩}(\underline{E_{r}})^{♯} on dN_{R}^{♭} by stackifying levelwise, so (\underline{𝔇𝔢𝔩}(\underline{E_{r}})^{♯})_{n} = (\underline{𝔇𝔢𝔩}(\underline{E_{r}})_{n})^{♯}.

Definition 6.27. By Lemma 5.17, MC(E_{r}(A)) is equivalent to the groupoid of pointed simplicial affine schemes X over A for which U_{∂}O(X) ≅ V̄ (A^{r}) ∈ c_{+}Mod(A). We then define

MC_{g}(E_{r}(A)) ⊂ MC(E_{r}(A))

to consist of simplicial schemes of the form BG, for G a group scheme. By Lemma 6.20, this inclusion of functors is formally étale.

We define 𝔇𝔢𝔩_{g}(E_{r}), \underline{MC}_{g}(\underline{E}_{r}), MC_{g}(E_{r}^{♯}) and so on similarly.

Proposition 6.28. The functor W̄ \underline{𝔇𝔢𝔩_{g}(\underline{E_{r}})^{♯}} → 𝕊 is representable by an almost finitely presented derived geometric 1-stack. Moreover,

W̄ \underline{𝔇𝔢𝔩_{g}(\underline{E_{r}})^{♯}} ≃ W̄ \underline{𝔇𝔢𝔩}_{g}(\underline{E_{r}})^{♯} ≃ \underline{Del}_{g}(\underline{E_{r}}^{♯}),

where the last is defined using the quasi-comonoid sheafification of Definition 5.49.

Proof. The proofs of Propositions 6.4 and 6.17 carry over. The only differences lie in a straightforward check that 𝔇𝔢𝔩_{g}(E_{r})^{♯} is locally of finite presentation, and in the calculation of cohomology groups.

For the comonad ⊥ := Symm^{+} ∘ F_{∂}U_{∂} on cNAlg(A), we get a canonical simplicial resolution ⊥_{•}S, given by ⊥_{n}S := ⊥^{n+1}S. For A ∈ Alg_{R}, the proof of [Pri3] Lemma 5.7 then shows that A ⊕ (⊥_{•}S)^{m} is a cofibrant resolution of A ⊕ S^{m} for all m, whenever S^{m} is projective as an A-module. If we set 𝕃_{•}^{⊥}(S) := Ω((A ⊕ ⊥_{•}S)/A), this means that the simplicial complex 𝕃_{•}^{⊥}(S)^{m} is a model for the cotangent complex of A ⊕ S^{m}.

If S is levelwise projective as an A-module, then by [Pri3] Lemma 5.8, 𝕃_{n}^{⊥}(S) is a projective object of cMod(S). Thus, for S ∈ E_{r}^{♯}(A), Lemma 5.47 gives that

D_{[S]}^{i}(\underline{Del}_{f}(\underline{E_{r}}^{♯}), M) ≅ 𝔼xt_{S}^{i+1}(𝕃_{•}^{⊥}(S), S ⊗_{A} M),

which satisfies the finiteness conditions of Theorem 1.19. □

Definition 6.29. Given a flat group scheme over a ring A, follow [Ill2] §2.5.1 in defining the A-complex χ^{G/A} by

χ^{G/A} := Le^{∗}𝕃^{G/A},

where e : Spec A → G is the unit of the group structure. This has a canonical G-action, and we write χ χ χ^{G/A} for the associated complex on BG.

As in [Ill2] §4.1, χ^{G/A} is perfect and concentrated in chain degrees 0, 1. We set ω^{G/A} := H_{0}(χ^{G/A}), with ω ω ω^{G/A} the associated sheaf on BG.

Definition 6.30. For a cosimplicial ring S, we make cMod(S) into a simplicial category by setting (for K ∈ 𝕊)

(M^{K})^{n} := (M^{n})^{K_{n}},

as an S^{n}-module. This has a left adjoint, which we denote by M ↦ M ⊗ K. Given a cofibration K ↪ L in 𝕊, we write M ⊗ (L/K) := (M ⊗ L)/(M ⊗ K).

Given M ∈ cMod(S), define \underline{M} to be the bicosimplicial complex given in horizontal level i by \underline{M}^{i•} = M ⊗ Δ^{i}. Let N_{c}\underline{M} we the cochain complex in cMod(S) given by taking the horizontal cosimplicial normalisation of Definition 4.6.

