REPRESENTABILITY OF DERIVED STACKS

J.P.PRIDHAM

Abstract. Lurie’s representability theorem gives necessary and sufficient conditions for a functor to be an almost finitely presented derived geometric stack. We establish several variants of Lurie’s theorem, making the hypotheses easier to verify for many applications. Provided a derived analogue of Schlessinger’s condition holds, the theorem reduces to verifying conditions on the underived part and on cohomology groups. Another simplification is that functors need only be defined on nilpotent extensions of discrete rings. Finally, there is a pre-representability theorem, which can be applied to associate explicit geometric stacks to dg-manifolds and related objects.

Contents

Introduction 1 1. Representability of derived stacks 3 1.1. Background 3 1.2. Tangent spaces and homogeneity 4 1.3. Finite presentation 7 1.4. Sheaves 8 1.5. Representability 10 1.6. Strong quasi-compactness 13 2. Complete simplicial and chain algebras 14 2.1. Nilpotent algebras 17 2.2. A nilpotent representability theorem 18 2.3. Covers 21 3. Pre-representability 21 3.1. Simplicial structures 21 3.2. Deriving functors 23 3.3. Representability 26 References 27

Introduction

Artin’s representability theorem ([Art]) gives necessary and sufficient conditions for a functor from R-algebras to groupoids to be representable by an algebraic Artin stack, locally of finite presentation. In his thesis, Lurie established a similar result not just for derived Artin 1-stacks, but for derived geometric Artin n-stacks. Explicitly, given a functor F : sAlg_{R} → 𝕊 from simplicial R-algebras to simplicial sets, [Lur] Theorems 7.1.6 and 7.5.1 give necessary and sufficient conditions for F to be representable by a derived geometric Artin n-stack, almost of finite presentation over R.

Lurie’s Representability Theorem is more natural than Artin’s in one important respect: in the derived setting, existence of a functorial obstruction theory is an automatic consequence of left-exactness. However, Lurie’s theorem can be difficult to verify for problems not explicitly coming from topology. The most basic difficulty can be showing that a functor is homotopy-preserving, or finding a suitable functor which is. It tends to be even more difficult to show that a functor is almost of finite presentation, or to verify that it is a hypersheaf. The purpose of this paper is to adapt the representability theorems in [Lur] and [TV], simplifying these criteria for a functor F : sAlg_{R} → 𝕊 to be a geometric n-stack.

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

In [Lur], the key exactness properties used were cohesiveness and infinitesimal cohesiveness. These are said to hold for a functor F : sAlg_{R} → 𝕊 if the maps

θ : F (A ×_{B} C) → F (A) ×_{F (B)}^{h} F (C)

to the homotopy fibre product are weak equivalences for all surjections (resp. nilpotent surjections) A ↠ B and C ↠ B. The key idea of this paper is to introduce a notion more in line with Schlessinger’s conditions ([Sch]). We say that F is homotopy-homogeneous if θ is a weak equivalence for all nilpotent surjections A ↠ B and arbitrary maps C → B.

The first major consequence is Theorem 1.23, showing that if F is homotopyhomogeneous, then it is almost finitely presented whenever the restriction π^{0}(F) : Alg_{H_{0}R} → 𝕊 and the cohomology theories D_{x}^{i}(F, −) of the tangent spaces of F at discrete points x are all finitely presented. This reduces the question to familiar invariants, since the cohomology groups are usually naturally associated to the moduli problem. Likewise, Proposition 1.32 shows that to ensure that a homotopy-homogeneous functor F is a hypersheaf, it suffices to check that π^{0}F is a hypersheaf and that the modules D_{x}^{i}(F, −) are quasi-coherent.

These results are applied to Proposition 1.33, which shows that with certain additional finiteness hypotheses on D_{x}^{i}(F), a cotangent complex and obstruction theory exist for F. This leads to Theorem 1.34, which replaces Lurie’s almost finite presentation condition with those of Theorem 1.23. We then obtain Corollary 1.36, which incorporates the further simplifications of Proposition 1.32.

A key principle in derived algebraic geometry is that the derived structure is no more than an infinitesimal thickening of the underived objects. For instance, every simplicial ring can be expressed as a composite of homotopy square-zero extensions of a discrete ring. Proposition 2.7 strictifies this result, showing that we can work with extensions which are nilpotent (rather than just homotopy nilpotent). This approach leads to Theorem 2.17, which shows how the earlier representability results can be reformulated for functors on dg or simplicial rings A for which A → H_{0}A is nilpotent, thereby removing the need for Lurie’s nilcompleteness hypothesis.

The last major result is Theorem 3.16, which shows how to construct representable functors from functors which are not even homotopy-preserving. The key motivation is Example 3.17, which constructs explicit derived geometric stacks from Kontsevich’s dg manifolds.

The structure of the paper is as follows.

In Section 1, we recall Lurie’s Representability Theorem, introduce homotopyhomogeneity, and establish the variants Theorem 1.34 and Corollary 1.36 of Lurie’s theorem. We also establish Proposition 1.38, which identifies weak equivalences between geometric derived n-stacks, and Proposition 1.40, which gives a functorial criterion for strong quasi-compactness.

Section 2 then introduces simplicial or dg algebras A for which A → H_{0}A is a nilpotent extension, showing in Theorem 2.17 how to re-interpret representability in terms of functor on such algebras.

Finally, Section 3 introduces the notion of homotopy-surjecting functors; these map square-zero acyclic extensions to surjections. For any such functor F, we construct another functor W̄ \underline{F}, and Proposition 3.10 shows that this is homotopy-preserving whenever F is homotopy-homogeneous and homotopy-surjecting. This leads to Theorem 3.16, which gives sufficient conditions on F for W̄ \underline{F} to be a derived geometric n-stack.

1. Representability of derived stacks

We denote the category of simplicial sets by 𝕊, the category of simplicial rings by sRing, and the category of simplicial R-algebras by sAlg_{R}. We let dg_{+}Alg_{R} be the category of differential graded-commutative R-algebras in non-negative chain degrees. The homotopy category Ho(C) of a category C is obtained by formally inverting weak equivalences.

1.1. Background. Given a simplicial ring R, a derived geometric n-stack over R is a functor

F : sAlg_{R} → 𝕊

satisfying many additional conditions. These are detailed in [TV] Chapter 2.2 or [Lur] §5.1. A more explicit characterisation in terms of certain simplicial cosimplicial rings is given in [Pri3] Theorem 4.15. However, for the purposes of this paper, these definitions are largely superfluous, since it will be enough to consider functors satisfying Lurie’s Representability Theorem:

Theorem 1.1. A homotopy-preserving functor F : sAlg_{R} → 𝕊 is a geometric derived n-stack which is almost of finite presentation if and only if

(1) The functor F commutes with filtered colimits when restricted to k-truncated objects of sAlg_{R}, for each k ≥ 0.

(2) For any discrete commutative ring A, the space F (A) is n-truncated.

(3) The functor F is a hypersheaf for the étale topology.

(4) The functor F is cohesive: for any pair A → C, B → C of surjective morphisms in sAlg_{R}, the induced map

F (A ×_{C} B) → F (A) ×_{F (C)}^{h} F (B)

is a weak equivalence.

(5) The functor F is nilcomplete: for any A ∈ sAlg_{R}, the natural map F (A) → \underleftarrow{lim}_{k}^{h} F (P_{k}A) is an equivalence, where \{P_{k}A\}_{k} denotes the Moore-Postnikov tower of A.

(6) Let B be a complete, discrete, local, Noetherian R-algebra, and 𝔪 ⊂ B the maximal ideal. Then the natural map F (B) → \underleftarrow{lim}_{n}^{h} F (B/m^{n}) is a weak equivalence.

(7) Let x ∈ F (C), where C is a (discrete) integral domain which is finitely generated as a π_{0}R-algebra. For each i, n, the tangent module

D_{x}^{n−i}(F, C) := π_{i}(F (C ⊕ C[−n]) ×_{F (C)}^{h} {x})

is a finitely generated C-module.

(8) R is a derived G-ring:

(a) π_{0}R is Noetherian,

(b) for each prime ideal 𝔭 ⊂ π_{0}R, the 𝔭(π_{0}R)_{𝔭}-adic completion of (π_{0}R)_{𝔭} is a geometrically regular π_{0}R-algebra, and

(c) for all n, π_{n}R is a finite π_{0}R-module.

(9) R admits a dualising module in the sense of [Lur] Definition 3.6.1. [For discrete rings, this is equivalent to a dualising complex. In particular, ℤ and Gorenstein local rings are all derived G-rings with dualising modules.]

Proof. [Lur] Theorem 7.5.1. □

Readers unfamiliar with the conditions of this theorem should not despair, since the conditions will be explained and considerably simplified over the course of this paper.

Remark 1.2. 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. Conversely, every geometric n-stack is (n + 2)-geometric.

We can summarise this by saying that for a derived geometric stack 𝔛 to be n-truncated means that 𝔛 → 𝔛^{S^{n+1}} is an equivalence, or equivalently that 𝔛 → 𝔛^{S^{n−1}} is representable by derived algebraic spaces. For 𝔛 to be n-geometric means that 𝔛 → 𝔛^{S^{n−1}} is representable by disjoint unions of derived affine schemes.

Theorem 1.34 takes the convention from [Lur], so “geometric derived n-stack” means “n-truncated derived geometric stack”.

1.2. Tangent spaces and homogeneity.

Definition 1.3. We say that a map A → B in sRing is a square-zero extension if it is surjective, and the kernel I is square-zero, i.e. satisfies I^{2} = 0.

Lemma 1.4. In Ho(sAlg_{R}), square-zero extensions A → B with kernel I correspond up to weak equivalence to the small extensions A of B by I in the sense of [Lur] Definition 3.3.1.

Proof. Given a square-zero extension A → B, observe that the kernel I is a simplicial B-module. Choose an inclusion i : I ↪ N of simplicial B-modules, with N acyclic, and set B̃ to be the simplicial algebra A ⊕_{I} N. Then B̃ → B is a trivial fibration, and if we let C = coker i, then

A = B̃ ×_{B⊕C} B.

Now we need only observe that ΩC ≃ I in the notation of [TV], so B̃ → B ⊕ C gives a homotopy derivation s : B → I[−1], with

A = B ⊕_{s} I := B ×_{id+s,B⊕I[−1],id+0}^{h} B,

so A → B is a small extension in Lurie’s sense.

Conversely, given a homotopy derivation s : B → M[−1], we may assume that B is cofibrant, so lift this to a morphism B → B ⊕ M[−1] of simplicial R-algebras. Taking a surjection f : N ↠ M[−1] of simplicial B-modules, with N acyclic, we see that

B ×_{id+s,B⊕M[−1],id+0}^{h} B ≃ B ×_{id+s,B⊕M[−1],id} (B ⊕ N),

since the right-hand map is a fibration. But this maps surjectively to B, with kernel I := ker f, which is a B-module, so square-zero. Moreover M ≃ I, so the respective square-zero extensions are by the same module. □

Remark 1.5. Given a simplicial ring B and a simplicial B-module M, we may define a derivation t : B ⊕ M[−1] → M[−1] given by 0 on B, and by the identity on M[−1]. The corresponding square-zero extension (B ⊕ M[−1]) ⊕_{t} M is equivalent to B. In particular, this means that B → B ⊕ M[−1] is weakly equivalent to a square-zero extension.

Definition 1.6. Say that a functor between model categories is homotopy-preserving if it maps weak equivalences to weak equivalences.

