DERIVED MODULI OF SCHEMES AND SHEAVES

J.P.PRIDHAM

Abstract. We describe derived moduli functors for a range of problems involving schemes and quasi-coherent sheaves, and give cohomological conditions for them to be representable by derived geometric n-stacks. Examples of problems represented by derived geometric 1-stacks are derived moduli of polarised projective varieties, derived moduli of vector bundles, and derived moduli of abelian varieties.

Introduction

The driving philosophy behind derived algebraic geometry is that for every moduli problem, there should be an associated derived moduli problem. Thanks to the framework of [TV2] and [Lur1], there is often an obvious candidate for such a functor for derived moduli of schemes:

Given a simplicial algebra A, we first form the ∞-category C(A) of derived geometric stacks 𝔛 homotopy-flat over Spec A, with the additional property that

𝔛 ×_{Spec A}^{h} Spec π_{0}A

is a scheme. We can impose additional restrictions (smoothness, properness, dimension) on 𝔛 by requiring that they apply to 𝔛×_{Spec A}^{h} Spec π_{0}A. We then consider the ∞-category G(A) ⊂ C(A) in which the only morphisms are weak equivalences. Finally, we define the derived moduli functor 𝔐(A) to be the nerve of the ∞-category G(A). There are variants of this construction for subschemes and so on.

Similarly, for derived moduli of sheaves or complexes on a scheme X, we can first set C(A) to be the ∞-category of quasi-coherent complexes ℱ (in the sense of [TV2]) on X × Spec A, such that the complex ℱ ⊗_{A}^{L} π_{0}A on X × Spec (π_{0}A) satisfies any additional constraints (bounded, fixed Euler characteristic, concentrated in degree 0, free of rank n). We then form G(A) ⊂ C(A) as before, and take the nerve.

We can even combine these approaches to consider moduli of polarised projective schemes. C(A) then consists of pairs (𝔛, ℱ), for 𝔛 a derived geometric stack homotopy-flat over Spec A and ℱ a quasi-coherent complex on 𝔛, with

(𝔛 ×_{Spec A}^{h} Spec π_{0}A, ℱ ⊗_{A}^{L} π_{0}A)

a projective scheme and an ample line bundle.

Lurie’s Representability Theorem provides explicit criteria to check whether such a functor is representable by a geometric derived stack. However, the framework of [TV2] and [Lur1] does not lend itself easily to calculations, making it difficult to verify that these functors satisfy the criteria of the representability theorem. Even to show that the cotangent complex governs first-order deformations of n-geometric stacks, the more concrete approach of Artin hypergroupoids is needed (as in [Pri3] §7.1).

Several examples of representable derived moduli functors have already been established, however. Both [TV2] and [Lur1] construct Hom-stacks, while [Lur1] addresses derived moduli of stable curves, as well as derived Picard stacks and derived Hilbert schemes.

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

[TV2] deals with derived moduli of local systems and of finite algebras for any operad, and [Toë1] describes derived moduli of stable maps.

The purpose of this paper is to establish representability more generally for problems involving schemes and quasi-coherent sheaves, building on the explicit characterisations of [Pri3]. In simplistic terms, we show that if the associated underived moduli problem is representable, then the derived moduli problem will be representable provided only that certain cohomology groups satisfy mild finiteness conditions. The main results are Theorem 3.32 for moduli of derived geometric stacks over a fixed base, Theorem 3.35 which has the additional datum of an augmentation, and Theorem 4.12 for moduli of quasi-coherent complexes.

In 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 there no universal family is given, so the derived moduli spaces lack functorial interpretations. In [Pri2], we use the results of this paper to compare these approaches, thereby giving explicit presentations for the derived moduli spaces constructed here.

The structure of the paper is as follows.

In Section 1 we review the framework of simplicial categories, giving several properties of the nerve construction from simplicial categories to simplicial sets. Section 2 then recalls a variant (Theorem 2.10) of Lurie’s Representability Theorem, and shows how the nerve construction transfers good properties from simplicial category-valued functors to simplicial set-valued functors. These results are then applied in the last two sections to a wide range of moduli problems.