Lemma 6.31. Given an affine group scheme G over A, with Γ(G, 𝒪_{G}) locally free of rank r over A, let S ∈ 𝔇𝔢𝔩(E_{r})^{♯}(A) be the associated cosimplicial ring

S^{n} := ker(Γ((BG)_{n}, 𝒪_{(BG)_{n})}) → A).

Then for M ∈ Mod_{A},

D_{[S]}^{i}(\underline{Del}_{g}(\underline{E_{r}}^{♯}), M) ≅ 𝔼xt_{BG}^{i}(𝕃^{BG/A}, ker(𝒪_{BG} ⊗_{A} M → M)).

In particular,

D_{[S]}^{i}(\underline{Del}_{g}(\underline{E_{r}}^{♯}), M) ≅ 𝔼xt_{BG}^{i+2}(χ χ χ^{G/A}, 𝒪_{BG} ⊗_{A} M)

for i ≥ 1. For low degrees, there is an exact sequence

0 → Hom_{BG}(ω ω ω^{G/A}, 𝒪_{BG} ⊗_{A} M) → Hom_{A}(ω^{G/A}, M) →
D_{[S]}^{−1}(\underline{Del}_{g}(\underline{E_{r}}^{♯}, M) → Ext_{BG}^{1}(χ χ χ^{G/A}, 𝒪_{BG} ⊗_{A} M) → Ext_{A}^{1}(χ^{G/A}, M) →
D_{[S]}^{0}(\underline{Del}_{g}(\underline{E_{r}}^{♯}, M) → Ext_{BG}^{2}(χ χ χ^{G/A}, 𝒪_{BG} ⊗_{A} M) → 0.

Proof. Write L := 𝕃_{•}^{⊥}(S) , defined as in the proof of Proposition 6.28, and recall that this is a projective object of cMod(S). Observe that in the terminology of [Pri8], Spec (A ⊕ S) is a derived fppf 1-hypergroupoid, and a derived Artin 1-hypergroupoid whenever G is smooth.

If G is smooth, then [Pri8] Proposition 7.21 shows that

𝔼xt_{S}^{i}(Tot N_{c}\underline{L}, S ⊗_{A} M) ≅ 𝔼xt_{BG}^{i}(𝕃^{BG/A}, ker(𝒪_{BG} ⊗_{A} M → M)),

where Tot is the total complex functor. For general G, the same formula holds, since [Pri8] Proposition 7.14 only uses the Artin hypothesis to prove that Tot N_{c}N^{s}\underline{L} is projective, while the descent argument from the proof of [Pri8] Proposition 7.13 works for all faithfully flat morphisms.

Now, [Pri8] Lemmas 2.18 and 7.4 combine to show that N_{c}^{i}\underline{L} is acyclic for i > 1, while N_{c}^{1}\underline{L} is the pullback to BG of the cotangent complex of a trivial relative derived 1-hypergroupoid. It then follows from [Pri8] Lemma 2.9 that there are canonical isomorphisms

Ext_{S}^{∗}(N_{c}^{1}\underline{L}, P) ≅ Ext_{S^{0}}^{∗}((N_{c}^{1}\underline{L})^{0}, P^{0})

for all P ∈ cMod(S). Since S^{0} = 0, this means that

𝔼xt_{S}^{i}(Tot N_{c}N^{s}\underline{L}, S ⊗_{A} M) ≅ 𝔼xt^{i}(L, S ⊗_{A} M).

Thus, combined with the proof of Proposition 6.28, we get

D_{[S]}^{i}(\underline{Del}_{g}(\underline{E_{r}}^{♯}), M) ≅ 𝔼xt_{BG}^{i}(𝕃^{BG/A}, ker(𝒪_{BG} ⊗_{A} M → M)).

Finally, 𝕃^{BG/A} ≃ χ χ χ^{G/A}[1], so the exact sequence 0 → S → A⊕S → A → 0 of S-modules gives the required long exact sequence. □

Proposition 6.32. For A ∈ sN_{R}^{♭}, the space \underline{Del}_{g}(\underline{E_{r}}^{♯})(A) is functorially weakly equivalent to the nerve W̄ 𝔐(A) of the ∞-groupoid 𝔐(A) of pointed derived geometric 1-stacks X over A for which X ⊗_{A}^{L} π_{0}A is weakly equivalent to the nerve of a flat rank r group scheme over π_{0}A.