Definition 1.7. We say that a functor

F : sAlg_{R} → 𝕊

is homotopy-homogeneous if for all square-zero extensions A → B and all maps C → B in sAlg_{R}, the natural map

F (A ×_{B} C) → F (A) ×_{F (B)}^{h} F (C)

to the homotopy fibre product is a weak equivalence.

Definition 1.8. Given a homotopy-preserving homotopy-homogeneous functor F : sAlg_{R} → 𝕊, a simplicial R-algebra A, and a point x ∈ F (A), define the tangent functor

T_{x}(F/R) : sMod_{A} → 𝕊

by

T_{x}(F/R)(M) := F (A ⊕ M) ×_{F (A)}^{h} {x}.

Lemma 1.9. If F satisfies the conditions of Definition 1.8, then up to non-canonical weak equivalence, T_{x}(F/R)(M) is an invariant of the class [x] ∈ π_{0}F (A).

Proof. Given a path γ : Δ^{1} → F (A), we have equivalences

T_{γ(0)}(F/R)(M) ≃ Δ^{1} ×_{γ,F (A)}^{h} F (A ⊕ M) ≃ T_{γ(1)}(F/R)(M),

so paths in F (A) give equivalences between stalks. Considering maps Δ^{2} → F (A), we see that these equivalences satisfy the cocycle condition up to homotopy, with the maps Δ^{n} → F (A) giving higher homotopies. Thus T_{(−)}(F/R)(M) forms a weak local coefficient system on F (A). □

Definition 1.10. Given a simplicial abelian group A_{•}, we denote the associated normalised chain complex 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).

Using the Eilenberg-Zilber shuffle product, normalisation N extends to a functor

N : sRing → dg_{+}Ring

from simplicial rings to differential graded rings in non-negative chain degrees.

By the Dold-Kan correspondence, normalisation gives an equivalence of categories between simplicial abelian groups and chain complexes in non-negative degrees. For any R ∈ sRing, this extends to an equivalence

sMod_{R} → dg_{+}Mod_{NR}

between simplicial R-modules and NR-modules in non-negatively graded chain complexes.

Definition 1.11. Given a chain complex V, let V [r] be the chain complex V [r]_{i} := V_{r+i}. Given a simplicial abelian group M and n ≥ 0, let M[−n] := N^{−1}(NM[−n]), where N^{−1} is inverse to the normalisation functor N.

For R ∈ sRing, observe that this extends to a functor [−n] : sMod_{R} → sMod_{R}. Note that π_{i}M[−n] = π_{i−n}M.

Lemma 1.12. For all F, A, M, x as in Definition 1.8, there is a natural abelian structure on π_{i}T_{x}F (M). Moreover, there are natural isomorphisms

π_{i}T_{x}(F/R)(M) ≅ π_{i+1}T_{x}F (M[−1]),

where homotopy groups are defined relative to the basepoint 0 given by the image of T_{x}(F/R)(0) → T_{x}(F/R)(M).

Proof. Addition in M gives a morphism

(A ⊕ M) ×_{A} (A ⊕ M) ≅ A ⊕ (M ⊕ M) → A ⊕ M,

so the corresponding map

F (A ⊕ M) ×_{F (A)}^{h} F (A ⊕ M) → F (A ⊕ M).

induces an abelian structure on π_{i}T_{x}F (M).

For the second part, observe that M = 0 ×_{M[−1]}^{h} 0, and that 0 → M[−1] is surjective (in the sense that it is surjective on π_{0}), so

F (A ⊕ M) ≃ F (A) ×_{F (A⊕M[−1])}^{h} F (A)

by homotopy-homogeneity, giving

T_{x}(F/R)(M) ≃ 0 ×_{T_{x}(F/R)(M[−1])}^{h} 0.

Thus π_{i}T_{x}(F/R)(M) ≅ π_{i+1}T_{x}(F/R)(M[−1]), as required. □

Definition 1.13. For all F, A, x as above, and all simplicial A-modules M, define

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

observing that this is well-defined, by Lemma 1.12.

Remark 1.14. Observe that if F is a derived geometric n-stack, and x : Spec A → F over Spec R, then D_{x}^{j}(F, M) = Ext_{A}^{j}(x^{∗}𝕃_{•}^{F/Spec R}, M), for 𝕃_{•}^{F/R} the cotangent complex of F over R.

Lemma 1.15. For F, A, x as above, with f : A → B a morphism of simplicial R-algebras, and M a simplicial B-module, there are natural isomorphisms

T_{x}(F/R)(f_{∗}M) ≃ T_{f_{∗}x}(F/R)(M),

and hence D_{x}^{j}(F, f_{∗}M) ≅ D_{f_{∗}x}^{j}(F, M).

Proof. This is just the observation that A ⊕ f_{∗}M = A ×_{B} (B ⊕ M), so F (A ⊕ f_{∗}M) ≃ F (A) ×_{F (B)}^{h} F (B ⊕ M). □

Lemma 1.16. If X : sAlg_{R} → 𝕊 is homotopy-preserving and homotopy-homogeneous, take an object A ∈ sAlg_{R} and an A-module M. Then there is a local coefficient system

D^{∗}(X, M)

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

Proof. This follows straightforwardly from the proof of Lemma 1.9. □

Proposition 1.17. If F : sAlg_{R} → 𝕊 is homotopy-preserving and homotopy-homogeneous, then for any square-zero extension e : I → A ⟶^{f} B in sAlg_{R}, there is a sequence of sets

π_{0}(F A) ⟶^{f_{∗}} π_{0}(F B) ⟶^{o_{e}} Γ(F B, D^{1}(F, 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, 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 over x is weakly equivalent to T_{x}(F/R, 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, x) \ar[r]^{o_{e}} & D_{y}^{1−n}(F, I) \ar[r]^{e_{∗}} & π_{n−1}(F A, y) \ar[r]^{f_{∗}} & ⋯ \\  & ⋯ \ar[r]^{f_{∗}} & π_{1}(F B, x) \ar[r]^{o_{e}} & D_{y}^{0}(F, I) \ar[r]^{−∗y} & π_{0}(F A).}

Proof. The proof of [Pri1] Theorem 1.45 carries over to this context. The main idea is that as in the proof of Lemma 1.4, there is a trivial fibration B̃ → B, and A = B̃ ×_{B⊕I[−1]} B, with B̃ → B ⊕ I[−1] a square-zero extension. By homotopy-homogeneity,

F (A) ≃ F (B̃) ×_{F (B⊕I[−1])}^{h} F (B),

and F (B̃) ≃ F (B) since F is homotopy-preserving.

The rest of the proof then follows by studying the long exact sequence of homotopy groups associated to the homotopy fibres of

F (B̃) → F (B ⊕ I[−1])

and of F A → F B, noting that F (A ×_{B} A) ≃ F (A) ×_{F (B)}^{h} F (B ⊕ I). □

1.3. Finite presentation.

Definition 1.18. Recall (e.g. from [GJ] Definition VI.3.4) that the Moore-Postnikov tower {P_{n}X} of a fibrant simplicial set X is given by

P_{n}X_{q} := Im (X_{q} → Hom(sk_{n}Δ^{q}, X)),

with the obvious simplicial structure. Here, sk_{n}K denotes the n-skeleton of K, the simplicial set generated by K_{≤n}.

The spaces P_{n}X form an inverse system X → … → P_{n}X → P_{n−1}X → …, with X = \underleftarrow{lim} P_{n}X, and

π_{q}P_{n}X = \begin{cases}π_{q}X & q ≤ n \\ 0 & q > n. \end{cases}

The maps P_{n}X → P_{n−1}X are fibrations. If X is reduced, then so is P_{n}X.

Definition 1.19. Define τ_{≤k}(sAlg_{R}) to be the full subcategory of sAlg_{R} consisting of objects A with A = P_{k}A, the kth Moore-Postnikov space.

Definition 1.20. Define the category τ_{≤k}Ho(sAlg_{R}) to be the full subcategory of Ho(sAlg_{R}) consisting of objects A with π_{i}A = 0 for i > k. Note that τ_{≤k}Ho(sAlg_{R}) is equivalent to the category Ho(τ_{≤k}(sAlg_{R})) obtained by localising τ_{≤k}(sAlg_{R}) at weak equivalences.

Definition 1.21. Recall from [Lur] Proposition 5.3.10 that a homotopy-preserving functor F : sAlg_{R} → 𝕊 is said to be almost of finite presentation if for all k and all filtered direct systems {A_{α}}_{α∈𝕀} in τ_{≤k}(sAlg_{R}), the map

\underrightarrow{lim}F (A_{α}) → F (\underrightarrow{lim} A_{α})

is a weak equivalence.

Definition 1.22. Given a functor F : sAlg_{R} → 𝕊, define π^{0}F : Alg_{π_{0}R} → 𝕊 by π^{0}F (A) = F (A).

Theorem 1.23. If a homotopy-preserving functor F : sAlg_{R} → 𝕊 is homotopyhomogeneous, then it is almost of finite presentation if and only if the following hold:

(1) the functor π^{0}F : Alg_{π_{0}R} → 𝕊 preserves filtered colimits;

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

Proof. Note that since π^{0}F preserves filtered colimits, Lemma 1.12 implies that the functors D_{x}^{i}(F, −) : Mod_{A} → Ab preserve filtered colimits for all i ≤ 0.

We need to prove that F preserves filtered homotopy colimits in the categories τ_{≤k}(sAlg_{R}). We prove this by induction on k, the case k = 0 following by hypothesis.

Take a filtered direct system {A_{α}} in τ_{≤k}(sAlg_{R}), with homotopy colimit A. Let B_{α} = P_{k−1}A_{α}, B = P_{k−1}A. Let M_{α} := π_{k}A_{α}, M := π_{k}A, and observe that these are π_{0}A_{α}- and π_{0}A-modules respectively.

Now, A_{α} → B_{α} and A → B are square-zero extensions up to homotopy (see for instance [TV] Lemma 2.2.1.1), coming from essentially unique homotopy derivations δ : B_{α} → M_{α}[−k − 1], with

A_{α} ≃ B_{α} ×_{id+δ,B_{α}⊕M_{α}[−k−1],id+0}^{h} B_{α} = B_{α} ×_{id+δ,π_{0}(A_{α})⊕M_{α}[−k−1],id+0}^{h} π_{0}(A_{α}).

Now, by Remark 1.5, the map π_{0}(A_{α}) → π_{0}(A_{α}) ⊕ M_{α}[−k − 1] is weakly equivalent to a square-zero extension. Thus, since F is homotopy-homogeneous,

F (A_{α}) ≃ F (B_{α}) ×_{id+δ,F (π_{0}(A_{α})⊕M_{α}[−k−1])}^{h} F (π_{0}(A_{α}))

and similarly for A.

We wish to show that θ : \underrightarrow{lim} F (A_{α}) → F (A) is a weak equivalence, and our inductive hypothesis gives \underrightarrow{lim} F (B_{α}) ≃ F (B). It therefore suffices to consider the homotopy fibre of θ over y ∈ F (B), which lifts to some ỹ ∈ F (B_{β}). If we let ỹ_{α} be the image of ỹ in F (B_{α}), this gives

θ_{y} : \underrightarrow{lim}\{ỹ_{α}\} ×_{id+δ,F (π_{0}(A_{α})⊕M_{α}[−k−1])}^{h} F (π_{0}(A_{α})) → \{y\} ×_{id+δ,F (π_{0}(A)⊕M[−k−1])}^{h} F (π_{0}(A)).