The main results of Section 3 are Corollaries 3.10 and 3.15, which deal with derived moduli of affine schemes, and Theorems 3.32 and 3.35, addressing derived moduli of (augmented) geometric n-stacks. These are applied to moduli of finite schemes (Example 3.11), of schemes (Example 3.36), of torsors (Example 3.38), of polarised schemes (Example 3.39), of abelian schemes (Example 3.45) and derived Hom-stacks (Example 3.37).

Section 4 deals, in Theorem 4.12 and Example 4.15, with moduli of quasi-coherent complexes. Lemma 4.16 then checks that the paper’s two possible approaches to derived moduli of vector bundles (via GL_{n}-torsors and via quasi-coherent complexes) give equivalent results, which was already known in the framework of [TV2].

Contents

Introduction 1 1. Simplicial categories 3 2. Derived moduli from simplicial categories 6 2.1. Background on representability 6 2.2. Formal quasi-smoothness 8 2.3. Homogeneity 9 2.4. Simplicial categories 9 3. Derived moduli of schemes 13 3.1. Derived moduli of affine schemes 13 3.2. Derived moduli of schemes 17 3.3. Examples 22 4. Derived moduli of quasi-coherent sheaves 27 4.1. Derived quasi-coherent sheaves 27 4.2. Moduli 29 References 32

1. Simplicial categories

Let 𝕊 denote the category of simplicial sets, and s𝕊 the category of bisimplicial sets.

Definition 1.1. Let sCat be the category of small simplicially enriched categories, which we will refer to as simplicial categories. Explicitly, an object C ∈ sCat consists of a set Ob C of objects, together with \underline{Hom}_{C}(x, y) ∈ 𝕊 for all x, y ∈ Ob 𝕊, equipped with an associative composition law and identities.

Definition 1.2. Given a simplicial category C, the category π_{0}C is defined to have the same objects as C, with morphisms

Hom_{π_{0}C}(x, y) = π_{0}\underline{Hom}_{C}(x, y).

Definition 1.3. Recall from [Ber] Theorem 1.1 that a morphism f : C → D in sCat is said to be a weak equivalence (a.k.a. an ∞-equivalence) whenever (W1) for any objects a_{1} and a_{2} in C, the map \underline{Hom}_{C}(a_{1}, a_{2}) → \underline{Hom}_{D}(fa_{1}, fa_{2}) is a weak

equivalence of simplicial sets; (W2) the induced functor π_{0}f : π_{0}C → π_{0}D is an equivalence of categories.

Definition 1.4. Given a simplicial category C, a morphism in C_{0} is said to be a homotopy equivalence if it becomes an isomorphism in π_{0}C.

Definition 1.5. Recall from [Ber] Theorem 1.1 that a morphism f : C → D in sCat is said to be a fibration whenever

(F1) for any objects a_{1} and a_{2} in C, the map \underline{Hom}_{C}(a_{1}, a_{2}) → \underline{Hom}_{D}(fa_{1}, fa_{2}) is a

fibration of simplicial sets; (F2) for any objects a_{1} ∈ C, b ∈ D, and homotopy equivalence e : fa_{1} → b in D, there

is an object a_{2} ∈ C and a homotopy equivalence d : a_{1} → a_{2} in C such that fd = e.

Definition 1.6. Given C ∈ sCat, define W̄ C := W̄ BC, where W̄ : s𝕊 → 𝕊 is the right adjoint to Illusie’s total Dec functor given by DEC (X)_{mn} = X_{m+n+1}, and B : Cat → 𝕊 is the nerve. Explicitly

(W̄ C)_{n} = ∐_{\underline{x}∈(Ob C)^{n+1}} C_{n−1}(x_{n−1}, x_{n}) × C_{n−2}(x_{n−2}, x_{n−1}) × … × C_{0}(x_{0}, x_{1}),

with operations

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

Remark 1.7. Another functor from bisimplicial to simplicial sets is the diagonal functor diag (Y)_{n} = Y_{nn}. In [CR], it is established that the canonical natural transformation

diag Y → W̄ Y

is a weak equivalence for all Y. When Y = BC, this transformation is given by

(y_{n}, h_{n−1}, y_{n−1}, …, h_{0}, y_{0}) ↦ (y_{n}, ∂_{0}h_{n−1}, ∂_{0}y_{n−1}, …, (∂_{0})^{n}h_{0}, (∂_{0})^{n}y_{0}).