Proof. We work along the same lines as Proposition 6.6. As a consequence of Proposition 6.28 and Remark 1.30, it suffices to construct a natural transformation

Φ : 𝔇𝔢𝔩_{g}(\underline{E_{r}})^{♯} → 𝔐

of ∞-groupoids, inducing equivalences on π^{0} and isomorphisms on D^{i} of the nerves.

Given an object of 𝔇𝔢𝔩_{g}(\underline{E_{r}})^{♯}(A), we get M ∈ c_{+}Mod(A_{0}), locally isomorphic (over A_{0}) to V̄_{+}(A_{0}^{r}), together with elements

ω_{n} ∈ Hom_{c_{+}Mod(A_{0})}(⊤^{n+1}M, M ⊗_{A_{0}} (A^{I^{n}})_{0}),

satisfying the Maurer–Cartan relations of Definition 5.4. For the free functor F : sc_{+}Mod(A) → scNAlg(A) from simplicial almost cosimplicial A-modules to non-unital simplicial cosimplicial commutative A-algebras, Definition 5.26 thus gives us a functor

β_{F}^{∗} : 𝔇𝔢𝔩(E_{r})^{♯}(A) → scNAlg(A).

We therefore get a functor ϕ : 𝔇𝔢𝔩(E_{h}(A)) → scAlg(A)↓A to the category of augmented simplicial cosimplicial commutative A-algebras, given by ω ↦ A ⊕ β_{F}^{∗} (ω). Moreover, it follows from Definition 5.26 that all objects in the image of ϕ are Reedy cofibrant. If A ∈ Alg_{R} and ω corresponds to a group scheme G, then arguing as in the proof of Proposition 6.6, ϕ(ω) is a cofibrant resolution of O(BG) as a simplicial augmented cosimplicial commutative A-algebra. Therefore for arbitrary A ∈ sN_{R}^{♭},

Spec (ϕ(ω) ⊗_{A}^{L} π_{0}A)

is a pointed fppf 1-hypergroupoid, so Spec ϕ(ω) is a pointed derived fppf 1-hypergroupoid.

We therefore define Φ(ω) to be the homotopy-fppf hypersheafification of Spec ϕ(ω). By [Pri8] Theorem 4.15, Φ(ω) is a pointed derived geometric fppf 1-stack whenever ω ∈ 𝔇𝔢𝔩_{g}(\underline{E_{r}})^{♯}(A). By [Toë] Theorem 0.1, this is the same as a derived geometric Artin 1-stack.

When A ∈ Alg_{R}, with ω corresponding to G, we have seen that Φ(ω) is just the classifying stack BG. For arbitrary A ∈ sN_{R}^{♭}, this means that Φ(ω) ⊗_{A}^{L} π_{0}A is of the form BG for some flat rank r group scheme G over π_{0}A. Thus Φ indeed gives a functor Φ : 𝔇𝔢𝔩_{g}(\underline{E_{r}})^{♯} → 𝔐.

The arguments above have shown that π^{0}Φ is an equivalence, since the space of group homomorphisms G → G′ corresponds to the space of pointed morphisms BG → BG′. To see that Φ gives isomorphisms

D_{ω}^{i}(B𝔇𝔢𝔩(E_{r})^{♯}, M) → D_{Φ(ω)}^{i}(W̄ 𝔐, M),

we combine Lemma 6.31 with [Pri6] Theorem 3.35. □

References

[Ane] Mathieu Anel. Grothendieck topologies from unique factorisation systems. arXiv:0902.1130v2 [math.AG], 2009.

[BK] A. K. Bousfield and D. M. Kan. Homotopy limits, completions and localizations. Lecture Notes in Mathematics, Vol. 304. Springer-Verlag, Berlin, 1972.

[CFK1] Ionuţ Ciocan-Fontanine and Mikhail Kapranov. Derived Quot schemes. Ann. Sci. École Norm. Sup. (4), 34(3):403–440, 2001.

[CFK2] Ionuţ Ciocan-Fontanine and Mikhail M. Kapranov. Derived Hilbert schemes. J. Amer. Math. Soc., 15(4):787–815 (electronic), 2002.

[CLM] Frederick R. Cohen, Thomas J. Lada, and J. Peter May. The homology of iterated loop spaces. Springer-Verlag, Berlin, 1976. Lecture Notes in Mathematics, Vol. 533.

[CR1] A. M. Cegarra and J. Remedios. Diagonal fibrations are pointwise fibrations. J. Homotopy Relat. Struct., 2(2):81–92, 2007.