Since we know that F (π_{0}(A)) = \underrightarrow{lim} F (π_{0}(A_{α})), it suffices to show that for the images x̃_{α} ∈ F (π_{0}A_{α}), x ∈ F (π_{0}A) of ỹ_{α}, y, the maps

\underrightarrow{lim} F (π_{0}(A_{α}) ⊕ M_{α}[−k − 1]) ×_{F (π_{0}(A_{α}))} \{x̃_{α}\} → F (π_{0}(A) ⊕ M[−k − 1]) ×_{F (π_{0}(A))} \{x\}

are equivalences. Taking homotopy groups, this becomes

\underrightarrow{lim} Dk+1−i ; x̃_{α}(F, M_{α}) → D_{x}^{k+1−i}(F, M),

which by Lemma 1.15 is

\underrightarrow{lim} Dk+1−i ; x̃(F, M_{α}) → Dk+1−i ; x̃(F, M),

for x̃ ∈ F (π_{0}A_{β}) the image of ỹ.

It will therefore suffice to show that the functors Di ; x̃(F, −) : Mod_{π_{0}A_{β}} → Ab preserve filtered colimits. If we express π_{0}A_{β} as a filtered colimit of finitely generated π_{0}R-algebras, then the condition that π^{0}F preserves filtered colimits allows us to write [x̃] = [f_{∗}z] ∈ π_{0}F (A), for z ∈ F (C)_{0}, with C a finitely-generated π_{0}R-algebra. Then

Di ; x̃(F, −) ≅ D_{f_{∗}z}^{i}(F, −) ≅ D_{z}^{i}(F, f_{∗}−),

which preserves filtered colimits by hypothesis. □

1.4. Sheaves.

Definition 1.24. Let RTot_{𝕊} : c𝕊 → 𝕊 be the derived total space functor from cosimplicial simplicial sets to simplicial sets, given by

RTot_{𝕊}X^{•} = holim_{\overleftarrow{n∈Δ}} X^{n},

as in [GJ] §VIII.1. Explicitly,

RTot_{𝕊}X^{•} = {x ∈ ∏_{n}(X^{n})^{Δ^{n}} : ∂_{X}^{i}x_{n} = (∂_{Δ}^{i})^{∗}x_{n+1}, σ_{X}^{i}x_{n} = (σ_{Δ}^{i})^{∗}x_{n−1}},

whenever X is Reedy fibrant. Homotopy groups of the total space are related to a spectral sequence given in [GJ] §VIII.1.

Definition 1.25. A morphism f : A → B in sRing is said to be étale if π_{0}f is étale and the maps π_{n}(A) ⊗_{π_{0}(A)} π_{0}(B) → π_{n}(B) are isomorphisms for all n. An étale morphism is said to be an étale covering if the morphism Spec π_{0}f : Spec π_{0}B → Spec π_{0}A is a surjection of schemes.

Definition 1.26. Given A ∈ sRing and B^{•} ∈ (sAlg_{A})^{Δ}, we may regard B as a cocontinuous functor B : 𝕊 → sAlg_{A}, determined by B^{n} = B(Δ^{n}). Then B^{•} is said to be Reedy cofibrant if the latching morphisms f_{n} : B(∂Δ^{n}) → B^{n} are cofibrations for all n ≥ 0 (where B(∂Δ^{0}) = B(∅) = A).

Definition 1.27. A Reedy cofibrant object B^{•} ∈ (sAlg_{A})^{Δ} is an étale hypercover if the latching morphisms are étale coverings. An arbitrary object C^{•} ∈ (sAlg_{A})^{Δ} is an étale hypercover if there exists a levelwise weak equivalence f : B^{•} → C^{•}, for B^{•} a Reedy cofibrant étale hypercover.

Definition 1.28. Given a simplicial hypercover A → B^{•}, and a presheaf 𝒫 over A, define the cosimplicial complex Č^{•}(B^{•}/A, 𝒫) by Č^{n}(B^{•}/A, 𝒫) = 𝒫(B^{n}).

Definition 1.29. A homotopy-preserving functor F : sAlg → 𝕊 is said to be a hypersheaf for the étale topology if it satisfies the following conditions.

(1) It preserves finite products up to homotopy; this means that for any finite (possibly empty) subset {A_{i}} of sAlg_{R}, the map

F (∏ A_{i}) → ∏ F (A_{i})

is a weak equivalence.

(2) For all étale hypercovers A → B^{•}, the natural map

F (A) → RTot Č(B^{•}/A, F)

is a weak equivalence, for Č as in Definition 1.28.

Remark 1.30. The same definition applies for functors Alg_{π_{0}R} → 𝕊. Given a groupoidvalued functor Γ : Alg_{π_{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]).

Definition 1.31. Say that a functor F : sAlg → B is nilcomplete if for any A ∈ sAlg_{R}, the natural map F (A) → \underleftarrow{lim}_{k}^{h} F (P_{k}A) to the homotopy limit is an equivalence.

Proposition 1.32. Take a homotopy-homogeneous nilcomplete homotopy-preserving functor F : sAlg → 𝕊. If

(1) π^{0}F : Alg_{π_{0}R} → 𝕊 is a hypersheaf, and

(2) for all A ∈ Alg_{π_{0}R}, all x ∈ F (A)_{0}, all A-modules M and all étale morphisms f : A → A′, the maps

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

(induced by Lemma 1.15) are isomorphisms, then F is a hypersheaf.

Proof. Take an étale hypercover f : A → B^{•}. The first observation to make is that P_{k}A → P_{k}B^{•} is also an étale hypercover. Assume inductively that

F (P_{k−1}A) → RTot Č(P_{k−1}B^{•}/P_{k−1}A, F)

is an equivalence (the case k = 1 following because π^{0}F is a hypersheaf). Now P_{k}A → P_{k−1}A is a square-zero extension up to homotopy (see for instance [TV] Lemma 2.2.1.1), coming from an essentially unique homotopy derivation δ : P_{k−1}A → (π_{k}A)[−k − 1], with

P_{k}A ≃ P_{k−1}A ×_{id+δ,π_{0}A⊕(π_{k}A)[−k−1]}^{h} π_{0}A.

Since F is homotopy-homogeneous and homotopy-preserving, this means that

F (P_{k}A) ≃ F (P_{k−1}A) ×_{F (π_{0}A⊕(π_{k}A)[−k−1])}^{h} F (π_{0}A).

For the inductive step, it suffices to show that for any point x ∈ π^{0}F (A), the homotopy fibres of F (P_{k}A) and of RTot Č(P_{k}B^{•}/P_{k}A, F) over x are weakly equivalent. From the expression above, we see that

F (P_{k}A)_{x} ≃ F (P_{k−1}A)_{x} ×_{T_{x}(F/R,(π_{k}A)[−k−1])}^{h} {0},

and the corresponding statement for the hypercover is

RTot Č(P_{k}B^{•}/P_{k}A, F) ≃ RTot (Č(P_{k−1}B^{•}/P_{k−1}A, F)_{fx} ×_{T_{fx}(F/R,(π_{k}B^{•})[−k−1])}^{h} {0})
≃ F (P_{k−1}A)_{x} ×_{RTot T_{fx}(F/R,(π_{k}B^{•})[−k−1])}^{h} {0},

using the inductive hypothesis and the fact the RTot commutes with homotopy fibre products.

This reduces the problem to showing that the map T_{x}(F/R, (π_{k}A)[−k − 1]) → RTot T_{fx}(F/R, (π_{k}B^{•})[−k − 1]) is a weak equivalence. Since the cohomology groups D^{∗} commute with étale base change, it follows that the map

RTot T_{x}(F/R, (π_{k}A)[−k − 1]) ⊗_{A} B^{•} → RTot T_{fx}(F/R, (π_{k}B^{•})[−k − 1])

is a weak equivalence. Since A → B^{•} is an étale hypercover (and hence an fpqc hypercover), the map

T_{x}(F/R, (π_{k}A)[−k − 1]) → RTot T_{x}(F/R, (π_{k}A)[−k − 1]) ⊗_{A} B^{•}

is also a weak equivalence, completing the inductive step.

Finally, since F is nilcomplete, we get

F (A) ≃ \underleftarrow{lim}_{k}^{h}F (P_{k}A)
RTot Č(B^{•}/A, F) ≃ \underleftarrow{lim}_{k}^{h}RTot Č(P_{k}B^{•}/P_{k}A, F),

which completes the proof. □

1.5. Representability.

Proposition 1.33. Take a Noetherian simplicial ring R, and a homotopy-preserving functor F : sAlg_{R} → 𝕊, satisfying the following conditions:

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

(2) F is homotopy-homogeneous, i.e. for all square-zero extensions A ↠ C and all maps B → C, the map

F (A ×_{C} B) → F (A) ×_{F (C)}^{h} F (B)

is an equivalence.

(3) F is nilcomplete, i.e. for all A, the map

F (A) → \underleftarrow{lim}^{h}F (P_{k}A)

is an equivalence, for {P_{k}A} the Postnikov tower of A.

(4) F is a hypersheaf for the étale topology.

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

(6) R admits a dualising module, in the sense of [Lur] Definition 3.6.1. Examples are anything admitting a dualising complex in the sense of [Har] Ch. V, such as ℤ or Gorenstein local rings, and any simplicial ring almost of finite presentation over a Noetherian ring with a dualising module.

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

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

Then there is an almost perfect cotangent complex 𝕃_{F/R} in the sense of [Lur].

Proof. This is an adaptation of [Lur] Theorem 7.4.1. After applying Theorem 1.23 to show that F is almost of finite presentation, the only difference is in condition (2), where we only consider square-zero extensions A → C (rather than all surjections), but also allow arbitrary maps B → C (rather than just surjections). The key observation is that we still satisfy the conditions of [Lur] Theorem 3.6.9, guaranteeing local existence of the cotangent complex, while Lemma 1.15 provides the required compatibility. □

Theorem 1.34. Let R be a derived G-ring admitting a dualising module, and F : sAlg_{R} → 𝕊 a homotopy-preserving functor. Then F is a geometric derived n-stack which is almost of finite presentation if and only if the conditions of Proposition 1.33 hold, and for all complete discrete local Noetherian π_{0}R-algebras A, with maximal ideal 𝔪, the map

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

is a weak equivalence.

Proof. This is essentially the same as [Lur] Theorem 7.5.1, by combining ibid. Theorem 7.1.6 with Proposition 1.33 (rather than ibid. Theorem 7.4.1).

Note that our revised condition (2) implies infinitesimal cohesiveness, since, for any square-zero extensions 0 → M → Ã → A → 0, we may set B to be the mapping cone (so B ≃ A), and consider the fibre product Ã ≃ B ×_{A⊕M[−1]}^{h} A.

To see that the revised condition (2) is necessary, we adapt [Lur] Proposition 5.3.7. It suffices to show that for any smooth surjective map U → F of n-stacks, the map

U(A) ×_{U(C)}^{h} U(B) → F (A) ×_{F (C)}^{h} F (B)

is surjective, for all square-zero extensions A ↠ C. Moreover, the argument of [Lur] Proposition 5.3.7 allows us to replace A ×_{C} B with an étale algebra over it, giving a local lift of a point x ∈ F (B) to u ∈ U(B). The problem then reduces to showing that

U(A) ×_{U(C)}^{h} U(B) → F (A) ×_{F (C)}^{h} U(B)

is surjective, but this follows from pulling back the surjection

U(A) → U(C) ×_{F (C)}^{h} F (A)

given by the smoothness of U → F. □

Remark 1.35. The Milnor exact sequence ([GJ] Proposition 2.15) gives a sequence

• → \underleftarrow{lim}_{r}^{1} π_{i+1}F (A/𝔪^{r}) → π_{i}(\underleftarrow{lim}^{h}F (A/𝔪^{r})) → \underleftarrow{lim}_{r} π_{i}F (A/𝔪^{r}) → •,

which is exact as groups for i ≥ 1 and as pointed sets for i = 0. Thus the condition of Theorem 1.34 can be rephrased to say that the map

f_{0} : π_{0}F (A) → \underleftarrow{lim}_{r} π_{0}F (A/𝔪^{r})

is surjective, that for all x ∈ F (A) the maps

f_{i,x} : π_{i}(F A, x) → \underleftarrow{lim}_{r} π_{i}(F (A/𝔪^{r}), x)

are surjective for all i ≥ 1 and that the resulting maps

ker f_{i,x} → \underleftarrow{lim}_{r}^{1} π_{i+1}(F (A/𝔪^{r}), x)

are surjective for all i ≥ 0.