This observation means that throughout this paper, W̄ can be replaced with diag , which is simpler to define. Our reasons for preferring W̄ are that it produces much smaller objects, as exploited in [DK]. For instance, if Y is a bisimplicial abelian group, then the Dold– Kan normalisation NW̄ Y (see Definition 2.1) is just the total complex of the Dold–Kan binormalisation NY.

The following results seem to be folklore, but lacking a suitable reference, we prove them afresh.

Proposition 1.8. If f : C → D is a fibration of simplicial categories, with π_{0}C and π_{0}D being groupoids, and B → D is an arbitrary morphism in sCat, then the homotopy fibre product

(W̄ C) ×_{(W̄ D)}^{h} (W̄ B)

is weakly equivalent to W̄ (C ×_{D} B).

Proof. Construct a category π_{n}C with objects Ob C and morphisms ∐_{f∈C_{0}(x,y)} π_{n}(C(x, y), f) for x, y ∈ Ob C. If we form the bisimplicial set BC, then π_{n}BC = Bπ_{n}C and we wish to verify the π_{∗}-Kan condition ([GJ] §IV.4) — that

Bπ_{n}C → BC_{0}

is a fibration.

If a functor F : A → A′ of categories is an isomorphism on objects, then BA → BA′ is a fibration if and only if

(1) F is full, and

(2) if a morphism h in A factors as F (h) = fg in A′, with either f or g lifting to f̃ or g̃ in A, then the other lifts uniquely in such a way that f̃g̃ = h.

Since π_{n}C → C_{0} has a canonical section (given by 0 ∈ π_{n}), it is full. Now, as π_{0}C is a groupoid, for all f ∈ C_{0}(x, y) the map f_{∗} : π_{n}\underline{End}_{C}(x) → π_{n}(\underline{Hom}_{C}(x, y), f) is an isomorphism. It therefore gives rise to an isomorphism ad_{f} : π_{n}\underline{End}_{C}(x) → π_{n}\underline{End}_{C}(y). We can write f̃ = f_{∗}a, g̃ = g_{∗}b and h = f_{∗}g_{∗}c, so f̃ and g̃ determine each other by the formula a = ad_{g}(cb^{−1}).

Now, (F2) implies that Bπ_{0}C → Bπ_{0}D is a Kan fibration, while (F1) implies that BC → BD is a pointwise fibration. Therefore the Bousfield–Friedlander Theorem ([GJ] Theorem IV.4.9) gives that

diag B(C ×_{D} B) → (diag BC) ×_{(diag BD)}^{h} (diag BB)

is a weak equivalence. Finally, [CR] shows that the canonical natural transformation diag BA → W̄ A is a weak equivalence for all A, which completes the proof. □

Definition 1.9. Given a category C, define c(C) to be the set of isomorphism classes in C.

Corollary 1.10. If C ∈ sCat is such that π_{0}C is a groupoid, then the homotopy fibre of W̄ C → c(π_{0}C) over x ∈ Ob C is just

W̄ \underline{End}_{C}(x).

Thus π_{n}(W̄ C, x) = π_{n−1}\underline{End}_{C}(x), where the basepoint of the \underline{End}_{C}(x) is taken to be the identity.

Proof. By using the model structure of [Ber] Theorem 1.1, there is a morphism g : C → C′ over c(C), with c(π_{0}C) = c(π_{0}C′), π_{∗}\underline{Hom}_{C′}(gx, gy) ≅ π_{∗}\underline{Hom}_{C}(x, y), and C′ → c(C) a fibration. Replacing C with C′, we may therefore assume that \underline{Hom}_{C}(x, y) is fibrant for all x, y.