[CR2] A. M. Cegarra and Josué Remedios. The relationship between the diagonal and the bar constructions on a bisimplicial set. Topology Appl., 153(1):21–51, 2005.

[DK] W. G. Dwyer and D. M. Kan. Homotopy theory and simplicial groupoids. Nederl. Akad. Wetensch. Indag. Math., 46(4):379–385, 1984.

[GJ] Paul G. Goerss and John F. Jardine. Simplicial homotopy theory, volume 174 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1999.

[Gro1] A. Grothendieck. Éléments de géométrie algébrique. III. étude cohomologique des faisceaux cohérents. I. Inst. Hautes Études Sci. Publ. Math., (11):167, 1961.

[Gro2] Alexander Grothendieck. Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert. In Séminaire Bourbaki, Vol. 6, pages Exp. No. 221, 249– 276. Soc. Math. France, Paris, 1995.

[Har1] Robin Hartshorne. Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20. Springer-Verlag, Berlin, 1966.

[Har2] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.

[HS] Mark Haiman and Bernd Sturmfels. Multigraded Hilbert schemes. J. Algebraic Geom., 13(4):725– 769, 2004.

[Ill1] Luc Illusie. Complexe cotangent et déformations. I. Springer-Verlag, Berlin, 1971. Lecture Notes in Mathematics, Vol. 239.

[Ill2] Luc Illusie. Cotangent complex and deformations of torsors and group schemes. In Toposes, algebraic geometry and logic (Conf., Dalhousie Univ., Halifax, N.S., 1971), pages 159–189. Lecture Notes in Math., Vol. 274. Springer, Berlin, 1972.

[LMB] Gérard Laumon and Laurent Moret-Bailly. Champs algébriques, volume 39 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2000.

[Lur] J. Lurie. Derived Algebraic Geometry. PhD thesis, M.I.T., 2004. www.math.harvard.edu/∼lurie/papers/DAG.pdf or http://hdl.handle.net/1721.1/30144.

[Mum] David Mumford. Lectures on curves on an algebraic surface. With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59. Princeton University Press, Princeton, N.J., 1966.

[Pri1] J. P. Pridham. Deformations of schemes and other bialgebraic structures. Trans. Amer. Math. Soc., 360(3):1601–1629, 2008.

[Pri2] J. P. Pridham. Pro-algebraic homotopy types. Proc. London Math. Soc., 97(2):273–338, 2008. arXiv math.AT/0606107 v8.

[Pri3] J. P. Pridham. Derived deformations of Artin stacks. arXiv:0805.3130v3 [math.AG], submitted, 2009.

[Pri4] J. P. Pridham. The homotopy theory of strong homotopy algebras and bialgebras. Homology, Homotopy Appl., 12(2):39–108, 2010. arXiv:0908.0116v2 [math.AG].

[Pri5] J. P. Pridham. Derived deformations of schemes. Comm. Anal. Geom., 20(3):529–563, 2012. arXiv:0908.1963v1 [math.AG].

[Pri6] J. P. Pridham. Derived moduli of schemes and sheaves. J. K-Theory, 10(1):41–85, 2012. arXiv:1011.2189v5 [math.AG].

[Pri7] J. P. Pridham. Representability of derived stacks. J. K-Theory, 10(2):413–453, 2012. arXiv:1011.2742v4 [math.AG].

[Pri8] J. P. Pridham. Presenting higher stacks as simplicial schemes. Adv. Math., to appear. arXiv:0905.4044v3 [math.AG].

[Qui1] Daniel Quillen. Rational homotopy theory. Ann. of Math. (2), 90:205–295, 1969.

[Qui2] Daniel Quillen. On the (co-) homology of commutative rings. In Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), pages 65–87. Amer. Math. Soc., Providence, R.I., 1970.

[Ser] Jean-Pierre Serre. Faisceaux algébriques cohérents. Ann. of Math. (2), 61:197–278, 1955. [Toë] Bertrand Toën. Descente fidèlement plate pour les n-champs d’Artin. Compos. Math., 147(5):1382–

1412, 2011. arXiv:0911.3554v2 [math.AG].

[TV] Bertrand Toën and Gabriele Vezzosi. Homotopical algebraic geometry. II. Geometric stacks and applications. Mem. Amer. Math. Soc., 193(902):x+224, 2008. arXiv math.AG/0404373 v7.

[Wei] Charles A. Weibel. An introduction to homological algebra. Cambridge University Press, Cambridge, 1994.