Now, we can say that an inverse system {G_{r}}_{r∈ℕ} of groups satisfies the Mittag-Leffler condition if for all r, the images Im (G_{s} → G_{r})_{s≥r} satisfy the descending chain condition. In that case, the usual abelian proof (see e.g. [Wei] Proposition 3.5.7) adapts to show that \underleftarrow{lim}^{1}\{G_{r}\}_{r} = 1.

Hence, if each system {Im (π_{i}(F (A/𝔪^{s}), x) → π_{i}(F (A/𝔪^{r}), x))}_{s≥r} satisfies the Mittag-Leffler condition for i ≥ 1, then the condition of Theorem 1.34 reduces to requiring that the maps

π_{i}F (A) → \underleftarrow{lim}_{r} π_{i}F (A/𝔪^{r})

be isomorphisms for all i.

Corollary 1.36. Let R be a derived G-ring admitting a dualising module (in the sense of [Lur] Definition 3.6.1) and F : sAlg_{R} → 𝕊 a homotopy-preserving functor. Then F is a geometric derived n-stack which is almost of finite presentation if and only if the following conditions hold

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

(2) F is homotopy-homogeneous, i.e. for all square-zero extensions A ↠ C and all maps B → C, the map

F (A ×_{C} B) → F (A) ×_{F (C)}^{h} F (B)

is an equivalence.

(3) F is nilcomplete, i.e. for all A, the map

F (A) → \underleftarrow{lim}^{h}F (P_{k}A)

is an equivalence, for {P_{k}A} the Postnikov tower of A.

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

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

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

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

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

the map

F (A) → \underleftarrow{lim}_{n}^{h}F (A/𝔪^{r})

is a weak equivalence (see Remark 1.35 for a reformulation).

Proof. If F is a derived n-stack of almost finite presentation, then the étale sheaf A′ ↦ D_{fx}^{i}(F, A′) on Y := Spec A is just

ℰxt_{𝒪_{Y}}^{i} (x^{∗}𝕃^{F/R}, 𝒪_{Y}),

which is necessarily quasi-coherent, as x^{∗}𝕃^{F/R} is equivalent to a complex of finitely generated locally free sheaves (for instance by the results of [Pri3] §5). Combined with Theorem 1.34, this ensures that all the conditions are necessary, once we note that conditions 5 and 6 are equivalent to π^{0}F : Alg_{π_{0}R} → 𝕊 preserving filtered colimits.

For the converse, we just need to show that F is a hypersheaf in order to ensure that it satisfies the conditions of Theorem 1.34. This follows almost immediately from Proposition 1.32, first noting that condition (7) above combines with almost finite presentation and exactness of the tangent complex to ensure that for all A ∈ Alg_{π_{0}R}, all x ∈ F (A)_{0}, all A-modules M and all étale morphisms f : A → A′, the maps

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

are isomorphisms. □

Remark 1.37. Although Corollary 1.36 seems more complicated than Theorem 1.34, since it has an extra condition, it is much easier to verify in practice. This is because F (A) is only n-truncated when A is discrete, so it is much easier to check that π^{0}F is a hypersheaf than to do the same for F.

Proposition 1.38. Take a morphism α : F → G of almost finitely presented geometric derived n-stacks a over R. Then α is a weak equivalence if and only if

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

(2) the maps D_{x}^{i}(F, A) → D_{αx}^{i}(G, A) are isomorphisms for all finitely generated integral domains A ∈ Alg_{π_{0}R}, all x ∈ F (A)_{0}, and all i > 0.

Proof. It suffices to show that 𝕃_{•}^{F/G} ≃ 0. For if this is the case, then [TV] Corollary 2.2.5.6 implies that α is étale. By applying [TV] Theorem 2.2.2.6 locally, it follows that an étale morphism α must be a weak equivalence whenever π^{0}α is so.

Now, 𝕃_{•}^{F/G} is the cone of α^{∗}𝕃^{G/R} → 𝕃^{F/R}, so we wish to show that this map is an equivalence locally. This is equivalent to saying that for all integral domains A ∈ π_{0}R, all π_{0}R-modules M, all x ∈ F (A) and all i, the maps

D_{x}^{i}(F, M) → D_{αx}^{i}(G, M)

are isomorphisms.

For i ≤ 0, these isomorphisms follow immediately from the hypothesis that π^{0}α be an equivalence. For i > 0, we first note that finite presentation of π^{0}F means that we may assume that A is finitely generated. We then have an almost perfect complex x^{∗}𝕃_{•}^{F/G} with the property that

Ext_{A}^{i}(x^{∗}𝕃_{•}^{F/G}, A) = 0

for all i, so Ext_{A}^{i}(x^{∗}𝕃_{•}^{F/G}, P) = 0 for all almost perfect A-complexes P (using nilcompleteness of F and G). In particular,

D_{x}^{i}(F/G, M) = Ext_{A}^{i}(x^{∗}𝕃_{•}^{F/G}, M) = 0

for all finite A-modules. Almost finite presentation of F and G now gives that D_{x}^{i}(F/G, M) = 0 for all A-modules M, completing the proof. □

1.6. Strong quasi-compactness.

Lemma 1.39. If S is a set of separably closed fields, and X = Spec (∏_{k∈S} k), then every surjective étale morphism f : Y → X of affine schemes has a section.

Proof. Since f is surjective, the canonical maps Spec k → X admit lifts to Y, for all k ∈ S, combining to give a map ∐_{k∈S} Spec k → Y. Since Y is affine, this is equivalent to giving a map X → Y, and this is automatically a section of f. □

Proposition 1.40. A morphism F → G of geometric m-stacks is strongly quasi-compact if and only if for all sets S of separably closed fields, the map

F (∏_{k∈S} k) → (∏_{k∈S} F (k)) ×_{(∏_{k∈S} G(k))}^{h} G(∏_{k∈S} k)

is a weak equivalence in 𝕊.

Proof. Let Z = Spec (∏_{k∈S} k), and fix an element g ∈ G(Z) . If F → G is strongly quasi-compact, then F ×_{G,g}^{h} Z is strongly quasi-compact, so by [Pri3] Theorem 4.7, there exists a simplicial affine scheme X whose sheafification X^{♯} is F ×_{G}^{h} Z. Now, Lemma 1.39 implies that Z is weakly initial in the category of étale hypercovers of Z, so (for instance by [Pri3] Corollary 4.10) X^{♯}(Z) ≃ X(Z). Now, since X is simplicial affine, it preserves arbitrary limits of rings, so

X(Z) ≅ ∏_{k∈S} X(k) ≅ ∏_{k∈S} X^{♯}(Z),

which proves that the condition is necessary.

To prove that the condition is sufficient, we need to show that for any affine scheme U and any morphism U → G, the homotopy fibre product F ×_{G}^{h} U is strongly quasi-compact. Since U is affine, it satisfies the condition, so F ×_{G}^{h} U will also, and so we may assume that G = U or even Spec ℤ.

Now, it follows (for instance from the proof of [Pri3] Theorem 4.7) that if an n-geometric stack F admits an n-atlas U → F, with U quasi-compact, and the diagonal F → F × F is strongly quasi-compact, then F is strongly quasi-compact.

We will proceed by induction on n (noting that we use n-geometric, as in Remark 1.2, rather than n-truncated). A 0-geometric stack F is a disjoint union of affine schemes, so is separated, and in particular its diagonal is strongly quasi-compact.

Assume that an n-geometric stack F has strongly quasi-compact diagonal and satisfies the condition above, and take an n-atlas V → F for V 0-geometric (where we interpret a 0-atlas as an isomorphism). Let S be a set of representatives of equivalence classes of geometric points of V, and set Z = Spec (∏_{k∈S} k). Since F satisfies the condition above,

F (Z) ≅ ∏_{k∈S} F (k),

so the points Spec k → V → F combine to define a map f : Z → F.

As V → F is an atlas, for some étale cover Z′ → Z, f lifts to a map f̃: Z′ → V. But Lemma 1.39 implies that Z′ → Z has a section, so we have a lifting f̃ : Z → V of f. Now, V = ∐_{α∈I} V_{α} is a disjoint union of affine schemes, and since Z is quasi-compact, there is some finite subset J ⊂ I with U := ∐_{α∈J} V_{α} containing the image of Z. But U is then quasi-compact, and U → F is surjective, hence an n-atlas, which completes the induction. □

Corollary 1.41. A morphism F → G of geometric derived stacks is strongly quasicompact if and only if for all sets S of separably closed fields, the map

F (∏_{k∈S} k) → (∏_{k∈S} F (k)) ×_{(∏_{k∈S} G(k))}^{h} G(∏_{k∈S} k)

is a weak equivalence in 𝕊.

Proof. The morphism F → G is strongly quasi-compact if and only if π^{0}F → π^{0}G is a strongly quasi-compact morphism of geometric stacks, so we apply Proposition 1.40. □

2. Complete simplicial and chain algebras

Proposition 2.1. Take a cofibrantly generated model category C. Assume that D is a complete and cocomplete category, equipped with an adjunction

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

with U preserving filtered colimits. If UF maps generating trivial cofibrations to weak equivalences, then D has a cofibrantly generated model structure in with a morphism f is a fibration or a weak equivalence whenever Uf is so.

This adjunction is a pair of Quillen equivalences if and only if the unit morphism A → UF A is a weak equivalence for all cofibrant objects A ∈ C.

Proof. To see that this defines a model structure on D, note that since U preserves filtered colimits, for any small object I ∈ C, the object F I is small in D, so we may apply [Hir] Theorem 11.3.2 to obtain the model structure on D.

Since U reflects weak equivalences, by [Hov] Corollary 1.3.16, the functors F ⊢ U form a pair of Quillen equivalences if and only if the morphisms Rη : A → RUF A are weak equivalences for all cofibrant A ∈ C. Since U preserves weak equivalences, the map UB → RUB is a weak equivalence for all B ∈ D. Thus the unit η : A → UF A is a weak equivalence if and only if Rη is so. □

Fix a Noetherian ring R.

Definition 2.2. Say that a simplicial R-algebra A is finitely generated if there are finite sets Σ_{q} ⊂ A_{q} of generators, closed under the degeneracy operations, with only finitely many elements of ⋃_{q} Σ_{q} being non-degenerate.

Define F GsAlg_{R} to be the category of finitely generated simplicial R-algebras. Define F Gdg_{+}Alg_{R} to be the category of finitely generated non-negatively graded chain R-algebras (if R is a ℚ-algebra).

Definition 2.3. Given A ∈ sAlg_{R}, define Â := \underleftarrow{lim}_{n} A/I_{A}^{n}, for I_{A} = ker(A → π_{0}A). Given A ∈ dg_{+}Alg_{R}, define Â := \underleftarrow{lim}_{n} A/I_{A}^{n}, for I_{A} = ker(A → H_{0}A).