Let D be the groupoid with objects Ob C, and a unique morphism x → y for any pair of objects with the same image in c(π_{0}C). There is therefore a canonical morphism C → D, and this is a fibration. Therefore the proposition gives

W̄ (C ×_{D} {x}) ≃ (W̄ C) ×_{(W̄ D)}^{h} {x}.

Since W̄ D ≃ c(π_{0}C), the right-hand side is just the homotopy fibre of W̄ C → c(π_{0}C). Meanwhile, C ×_{D} \{x\} is just the simplicial category \underline{End}_{C}(x) with unique object x and morphisms \underline{Hom}_{C}(x, x), as required. □

Corollary 1.11. If C and D are simplicial categories for which π_{0}C and π_{0}D are groupoids, then a morphism f : C → D is an ∞-equivalence if and only if

W̄ C → W̄ D

is a weak equivalence.

Proof. We have just seen that f is a W̄ -equivalence if and only if c(π_{0}C) → c(π_{0}D) is an isomorphism, and π_{n}\underline{End}_{C}(x) → π_{n}\underline{End}_{D}(fx) is an isomorphism for all n ≥ 0. This is precisely the same as saying that f is an ∞-equivalence. □

Corollary 1.12. If C → D and B → D are arbitrary morphisms in sCat, with π_{0}C and π_{0}D groupoids, then

W̄ (C ×_{D}^{h} B) → (W̄ C) ×_{(W̄ D)}^{h} (W̄ B)

is a weak equivalence, where the homotopy fibre product C ×_{D}^{h} B is taken in the model structure of [Ber].

In particular, if π_{0}B is also a groupoid, then the fundamental groupoid π_{f}((W̄ C) ×_{(W̄ D)}^{h} (W̄ B)) is equivalent to the 2-fibre product

(π_{0}C) ×_{(π_{0}D)}^{(2)} (π_{0}B),

while for any object α of C ×_{D}^{(2)} B given by x ∈ Ob C, y ∈ Ob B and f ∈ D_{0}(x̄, ȳ), we have

π_{n}((W̄ C) ×_{(W̄ D)}^{h} (W̄ B), α) ≅ π_{n−1}(\underline{End}_{C}(x) ×_{f_{∗},\underline{Hom}_{D}(x̄,ȳ),f^{∗}}^{h} \underline{End}_{B}(y)).

Proof. Factorise C → D as C → C′ → D, a weak equivalence followed by a fibration. By Corollary 1.11,

(W̄ C) ×_{(W̄ D)}^{h} (W̄ B) → (W̄ C′) ×_{(W̄ D)}^{h} (W̄ B)

is a weak equivalence. By Proposition 1.8, the latter is weakly equivalent to W̄ (C′ ×_{D} B), which is just W̄ (C ×_{D}^{h} B), since sCat is right proper.

The description of the fundamental groupoid follows from the observation that for A ∈ sCat and a discrete groupoid Γ, there are canonical isomorphisms

Hom_{𝕊}(W̄ A, BΓ) ≅ Hom_{sCat}(A, Γ) ≅ Hom_{Cat}(π_{0}A, Γ).

For C′ → D a fibration, the map

π_{0}(C′ ×_{D} B) = (π_{0}C′) ×_{(π_{0}D)} (π_{0}B) → (π_{0}C′) ×_{(π_{0}D)}^{(2)} (π_{0}B)

is an equivalence, and the latter is equivalent to (π_{0}C) ×_{(π_{0}D)}^{(2)} (π_{0}B).

Finally, for α := (x, f, y) as above, also write x for the image of x in C′. Since C′ → D lifts homotopy equivalences, there is a morphism g : x → z in C′ mapping to f in D. Then α defines the same element in c(π_{0}C′) ×_{(π_{0}D)}^{(2)} (π_{0}B) as α′ := (z, id, y), so we need only describe homotopy groups based at α′.