Definition 2.4. Define { \atop F GsAlg_{R}} to be the full subcategory of sAlg_{R} consisting of objects of the form Â, for A ∈ F GsAlg_{R}. Define { \atop F Gdg_{+}Alg_{R}} to be the full subcategory of dg_{+}Alg_{R} consisting of objects of the form Â, for A ∈ F Gdg_{+}Alg_{R}

Lemma 2.5. The categories { \atop F GsAlg_{R}} and { \atop F Gdg_{+}Alg_{R}} contain all finite colimits.

Proof. The initial object is R̂ (which equals R whenever R is discrete), and the cofibre coproduct of A ← B → C is given by

A ˆ⊗_{B}C := { \atop A ⊗_{B} C}.

□

Proposition 2.6. For C = { \atop F GsAlg_{R}} or { \atop F Gdg_{+}Alg_{R}}, the category ind(C) is equivalent to the category of left-exact functors F : C^{opp} → Set, i.e. functors for which

(1) F (R̂) is the one-point set, and

(2) the map

F (A ˆ⊗_{B}C) → F (A) ×_{F (B)} F (C)

is an isomorphism for all diagrams A ← B → C. The equivalence is given by sending a direct system {A_{α}}_{α} to the functor F (B) =

\underrightarrow{lim}_{α} Hom_{C}(B, A_{α}).

Proof. For A ∈ C, a subobject of A^{opp} ∈ C^{opp} is just a surjective map A → B in C, or equivalently a simplicial (resp. dg) ideal of A. Since A is Noetherian, it satisfies ACC on such ideals, and hence A^{opp} satisfies DCC on strict subobjects. Therefore C^{opp} is an Artinian category containing all finite limits, so the required result is given by [Gro], Corollary to Proposition 3.1. □

Proposition 2.7. There are cofibrantly generated model structures on the categories ind({ \atop F GsAlg_{R}}) and ind({ \atop F Gdg_{+}Alg_{R}}) in which a morphism f : \{A_{α}\}_{α} → \{B_{β}\}_{β} is a fibration or a weak equivalence whenever the corresponding map

\underrightarrow{lim} f : \underrightarrow{lim}_{α} A_{α} → \underrightarrow{lim}_{β} B_{β}

in sAlg_{R} or dg_{+}Alg_{R} is so.

For these model structures, the functors

U : ind({ \atop F GsAlg_{R}}) → sAlg_{R}
U : ind({ \atop F Gdg_{+}Alg_{R}}) → dg_{+}Alg_{R}

given by U(\{A_{α}\}_{α}) = \underrightarrow{lim}_{α} A_{α} are right Quillen equivalences.

Proof. We begin by showing that ind({ \atop F GsAlg_{R}}) and ind({ \atop F Gdg_{+}Alg_{R}}) are complete and cocomplete. By Lemma 2.5, they contain finite colimits, and the proof of [Isa] Proposition 11.1 then ensures that they contain arbitrary coproducts, and hence arbitrary colimits. It follows immediately from Proposition 2.6 that the categories contain arbitrary limits, since any limit of left-exact functors is left-exact.

We need to establish that the functors U have left adjoints. Since R is Noetherian, finitely generated objects over R are finitely presented, so the functors

\underrightarrow{lim} : ind(F GsAlg_{R}) → sAlg_{R}
\underrightarrow{lim} : ind(F Gdg_{+}Alg_{R}) → dg_{+}Alg_{R}

are equivalences of categories. The left adjoints

F : ind(F GsAlg_{R}) → ind({ \atop F GsAlg_{R}})
F : ind(F Gdg_{+}Alg_{R}) → ind({ \atop F Gdg_{+}Alg_{R}})

to U are thus given by {A_{α}}_{α} ↦ {Â_{α}}_{α}.

It is immediate that U preserves filtered colimits, so we may apply Proposition 2.1 to construct the model structures. It only remains to show that U is a Quillen equivalence. By Proposition 2.1, we need only show that, for any cofibrant A ∈ sAlg_{R} or A ∈ dg_{+}Alg_{R}, the map

A → UF A

is a weak equivalence. If we write A = \underrightarrow{lim}_{α} A_{α}, for A_{α} ∈ F GsAlg_{R} (or A_{α} ∈ F Gdg_{+}Alg_{R}), then

UF A = \underrightarrow{lim}_{α} Â_{α}.

Thus it suffices to show that for A ∈ F GsAlg_{R} (or A ∈ F Gdg_{+}Alg_{R}), the map A → Â is a weak equivalence. If A ∈ F GsAlg_{R}, then each A_{n} is Noetherian, so [Pri3] Theorem 6.6 gives the required equivalence. If A ∈ F Gdg_{+}Alg_{R}, then A_{0} is Noetherian and each A_{n} is a finite A_{0}-module, so [Pri3] Lemma 6.37 gives the required equivalence. □

Lemma 2.8. The category ind({ \atop F GsAlg_{R}}) (resp. ind({ \atop F Gdg_{+}Alg_{R}}) ) is equivalent to a full subcategory C of sAlg_{R} (resp. dg_{+}Alg_{R}). If I_{A} = ker(A → H_{0}A) , then A is an object of C if and only if it contains the I_{A}-adic completions of all its finitely generated subalgebras.

Proof. It is immediate that A satisfies the condition above if and only if A = UF A for the functors U and F from the proof of Proposition 2.7. Thus we need only show that the functor U : ind({ \atop F GsAlg_{R}}) → F sAlg given by \{A_{α}\} ↦ \underrightarrow{lim}_{α} A_{α} is full and faithful.

It suffices to show that for A ∈ { \atop F GsAlg_{R}} and B ∈ ind({ \atop F GsAlg_{R}}), Hom(A, \underrightarrow{lim} B_{β}) = \underrightarrow{lim}_{β} Hom(A, B_{β}).

To do this, recall that A = Â′ for some finitely generated A′, and express A as \underrightarrow{lim} A_{α}, for A′ ⊂ A_{α} ∈ F GsAlg_{R}. Then

Hom(A, \underrightarrow{lim} B_{β}) = \underleftarrow{lim}_{α} Hom(A_{α}, \underrightarrow{lim} B_{β})
= \underleftarrow{lim}_{α} \underrightarrow{lim}_{β} Hom(A_{α}, B_{β})
= \underleftarrow{lim}_{α} \underrightarrow{lim}_{β} Hom(Â_{α}, B_{β}),

but Â_{α} = A, giving the required result. □

2.1. Nilpotent algebras.

Definition 2.9. Say that a surjection A → B in dg_{+}Alg_{R} (resp. sAlg_{R}) is a little extension if the kernel K satisfies I_{A} · K = 0. Say that an acyclic little extension is tiny if K (resp. NK) is of the form cone(M)[−r] for some H_{0}A-module (resp. π_{0}A-module) M.

Note that acyclic little extensions are necessarily square-zero, but that arbitrary little extensions need not be.

Definition 2.10. 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.

Lemma 2.11. Every surjective weak equivalence f : A → B in dg_{+}N_{R}^{♭} (resp. sN_{R}^{♭}) factors as a composition of tiny acyclic extensions.

Proof. We first prove this for dg_{+}N_{R}^{♭}. Let K = ker(f), and observe that the good truncations

(τ_{≥r}K)_{i} = \begin{cases}K_{i} & i > r \\ Z_{r}K & i = r \\ 0 & i < r \end{cases}

are also dg ideals in A. Since A is concentrated in degrees [0, d] for some d, we get a factorisation of f into acyclic surjections

A = A/(τ_{≥d}K) → A/(τ_{≥(d−1)}K) → … → A/(τ_{≥0}K) = B.

We therefore reduce to the case where K is concentrated in degrees r, r + 1.

Let s be least such that K_{r} · I_{A}^{s} = 0; if s = 1 then f is already a tiny acyclic extension. We will proceed by induction on s. Since K ↠ (K/I_{A}K), we have H_{r}(K/I_{A}K) = 0. This means that the inclusion τ_{>r}(K/I_{A}K) → (K/I_{A}K) is a quasi-isomorphism of ideals in A. If we set B′ := (A/I_{A}K)/(τ_{>r}K/I_{A}K) and K′′ := ker(A → B′), then I_{A} · K′′ = 0 so f′′ : B′ → B is an acyclic little extension. In fact, for M := (K/I_{A} · K)_{r}, we have K′′ = cone(M)[−r], so f′′ is a tiny acyclic extension.

Now, for K′ := ker(f′ : A → B′) we have K′_{r} = (I_{A}K)_{r}, so K′_{r} ·I_{A}^{s−1} = 0, so by induction f′ factors as a composition of tiny acyclic extensions. This completes the inductive step.

Finally, for f : A → B in sN_{R}^{♭} , normalisation gives an equivalence of categories between simplicial A-modules and non-negatively graded dg NA-modules. In particular, it gives an equivalence between the categories of ideals, and hence quotients of A correspond to quotients of NA. If Nf is a tiny acyclic extension, then so is f, since NK is automatically an H_{0}NA-module, and H_{0}NA = π_{0}A. The proof above expresses NA → NB as a composition of tiny acyclic extensions, which thus yields such an expression for A → B. □

Definition 2.12. Define { \atop F GsAlg_{R}}^{♭} (resp. { \atop F Gdg_{+}Alg_{R}}^{♭}) to be the full subcategory of { \atop F GsAlg_{R}} (resp. { \atop F Gdg_{+}Alg_{R}}) consisting of objects A for which A_{i} = 0 (resp. N_{i}A = 0) for all i ≫ 0.

Lemma 2.13. For any surjective weak equivalence f : A → B in { \atop F GsAlg_{R}}^{♭} (resp.

{ \atop F Gdg_{+}Alg_{R}}^{♭}), the associated morphism

{A/I_{A}^{n}} → {B/I_{B}^{n}}

in pro(dg_{+}N_{R}^{♭}) (resp. pro(sN_{R}^{♭})) is isomorphic to an inverse limit of surjective weak equivalences in dg_{+}N_{R} (resp. sN_{R}).

Proof. With reasoning as at the end of Lemma 2.11, it suffices to prove this for

{ \atop F Gdg_{+}Alg_{R}}^{♭}. The first observation to make is that if f and g are composable morphisms satisfying the conclusions of this lemma, then fg also satisfies the conclusions. Let K = ker(f); since A is concentrated in degrees [0, d] for some d, we get a factorisation of f into acyclic surjections

A = A/(τ_{≥d}K) → A/(τ_{≥(d−1)}K) → … → A/(τ_{≥0}K) = B,

and therefore reduce to the case where K is concentrated in degrees r, r + 1.

Set I := I_{A} and J := I_{B}; we now define a dg ideal I(n)′ ⊲ A to be generated by I^{n} and K_{r+1} ∩ d^{−1}(I^{n}), and set A(n)′ := A/I(n)′. There is a surjection A(n)′ → B/J^{n}, with kernel K/(K ∩ I(n)′). This is given by

(K/K ∩ I(n)′)_{i} = \begin{cases}K_{r}/(K ∩ I^{n})_{r} & i = r \\ K_{r+1}/(K_{r+1} ∩ d^{−1}(I^{n})) & i = r + 1 \\ 0 & i ≠ r, r + 1. \end{cases}

Since d : K_{r+1} → K_{r} is an isomorphism, so is d : K_{r+1} ∩ d^{−1}I^{n} → (K ∩ I^{n})_{r}, which means that H_{∗}(K/K ∩ I(n)′) = 0, so A(n′) → B/J^{n} is a weak equivalence.

Thus it only remains to show that the pro-objects {A/I^{n}}_{n} and {A/I(n)′}_{n} are isomorphic. Since I^{n} ⊂ I(n)′, there is an obvious morphism A/I^{n} → A/I(n)′, and it remains to construct an inverse in the pro-category, Observe that A_{0} is a Noetherian ring, and that (I^{r})_{r} and K_{r} are finitely generated A_{0}-modules.

Now, (K ∩ I^{n})_{r} = K_{r} ∩ I_{0}^{n−r}(I^{r})_{r} for all n ≥ r. By the Artin–Rees Lemma ([Mat] Theorem 8.5), there exists some c ≥ r such that for all n ≥ c, this is

I_{0}^{n−c}(K_{r} ∩ I_{0}^{c−r}(I^{r})_{r}) = I_{0}^{n−c}(K_{r} ∩ (I^{c})_{r}).

Thus K_{r+1} ∩ d^{−1}(I^{n}) is just I_{0}^{n−c}K_{r+1} ∩ d^{−1}(I^{c}). Therefore I(n)′ ⊂ I^{n−c}, so giving maps A/I(n)′ → A/I^{n−c}, and hence the required inverse in the pro-category. □

2.2. A nilpotent representability theorem. Let dN_{R}^{♭} (or simply dN^{♭}) be either of the categories sN_{R}^{♭} or dg_{+}N_{R}^{♭}.

Remark 2.14. Note that the constructions of §1.2 carry over to the categories dN_{R}^{♭}, since they are closed under fibre products.

Lemma 2.15. Given a weak equivalence f : A → B between fibrant objects in a right proper model category C, there exists a diagram

\xymatrix{ & & B \\ A \ar[r]^{i} & C \ar@{-->}[ur]^{g_{1}} \ar[dr]_{g_{0}} \\ & & A} ,

such that g_{0}, g_{1} are trivial fibrations, g_{1} ∘ i = f and g_{0} ∘ i = id.

Proof. Let C := A ×_{f,B,ev_{0}} B^{I}, for B^{I} the path object of B, and let g_{0} be given by projection onto A. The projection C → B^{I} is the pullback of A → B along the fibration B^{I} → B, so is a weak equivalence by right properness. Define g_{1} to be the composition of this with the trivial fibration ev_{1} : B^{I} → B. The projection g_{0} is the pullback of the trivial fibration ev_{0} : B^{I} → B along f, so is a trivial fibration.

It only remains to show that g_{1} is a fibration. Since B^{I} → B × B is a fibration, pulling back along f shows that (g_{0}, g_{1}) : C → A × B is a fibration, and since A is fibrant, we deduce that A × B → B is a fibration, so g_{1} must be a fibration. □

Lemma 2.16. If a homotopy-preserving functor F : dN_{R}^{♭} → 𝕊 is homotopy-homogeneous, then it is almost of finite presentation if and only if the following hold:

(1) the functor π^{0}F : Alg_{π_{0}R} → 𝕊 preserves filtered colimits;

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

Proof. This is essentially the same as Theorem 1.23 — we need only show that any squarezero extension A → B in sN_{R}^{♭} (resp. dg_{+}N_{R}^{♭}) is of the form A = B ×_{B⊕M} B̃, for B̃ → B a weak equivalence, and some derivation B → M. Now just note that such an expression is constructed in the proof of Lemma 1.4. □

Theorem 2.17. Let R be a derived G-ring admitting a dualising module (in the sense of [Lur] Definition 3.6.1) and take a functor F : dN_{R}^{♭} → 𝕊. Then F is the restriction of an almost finitely presented geometric derived n-stack F′ : dAlg_{R} → 𝕊 if and only if the following conditions hold

(1) F maps tiny acyclic extensions to weak equivalences.

(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 homotopy-homogeneous, i.e. for all square-zero extensions A ↠ C and all maps B → C, the map

F (A ×_{C} B) → F (A) ×_{F (C)}^{h} F (B)

is an equivalence.

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

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

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

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

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

the map

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

is a weak equivalence (see Remark 1.35 for a reformulation). Moreover, F′ is uniquely determined by F (up to weak equivalence).

Proof. We will deal with the simplicial case. Since normalisation gives an equivalence N : sN_{R}^{♭} → dg_{+}N_{R}^{♭} when R is a ℚ-algebra, the dg case is entirely similar.

First observe that F extends to a functor F̂ : pro(sN_{R}^{♭}) → 𝕊, given by F̂({A^{(i)}}_{i∈I}) = \underleftarrow{lim}_{i∈I}^{h} F (A^{(i)}).

Define F′ as follows. For any A ∈ sAlg_{R}, write A = \underrightarrow{lim} A_{α}, for A_{α} ∈ F GsAlg_{R}, and set

F′(A) := \underleftarrow{lim}_{k}^{h} \underrightarrow{lim}_{α} F̂(\{P_{k}A_{α}/I_{A_{α}}^{n}\}_{n∈ℕ}).

We first show that F′ is homotopy-preserving; it follows from Lemma 2.11 and the proof of Proposition 2.7 that F is homotopy-preserving. Note that the formula for F′ defines a functor F′′ on ind({ \atop F GsAlg_{R}}), and that F′ is the composition of F′′ with the derived left Quillen functor of Proposition 2.7. By the proof of Lemma 2.15, it suffices to show that F′′ maps trivial fibrations to weak equivalences. Any such morphism is isomorphic to one of the form {A_{α}}_{α} → {B_{α}}_{α}, where each A_{α} → B_{α} is a surjective weak equivalence in { \atop F GsAlg_{R}}. Note that P_{k}A_{α} → P_{k}B_{α} is also a surjective weak equivalence, so we may apply Lemma 2.13, which implies that

F̂({P_{k}A_{α}/I_{A_{α}}^{n}}_{n∈ℕ}) → F̂({P_{k}B_{α}/I_{B_{α}}^{n}}_{n∈ℕ})

is a weak equivalence, since F is homotopy-preserving. Thus F′′ (and hence F′) is homotopy-preserving.

If A ∈ sN_{R}^{♭}, note that

F′(A) = \underrightarrow{lim}_{α} F (A_{α}) ≃ F (A),

by nilpotence and almost finite presentation, respectively, noting that as in the proof of Theorem 1.34, conditions (5), (6) and (8) ensure almost finite presentation of F. Thus F ≃ F′|_{sN_{R}^{♭}}; in particular, this ensures that D_{x}^{i}((F′), M) ≅ D_{x}^{i}(F, M).

Since P_{k}A = \underrightarrow{lim} P_{k}A_{α} (for A_{α} as above), it follows immediately that F is nilcomplete. Likewise, π^{0}F automatically preserves filtered colimits, as do the functors D_{x}^{i}(F, −) : Mod_{A} → Ab. Therefore F′ satisfies the conditions of Corollary 1.36.

Finally, it remains to show that F′ is uniquely determined by F. Assume that we have some geometric derived stack G : sAlg_{R} → 𝕊, almost of finite presentation, with G|_{sN_{R}^{♭}} ≃ F. Then, since G is nilcomplete and almost of finite presentation, we must have

G(A) ≃ \underleftarrow{lim}_{k}^{h}G(P_{k}A)
≃ \underleftarrow{lim}_{k}^{h} \underrightarrow{lim}_{α} G(P_{k}A_{α})
≃ \underleftarrow{lim}_{k}^{h} \underrightarrow{lim}_{α} G(P_{k}Â_{α}),

where we write A = \underrightarrow{lim}_{α} A_{α} as a filtered colimit of finitely generated subalgebras, and the final isomorphism comes from the weak equivalence A_{α} → Â_{α} of [Pri3] Theorem 6.6.

Now, if we take an inverse system {B_{i}}_{i} in sAlg in which the morphisms B_{i} → B_{j} induce isomorphisms π_{0}B_{i} → π_{0}B_{j}, then G(\underleftarrow{lim}^{h} B_{i}) ≃ \underleftarrow{lim}^{h} G(B_{i}) (as G is a geometric derived stack, so has an atlas as in [Pri3] Theorem 4.7). In particular,

G(P_{k}Â_{α}) = G(\underleftarrow{lim}_{n} P_{k}Â_{α}/(I_{A_{α}}^{n}))
≃ \underleftarrow{lim}_{n}^{h}G(P_{k}Â_{α}/(I_{A_{α}}^{n}))
= \underleftarrow{lim}_{n}^{h}F (P_{k}Â_{α}/(I_{A_{α}}^{n}))
= F̂(P_{k}Â_{α}).

Thus

G(A) ≃ \underleftarrow{lim}_{k}^{h} \underrightarrow{lim}_{α} F̂(P_{k}Â_{α}),

as required. □

Remark 2.18. Note that if we replace dN_{R}^{♭} with sN_{R} or dg_{+}N_{R}, then the theorem remains true, provided we impose the additional condition that F be nilcomplete, in the sense that for all A, the map F (A) → \underleftarrow{lim}_{k}^{h} F (P_{k}A) is a weak equivalence.

2.3. Covers. We end this section with a criterion which allows us to verify the key representability properties on formally étale covers.

Definition 2.19. A transformation α : F → G of functors F, G : dN^{♭} → 𝕊 is said to be homotopy formally étale if for all square-zero extensions A → B, the map

F (A) → F (B) ×_{G(B)}^{h} G(A)

is an equivalence.

Proposition 2.20. Let α : F → G be a homotopy formally étale morphism of functors F, G : dN^{♭} → 𝕊. If G is homotopy-homogeneous (resp. homotopy-preserving), then so is F. Conversely, if α is surjective (in the sense that π_{0}F (A) ↠ π_{0}G(A) for all A) and F is homotopy-homogeneous (resp. homotopy-preserving), then so is G.

Proof. Take a square-zero extension A → B, and a morphism C → B, noting that A ×_{B} C → C is then another square-zero extension. Since α is homotopy formally étale,

F (A ×_{B} C) ≃ G(A ×_{B} C) ×_{GC}^{h} F C
F A ×_{F B}^{h} F C ≃ [G(A) ×_{G(B)}^{h} F (B)] ×_{F B}^{h} F C
≃ G(A) ×_{G(B)}^{h} F C
≃ (GA ×_{GB}^{h} GC) ×_{GC}^{h} F C.

Thus homogeneity of G implies homogeneity of F, and if π_{0}F C → π_{0}GC is surjective for all C, then homogeneity of F implies homogeneity of G.

Now take a tiny acyclic extension A → B in dN^{♭}. Since α is homotopy formally étale,

F (A) ≃ G(A) ×_{G(B)}^{h} F (B),

so if G is homotopy-preserving, then F maps tiny acyclic extensions to weak equivalences. By Lemma 2.11 and the proof of Lemma 2.15, this implies that F is homotopy-preserving. If π_{0}F (B) → π_{0}G(B) is surjective for all B, then the converse holds. □

3. Pre-representability

3.1. Simplicial structures.

Definition 3.1. Define simplicial structures (in the sense of [GJ] Definition II.2.1) on sAlg_{R} and ind({ \atop F GsAlg_{R}}) as follows. For A ∈ sAlg_{R} and K ∈ 𝕊, A^{K} is defined by

(A^{K})_{n} := Hom_{𝕊}(K × Δ^{n}, A).

Then for A ∈ ind({ \atop F GsAlg_{R}}), A^{K} is uniquely determined via Lemma 2.8 by the property that U(A^{K}) = (UA)^{K}.

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

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

We need to check that this is well-defined:

Lemma 3.2. For A ∈ ind({ \atop F GsAlg_{R}}) and K ∈ 𝕊, we have A^{K} ∈ ind({ \atop F GsAlg_{R}}). Moreover, if A → π_{0}A is a nilpotent extension, then so is A^{K} → π_{0}(A^{K}).

Proof. A^{K} can be expressed as the limit

lim_{\overleftarrow{(Δ^{n}⟶^{f}K)∈Δ↓K}} A^{Δ^{n}};

since the inclusion functor U : ind({ \atop F GsAlg_{R}}) → sAlg_{R} is a right adjoint, it preserves arbitrary limits, so it suffices to show that A^{Δ^{n}} ∈ ind({ \atop F GsAlg_{R}}).