By Corollary 1.10,

π_{n}(W̄ (C′ ×_{D} B), α′) = π_{n−1}(\underline{End}_{C′}(z) ×_{\underline{End}_{D}(ȳ)} \underline{End}_{B}(y), α′),

and we note that \underline{End}_{C′}(z) → \underline{End}_{D}(ȳ) is a Kan fibration, since C′ → D is a fibration. Now, since g and f are homotopy equivalences, they induce weak equivalences

g^{∗} : \underline{End}_{C′}(z) → \underline{Hom}_{C′}(x, z)
g_{∗} : \underline{End}_{C′}(x) → \underline{Hom}_{C′}(x, z)
f^{∗} : \underline{End}_{D}(ȳ) → \underline{Hom}_{D}(x̄, ȳ),

giving weak equivalences

\underline{End}_{C′}(z)×_{\underline{End}_{D}(ȳ)}\underline{End}_{B}(y) → \underline{Hom}_{C′}(x, z)×_{\underline{Hom}_{D}(x̄,ȳ)}^{h}\underline{End}_{B}(y) ← \underline{End}_{C′}(x)×_{f_{∗},\underline{Hom}_{D}(x̄,ȳ)}^{h}\underline{End}_{B}(y).

Finally, the weak equivalence C → C′ gives a weak equivalence \underline{End}_{C}(x) → \underline{End}_{C′}(x), yielding the expression required. □

2. Derived moduli from simplicial categories

2.1. Background on representability. Denote the category of simplicial rings by sRing, the category of simplicial R-algebras by sAlg_{R}, and the category of simplicial R-modules by sMod_{R}. If R is a (dg) ℚ-algebra, we let dg_{+}Alg_{R} be the category of differential gradedcommutative 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 2.1. 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 : sAlg_{R} → dg_{+}Alg_{R}.

When R is a ℚ-algebra, this functor is a right Quillen equivalence.

Definition 2.2. 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. For A ∈ dAlg_{R}, write I_{A} := ker(A → H_{0}A).

Definition 2.3. 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.

Definition 2.4. Say that a functor F : dN^{♭} → C to a model category C is homotopypreserving if for all tiny acyclic extensions A → B, the map

F (A) → F (B)

is a weak equivalence.

Definition 2.5. We say that a functor

F : dN^{♭} → C

to a model category C is homotopy-homogeneous if for all square-zero extensions A → B and all maps C → B in dN^{♭}, the natural map

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

to the homotopy fibre product is a weak equivalence.

Definition 2.6. Given a homotopy-preserving homotopy-homogeneous functor F : dN^{♭} → 𝕊, A ∈ dN^{♭}, and a point x ∈ F (A), define

T_{x}(F/R) : dMod_{A} → 𝕊

by

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

Definition 2.7. Given F, A, x as above, and M ∈ dMod_{A}, define

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

noting that this is well-defined by [Pri1] Lemma 1.12.

Proposition 2.8. If F : sAlg_{R} → 𝕊 is homotopy-preserving and homotopy-homogeneous, then for any square-zero extension e : I → A ⟶^{f} B in C, there is a sequence of sets

π_{0}(F A) ⟶^{f_{∗}} π_{0}(F B) ⟶^{o_{e}} Γ(F B, D^{1}(F/R, 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/R, 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/R, 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/R, I) \ar[r]^{−∗y} & π_{0}(F A).}

Proof. This is [Pri1] Proposition 1.17 □

Definition 2.9. Given a functor F : dN_{R}^{♭} → 𝕊, define π^{0}F : Alg_{π_{0}R} → 𝕊 by π^{0}F (A) = F (A).

The following variant of Lurie’s Representability Theorem is [Pri1] Theorem 2.17:

Theorem 2.10. Let R be a derived G-ring admitting a dualising module (in the sense of [Lur1] 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 is homotopy-preserving.

(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.

(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/R), −) : 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/R, 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. Moreover, F′ is uniquely determined by F (up to weak equivalence).