Write A := \underrightarrow{lim}_{α} A_{α}, for A_{α} ∈ { \atop F GsAlg_{R}}. Since Δ^{n} is finite, we have A^{Δ^{n}} = \underrightarrow{lim}_{α} A_{α}^{Δ^{n}},

so we may assume that A ∈ { \atop F GsAlg_{R}}.

The exact sequence 0 → I_{A} → A → π_{0}A → 0 gives an exact sequence 0 → I_{A}^{Δ^{n}} → A^{Δ^{n}} → π_{0}A → 0 (as (π_{0}A)^{Δ^{n}} = π_{0}A, since Δ^{n} is connected). Since Δ^{n} is contractible, π_{0}(I_{A}^{Δ^{n}}) = π_{0}(I_{A}) = 0, so I_{A^{Δ^{n}}} = I_{A}^{Δ^{n}}. Hence

\underleftarrow{lim}_{m}(A^{Δ^{n}}/I_{A^{Δ^{n}}}^{m}) = \underleftarrow{lim}_{m}(A^{Δ^{n}}/(I_{A}^{Δ^{n}})^{m}) = \underleftarrow{lim}(A/I_{A}^{m})^{Δ^{n}} = A^{Δ^{n}},

so A^{Δ^{n}} ∈ { \atop F GsAlg_{R}}.

Finally, if I_{A}^{m} = 0, then (I_{A}^{Δ^{n}})^{m} = 0, so I_{A^{Δ^{n}}}^{m} = 0 for all n, and hence I_{A^{K}}^{m} = 0 for all K ∈ 𝕊. □

In fact, this makes ind({ \atop F GsAlg_{R}}) into a simplicial model category in the sense of [GJ] Ch. II (with U : ind({ \atop F GsAlg_{R}}) → sAlg_{R} becoming a simplicial right Quillen equivalence). Although the same is not true for dg_{+}Alg_{R} or ind({ \atop F Gdg_{+}Alg_{R}}), we now show that they carry compatible weak simplicial structures.

Definition 3.3. Explicitly, we say that a model category C has a weak simplicial structure if we have the following data:

(1) a functor \underline{Hom}_{C} : C^{opp} × C → 𝕊 such that \underline{Hom}_{C}(A, B)_{0} = Hom_{C}(A, B).

(2) a functor (f𝕊)^{opp} × C → C (where f𝕊 is the category of finite simplicial sets), denoted by (K, B) ↦ B^{K}, with natural isomorphisms

Hom_{C}(A, B^{K}) ≅ Hom_{𝕊}(K, \underline{Hom}_{C}(A, B)).

These must satisfy the property (known as SM7) that if i : A → B is a cofibration in C, and p : X → Y a fibration, then

\underline{Hom}_{C}(B, X) → \underline{Hom}_{C}(A, X) ×_{\underline{Hom}_{C}(A,Y)} \underline{Hom}_{C}(B, Y)

is a fibration in 𝕊 which is trivial whenever either i or p is a weak equivalence.

This means that C satisfies all of the axioms of a simplicial model category from [GJ] Ch. II except for conditions (2) and (3) of Definition II.2.1 (which require that for all objects A ∈ C, the functors \underline{Hom}_{C}(A, −) : C → 𝕊 and \underline{Hom}_{C}(−, A) : C^{opp} → 𝕊 have left adjoints).

Note that this is enough to ensure that C is still a simplicial model category in the sense of [Qui].

Lemma 3.4. The model categories dg_{+}Alg_{R} and ind({ \atop F Gdg_{+}Alg_{R}}) carry weak simplicial structures.

Proof. First set Ω_{n} = Ω(Δ^{n}) to be the cochain 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}. These fit together to form a simplicial complex Ω_{•} of DG-algebras, and we define A^{Δ^{n}} as the good truncation A^{Δ^{n}} := τ_{≥0}(A⊗Ω_{n}). Note that this construction only commutes with finite limits, so only extends to define A^{K} for finite simplicial sets K, and does not have a left adjoint.

For A ∈ { \atop F Gdg_{+}Alg_{R}}, we replace A^{K} with its completion over H_{0}(A^{K}), and extend this construction to ind({ \atop F Gdg_{+}Alg_{R}}) in the obvious way.

That these have the required properties follows because the matching maps Ω_{n} → M_{n}Ω = Ω(∂Δ^{n}) are surjective. Explicitly,

M_{n}Ω ≅ Ω_{n}/(t_{0} ⋯ t_{n}, ∑_{i} t_{0} ⋯ t_{i−1}(dt_{i})t_{i+1} ⋯ t_{n}).

□

Definition 3.5. Although the categories sN_{R}^{♭} and dg_{+}N_{R}^{♭} are not model categories, we endow them with weak simplicial structures inherited from sAlg_{R} and dg_{+}Alg_{R}, respectively. The key observation is that for K ∈ f𝕊 and A ∈ dN^{♭}, the object A^{K} lies in dN^{♭}.

3.2. Deriving functors.

Definition 3.6. Given a functor F : dN^{♭} → 𝕊, we define a functor \underline{F} : dN^{♭} → s𝕊 to the category of bisimplicial sets by

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

For a functor F : C → Set, we will abuse notation by also writing F : C → 𝕊 for the composition C ⟶^{F} Set → 𝕊.

Proposition 3.7. If F : dN^{♭} → 𝕊 is homotopy-homogeneous, then for A → B an acyclic little extension in dN^{♭} and K ∈ 𝕊 finite, the map

F (A^{K}) → (M_{K}^{h} \underline{F} (A)) ×_{(M_{K}^{h} \underline{F} (B))}^{h} F (B^{K})

is a weak equivalence in 𝕊, where M_{K}^{h} denotes the Reedy homotopy K-matching object.

Proof. We prove this by induction on the dimension of K. If K is dimension 0 (i.e. discrete), then the map is automatically an equivalence, as

M_{K}^{h} \underline{F} (A) = \underline{F} (A)_{0}^{K} = F (A^{K}).

Now assume the statement holds for all finite simplicial sets of dimension < n, take K of dimension n, and let K′ := sk_{n−1}K, the (n − 1)-skeleton. Thus there is a pushout square

(∂Δ^{n} × N_{n}K) ⊔ (Δ^{n} × L_{n}K) ⟶ Δ^{n} × K_{n}
↓  ↓
K′  ⟶ K,

where L_{n}K is the nth latching object and N_{n}K = K_{n} − L_{n}K. Hence we have a pullback square

A^{K}  ⟶  B^{K} ×_{B^{K′}} A^{K′}
↓  ↓
B^{K} ×_{B^{(Δ^{n}×Kn)}} A^{(Δ^{n}×K_{n})} ⟶ B^{K} ×_{[B^{(∂Δ^{n}×NnK)}×B^{(Δ^{n}×LnK)}]} [A^{(∂Δ^{n}×N_{n}K)} × A^{(Δ^{n}×L_{n}K)}].

Now, since A → B is an acyclic little extension, the map A^{Δ^{n}} → A^{∂Δ^{n}} ×_{B^{∂Δ^{n}}} B^{Δ^{n}} is a square-zero extension, so the bottom map in the diagram above is a square-zero extension, giving a homotopy pullback square

F (A^{K})  ⟶  F (B^{K}) ×_{F (B^{K′})}^{h} F (A^{K′})
↓  ↓
F (B^{K}) ×_{F (B^{Δ^{n}})^{Kn}}^{h} F (A^{Δ^{n}})^{K_{n}} ⟶ F (B^{K}) ×_{[F (B^{∂Δ^{n}})^{NnK}×F (B^{Δ^{n}})^{LnK}]}^{h} [F (A^{∂Δ^{n}})^{N_{n}K} × F (A^{Δ^{n}})^{L_{n}K}].

Here, the top right isomorphism comes from A^{K′} ↠ B^{K′}, the bottom left from A^{Δ^{n}} ↠ B^{Δ^{n}}, and the bottom right from A^{Δ^{n}} ↠ B^{Δ^{n}} and from A^{∂Δ^{n}} ↠ B^{∂Δ^{n}}; these are all square-zero extensions and F is homotopy-homogeneous.

By induction (using F (A^{K′}) ≃ (M_{K′}^{h}\underline{F} (A)) ×_{(M_{K′}^{h} \underline{F} (B))}^{h} F (B^{K′}) and F (A^{∂Δ^{n}}) ≃

(M_{n}^{h}\underline{F} (A)) ×_{(M_{n}^{h}\underline{F} (B))} F (B^{∂Δ^{n}})), we can rewrite this as saying that the following square is a homotopy pullback

F (A^{K})  ⟶  F (B^{K}) ×_{M_{K′}^{h} \underline{F} (B)}^{h} M_{K′}^{h}\underline{F} (A)
↓  ↓
F (B^{K}) ×_{F (B^{Δ^{n}})^{Kn}}^{h} F (A^{Δ^{n}})^{K_{n}} ⟶ F (B^{K}) ×_{[M_{n}^{h}\underline{F} (B)^{NnK}×F (B^{Δ^{n}})^{LnK}]} [M_{n}^{h}\underline{F} (A)^{N_{n}K} × F (A^{Δ^{n}})^{L_{n}K}].

Now just observe that this pullback defines F (B^{K}) ×_{M_{K}^{h} \underline{F} (B)} M_{K}^{h} \underline{F} (A), as required. □

Definition 3.8. Say that a functor F : dN^{♭} → 𝕊 is homotopy-surjecting if for all tiny acyclic extensions A → B, the map

π_{0}F (A) → π_{0}F (B)

is surjective.

Definition 3.9. 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 [CR], it is established that the canonical natural transformation

diag X → W̄ X

from the diagonal is a weak equivalence for all X.

Corollary 3.10. If a homotopy-homogeneous functor F : dN → 𝕊 is homotopy-surjecting, then the functor W̄ \underline{F} : dN → 𝕊 is homotopy-preserving.

Proof. Consider the homotopy matching maps (for the Reedy model structure on bisimplicial sets)

\underline{F} (A)_{n} → \underline{F} (B)_{n} ×_{M_{∂Δ^{n}}^{h} \underline{F} (B)} M_{∂Δ^{n}}^{h}\underline{F} (A)

of

\underline{F} (A) → \underline{F} (B),

for an acyclic little extension A → B. By Lemma 2.11, we may replace tiny acyclic extensions with little acyclic extensions in the definition of homotopy-surjections.

By Proposition 3.7, the map above is weakly equivalent to

F (A′) → F (B′),

where A′ = A^{Δ^{n}}, B′ = B^{Δ^{n}} ×_{B^{∂Δ^{n}}} A^{∂Δ^{n}}. Now, A′ → B′ is a little acyclic extension, so the homotopy matching maps of \underline{F} (A) → \underline{F} (B) are surjective on π_{0} (as α is homotopysurjecting).

For any Reedy fibrant replacement f : R → \underline{F} (B) of \underline{F} (A) → \underline{F} (B), the homotopy matching maps must also be surjective on π_{0}. However, for Reedy fibrations, matching objects model homotopy matching objects, so f is a Reedy surjective fibration, and hence a horizontal levelwise trivial fibration (the matching maps being surjective). It is therefore a diagonal weak equivalence by [GJ] Proposition IV.1.7, and [CR] then shows that W̄ f is also a weak equivalence. Lemma 2.15 then implies that W̄ F preserves all weak equivalences.

□ Proposition 3.11. If F : dN^{♭} → 𝕊 is homotopy-homogeneous, then for A → B a little extension in dN^{♭} and K a contractible finite simplicial set, the map

F (A^{K}) → (M_{K}^{h} \underline{F} (A)) ×_{(M_{K}^{h} \underline{F} (B))}^{h} F (B^{K})

is a weak equivalence in 𝕊.

Proof. We adapt the proof of Proposition 3.7, proceeding by induction on the dimension of K. If K is of dimension 0, the statement is automatically true.

For any contractible finite simplicial set K, any morphism Δ^{0} → K can be expressed as an iterated pushout of anodyne extensions Λ^{m,k} → Δ^{m}. In particular, if K has dimension n, there is a contractible simplicial set K′ ⊂ K of dimension n − 1, with the map K′ → K an iterated pushout of the maps Λ^{n,k} → Δ^{n} for various k. The proposition holds by induction for K′ and Λ^{n,k}, and is automatically satisfied by Δ^{n}.

Since the map A^{Δ^{n}} → B^{Δ^{n}} ×_{B^{Λ^{n},k}} A^{Λ^{n},k} is an acyclic little extension, the proof of Proposition 3.7 adapts to show that the proposition is satisfied by K, as required. □

Corollary 3.12. If a homotopy-homogeneous functor F : dN → 𝕊 is homotopy-surjecting, then the functor W̄ \underline{F} : dN → 𝕊 is homotopy-homogeneous.

Proof. Take a square-zero little extension A → B; by Proposition 3.11, the relative homotopy partial matching object

M_{Λ^{n,k}}^{h} \underline{F} (A) ×_{M_{Λ^{n,k}}^{h} \underline{F} (B)}^{h} \underline{F} (B)_{n}

is

F (A^{Λ^{n,k}}) ×_{F (B^{Λ^{n,k}})}^{h} F (B^{Δ^{n}}).

Since A^{Δ^{n}} → A^{Λ^{n,k}} ×_{B^{Λ^{n,k}}} B^{Δ^{n}} is an acyclic little extension, homotopy-surjectivity of F thus implies that the homotopy partial matching map

\underline{F} (A)_{n} → M_{Λ^{n,k}}^{h} \underline{F} (A) ×_{M_{Λ^{n,k}}^{h} \underline{F} (B)}^{h} \underline{F} (B)_{n}

gives a surjection on π_{0}.

If we take a Reedy fibrant replacement R for \underline{F} (A) over \underline{F} (B), this says that

R_{n} → M_{Λ^{n,k}}R ×_{M_{Λ^{n,k}} \underline{F} (B)} \underline{F} (B)_{n}

is surjective on π_{0} — since it is (automatically) a fibration, this implies that it is surjective levelwise.

Thus f : R → \underline{F} (B) is a Reedy fibration and a horizontal levelwise Kan fibration, so [GJ] Lemma IV.4.8 implies that diag f is a fibration, so for any map C → B,

(diag \underline{F} (A)) ×_{(diag \underline{F} (B))}^{h} (diag \underline{F} (C)) ≃ (diag R) ×_{(diag \underline{F} (B))} (diag \underline{F} (C))
= diag (R ×_{(\underline{F} (B)} \underline{F} (C))
≃ diag (\underline{F} (A) ×_{\underline{F} (B)}^{h} \underline{F} (C))
≃ diag \underline{F} (A ×_{B} C),

the penultimate equivalence following because R → \underline{F} (B) is a Reedy fibrant replacement for \underline{F} (A), and the final one because F is homotopy-homogeneous and A → B is squarezero.

Finally, [CR] shows that W̄ X and diag X are weakly equivalent for all X, so

(W̄ \underline{F} (A)) ×_{(W̄ \underline{F} (B))}^{h} (W̄ \underline{F} (C)) ≃ W̄ \underline{F} (A ×_{B} C).

Any square-zero extension A → B in dN with kernel K can be expressed as the composition of the little extensions A/(I_{A}^{n+1}K) → A/(I_{A}^{n}K), making W̄ \underline{F} homotopyhomogeneous. □

Lemma 3.13. For a homotopy-preserving functor F : dN → 𝕊, the natural transformation F → W̄ \underline{F} is a weak equivalence.

Proof. The transformation comes from applying W̄ to the maps F (A) → \underline{F} (A) of bisimplicial sets coming from the canonical maps A → A^{Δ^{n}}.

Since A → A^{Δ^{n}} is a weak equivalence, the maps F (A) → \underline{F} (A) are also weak equivalences levelwise, so F = W̄ F → W̄ \underline{F} is a weak equivalence (as W̄ sends levelwise weak equivalences to weak equivalences). □

3.3. Representability.

Definition 3.14. Given a homotopy-surjecting homotopy-homogeneous functor F : dN_{R}^{♭} → 𝕊, A ∈ Alg_{H_{0}R}, x ∈ F_{0}(A), and an A-module M, define D_{x}^{i}(F, M) as follows.

For i ≤ 0, set

D_{x}^{i}(F, M) := π_{−i}(F (A ⊕ M) ×_{F (A)}^{h} {x}).

For i > 0, set

D_{x}^{i}(F, M) := π_{0}(F (A ⊕ M[−i]) ×_{F (A)}^{h} {x})/π_{0}(F (A ⊕ cone(M)[1 − i]) ×_{F (A)}^{h} {x}).

Note that homotopy-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.

Lemma 3.15. For all F, A, M as above, there are canonical isomorphisms

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

where the group on the left-hand side is defined as in Definition 3.14, and that on the right as in Definition 1.13.

In particular, if F is homotopy-preserving, then Definitions 3.14 and 1.13 are consistent.

Proof. We begin by noting that W̄ \underline{F} is indeed homotopy-preserving and homotopyhomogeneous, by Corollaries 3.10 and 3.12. Since F (A) = W̄ \underline{F} (A) for all A ∈ Alg_{H_{0}R}, it follows immediately that D_{x}^{i}(F, M) ≅ D_{x}^{i}(W̄ \underline{F} , M) for all i ≤ 0. Now for i > 0,

D_{x}^{i}(W̄ \underline{F} , M) = π_{0}(W̄ (T_{x}(\underline{F} /R)(M[−i]))
= π_{0}(T_{x}(F/R)(M[−i]))/π_{0}(F ((A ⊕ M[−i])^{Δ^{1}} ×_{A^{Δ^{1}}} A) ×_{F (A)}^{h} {x})
= π_{0}(T_{x}(F/R)(M[−i]))/π_{0}(T_{x}(F/R)(M[−i]^{Δ^{1}})),

where the quotient is taken by the map ∂_{0} −∂_{1} coming from the projections M^{Δ^{1}} → M. If dN_{R}^{♭} = sN_{R}^{♭}, then M[−i]^{Δ^{1}} = M[−i]⊕cone(M)[1−i], so D_{x}^{i}(W̄ \underline{F} , M) ≅ D_{x}^{i}(F, M). When dN_{R}^{♭} = dg_{+}N_{R}^{♭}, we have more work to do. In this case, M[−i]^{Δ^{1}} = τ_{≥0}(M[−i] ⊗ Ω_{n}). The key observation to make is that M[−i] ⊕ cone(M)[1 − i] can be expressed as a retract of τ_{≥0}(M[−i] ⊗ Ω_{n}) over M ⊕ M given by m ⊗ 1 ↦ (m, 0), m ⊗ x_{0}^{n} ↦ (0, m) for n > 0, and m ⊗ x_{0}^{n}dx_{0} ↦ (0, dm/(n + 1)). Thus

(∂_{0} − ∂_{1}) : π_{0}(T_{x}(F/R)(M[−i]^{Δ^{1}}) → π_{0}(T_{x}(F/R)(M[−i]))

has the same image as π_{0}(T_{x}(F/R)(cone(M)[1 − i]) → π_{0}(T_{x}(F/R)(M[−i])), so D_{x}^{i}(W̄ \underline{F} , M) ≅ D_{x}^{i}(F, M).

Finally, if F is homotopy-preserving, then Lemma 3.13 shows that the map F → W̄ \underline{F} is a weak equivalence, making the definitions consistent. □

Theorem 3.16. Let R be a derived G-ring admitting a dualising module (in the sense of [Lur] Definition 3.6.1) and take a functor F : dN_{R}^{♭} → 𝕊 satisfying the following conditions.

(1) F is homotopy-surjecting, i.e. it maps tiny acyclic extensions to surjections (on π_{0}).

(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 homotopy-homogeneous, i.e. for all square-zero extensions A ↠ C and all maps B → C, the map

F (A ×_{C} B) → F (A) ×_{F (C)}^{h} F (B)

is an equivalence.

(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, −) : 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, A) are all finitely generated A-modules.

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

the map

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

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

Proof. By Corollaries 3.10 and 3.12, W̄ \underline{F} is homotopy-preserving and homotopyhomogeneous. Since π^{0}F = π^{0}\underline{F}, the map π^{0}F → π^{0}W̄ \underline{F} is a weak equivalence. Lemma 3.15 then shows that D_{x}^{i}(F, M) ≅ D_{x}^{i}(W̄ \underline{F} , M), so W̄ F satisfies all the conditions of Theorem 2.17. □

Example 3.17. If X is a dg manifold (in the sense of [CFK1]), then the functor X : dg_{+}N_{R}^{♭} → Set given by X(A) = Hom(Spec A, X) satisfies the conditions of Theorem 3.16, so \underline{X} : dg_{+}N_{R}^{♭} → 𝕊 is a geometric derived 0-stack.

In fact, \underline{X} is just the hypersheafification of X. This follows because \underline{X} is a geometric derived 0-stack, so \underline{X}^{♯} = \underline{X}, and there is thus a map f : X^{♯} → \underline{X}. Since X^{♯} is a geometric derived 0-stack (as can be shown for instance by observing that it is equivalent to the derived stack Gpd(X)^{♯} of [Pri3] §6.4), Proposition 1.38 implies that f must be an equivalence.

This example will be adapted further in [Pri2], constructing geometric derived n-stacks from DG Lie algebras similar to those used in [CFK2] and [CFK1].

References

[Art] M. Artin. Versal deformations and algebraic stacks. Invent. Math., 27:165–189, 1974.

[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.

[CR] 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.

[GJ] Paul G. Goerss and John F. Jardine. Simplicial homotopy theory, volume 174 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1999.

[Gro] Alexander Grothendieck. Technique de descente et théorèmes d’existence en géométrie algébrique. II. Le théorème d’existence en théorie formelle des modules. In Séminaire Bourbaki, Vol. 5, pages Exp. No. 195, 369–390. Soc. Math. France, Paris, 1995.

[Har] 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.

[Hir] Philip S. Hirschhorn. Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2003.

[Hov] Mark Hovey. Model categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1999.

[Isa] Daniel C. Isaksen. A model structure on the category of pro-simplicial sets. Trans. Amer. Math. Soc., 353(7):2805–2841 (electronic), 2001.

[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.

[Mat] Hideyuki Matsumura. Commutative ring theory. Cambridge University Press, Cambridge, second edition, 1989. Translated from the Japanese by M. Reid.

[Pri1] J. P. Pridham. Unifying derived deformation theories. Adv. Math., 224(3):772–826, 2010. arXiv:0705.0344v5 [math.AG].

[Pri2] J. P. Pridham. Constructing derived moduli stacks. Geom. Topol., to appear. arXiv:1101.3300v2 [math.AG].

[Pri3] J. P. Pridham. Presenting higher stacks as simplicial schemes. Adv. Math., to appear. arXiv:0905.4044v3 [math.AG].

[Qui] Daniel G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin, 1967.

[Sch] Michael Schlessinger. Functors of Artin rings. Trans. Amer. Math. Soc., 130:208–222, 1968.

[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.