Remark 2.11. 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 final condition above 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 final condition of the theorem reduces to requiring that the maps

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

be isomorphisms for all i.

Remark 2.12. Note that there are slight differences in terminology between [TV2] and [Lur1]. 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 [TV2], and it follows easily that every n-geometric stack in [TV2] is n-truncated. Conversely, every geometric n-stack is (n + 2)-geometric.

Theorem 2.10 takes the convention from [Lur1], so “geometric derived n-stack” means “n-truncated derived geometric stack”.

2.2. Formal quasi-smoothness.

Definition 2.13. Say that a natural transformation η : F → G of functors F, G : dN^{♭} → C to a model category C is homotopic if for all tiny acyclic extensions A → B, the map

F (A) → F (B) ×_{G(B)} G(A)

is a trivial fibration. Say that F is homotopic if F → • is so, where • denotes the final object of C.

Definition 2.14. Say that a natural transformation η : F → G of functors F, G : dN^{♭} → C to a model category C is formally quasi-presmooth if for all square-zero extensions A → B, the map

F (A) → F (B) ×_{G(B)} G(A)

is a fibration.

Say that η is formally quasi-smooth if it is formally quasi-presmooth and homotopic.

Definition 2.15. Say that a natural transformation η : F → G of functors F, G : dN^{♭} → 𝕊 is formally presmooth (resp. formally smooth) if it is formally quasi-presmooth (resp. formally quasi-smooth) and for all square-zero extensions A → B, the map

F (A) → F (B) ×_{G(B)} G(A)

is surjective on π_{0}.

Definition 2.16. 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.

2.3. Homogeneity.

Definition 2.17. 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 2.18. 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. 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 formally étale,

F (A ×_{B} C) ≅ G(A ×_{B} C) ×_{GC} F C,

and

F A ×_{F B} F C ≅ [GA ×_{GB} F B] ×_{F B} F C
= GA ×_{GB} F C
= [GA ×_{GB} GC] ×_{GC} 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. □

2.4. Simplicial categories.

Definition 2.19. Given functors A ⟶^{F} B ⟵^{G} C between categories, define the 2-fibre product A ×_{B}^{(2)} C as follows. Objects of A ×_{B}^{(2)} C are triples (a, θ, c), for a ∈ A, c ∈ C and θ : F a → Gc an isomorphism in B. A morphism in A ×_{B}^{(2)} C from (a, θ, c) to (a′, θ′, c′) is a pair (f, g), where f : a → a′ is a morphism in A and g : c → c′ a morphism in C, satisfying the condition that

Gg ∘ θ = θ′ ∘ F f.

Remark 2.20. This definition has the property that A ×_{B}^{(2)} C is a model for the 2-fibre product in the 2-category of categories. However, we will always use the notation A ×_{B}^{(2)} C to mean the specific model of Definition 2.19, and not merely any equivalent category.

Also note that

A ×_{B}^{(2)} C = (A ×_{B}^{(2)} B) ×_{B} C

Definition 2.21. Given functors A ⟶^{F} B ⟵^{G} C between simplicial categories, define the 2-fibre product A ×_{B}^{(2)} C as follows. Objects of A ×_{B}^{(2)} C are triples (a, θ, c), for a ∈ A, c ∈ C and θ : F a → Gc an isomorphism in B_{0}. Morphisms are given by setting

\underline{Hom}_{A×_{B}^{(2)}C}((a, θ, c), (a′, θ′, c′)) := \{(f, g) ∈ \underline{Hom}_{A}(a, a′) × \underline{Hom}_{C}(c, c′) : Gg ∘ θ = θ′ ∘ F f\}.

Definition 2.22. Say that a morphism F : A → B in sCat is a (trivial) 2-fibration if A ×_{B}^{(2)} B → B is a (trivial) fibration in the sense of Definition 1.5. Explicitly, F is a 2-fibration if (F1) for any objects a_{1} and a_{2} in A, the map \underline{Hom}_{A}(a_{1}, a_{2}) → \underline{Hom}_{B}(F a_{1}, F a_{2}) is a

fibration of simplicial sets; (F2) for any objects a_{1} ∈ A, b ∈ B, and any homotopy equivalence e : F a_{1} → b in

B, there is an object a_{2} ∈ C, a homotopy equivalence d : a_{1} → a_{2} in C and an isomorphism θ : F a_{2} → b such that θ ∘ F d = e.

Likewise, F is a trivial 2-fibration if (TF1) for any objects a_{1} and a_{2} in A, the map \underline{Hom}_{A}(a_{1}, a_{2}) → \underline{Hom}_{B}(F a_{1}, F a_{2}) is a

trivial fibration of simplicial sets; (TF2) F_{0} : A_{0} → B_{0} is essentially surjective on objects.

Definition 2.23. Say that a natural transformation η : D → E of functors D : dN^{♭} → sCat is 2-homotopic if for all tiny acyclic extensions A → B, the map

D(A) → D(B) ×_{E(B)}^{(2)} E(A)

is a trivial 2-fibration. Say that D is 2-homotopic if D → • is so.

Lemma 2.24. If D is 2-homotopic, then D is homotopy-preserving.

Proof. By [Pri1] Lemma 2.11 and the proof of [Pri1] Proposition 2.7, it suffices to show that D maps tiny acyclic extensions A → B to weak equivalences. Since D_{0}(A) → D_{0}(B) is essentially surjective on objects, so is π_{0}D(A) → π_{0}D(B). On morphisms, we have \underline{Hom}_{D(A)}(x, y) → \underline{Hom}_{D(B)}(x, y) a trivial fibration (and hence a weak equivalence) for all x, y ∈ D(A), so D(A) → D(B) is indeed a weak equivalence. □

Definition 2.25. Say that a natural transformation η : D → E of functors D, E : dN^{♭} → sCat is formally 2-quasi-presmooth if for all square-zero extensions A → B, the map

D(A) → D(B) ×_{E(B)}^{(2)} E(A)

is a 2-fibration. If η is also 2-homotopic, then we say that η is formally 2-quasi-smooth.

Say that D is formally 2-quasi-presmooth (resp. formally 2-quasi-smooth) if D → • is so.

Definition 2.26. Say that a functor F : dN^{♭} → sCat is 2-homogeneous if for all squarezero extensions A → B, the map

F(A ×_{B} C) → F(A) ×_{F(B)}^{(2)} F(C)

is essentially surjective on objects (for F_{0}), and an isomorphism on \underline{Hom}-spaces.

Lemma 2.27. If D : dN^{♭} → sCat is 2-homogeneous and formally 2-quasi-presmooth, then D is homotopy-homogeneous.

Proof. Take a square-zero extension A → B and a map C → B. We first need to show that

θ : π_{0}D(A ×_{B} C) → π_{0}(D(A) ×_{D(B)}^{h} D(C))

is essentially surjective on objects.

Any object of the right-hand side is represented by objects x ∈ D(A), y ∈ D(B), and a homotopy equivalence f : x̄ → ȳ in D(B). Since D is formally 2-quasi-presmooth, we may lift f to some map f̃: x → z in D(A)_{0} with g : z̄ ≅ ȳ in D(B)_{0}. Since D is 2-homogeneous, there must then be an object t ∈ D(A ×_{B} C) whose image in D(A)_{0} ×_{D(B)_{0}}^{(2)} D(C)_{0} is

isomorphic to (z, g, y). Since f̃ gives a homotopy equivalence from this to (x, f, y), we have shown that θ is essentially surjective on objects.

For morphisms, we have

\underline{Hom}_{D(A×_{B}C)}(t, t′) ≅ \underline{Hom}_{D(A)}(t, t′) ×_{\underline{Hom}_{D(B)}(t,t′)} \underline{Hom}_{D(C)}(t, t′)
≃ \underline{Hom}_{D(A)}(t, t′) ×_{\underline{Hom}_{D(B)}(t,t′)}^{h} \underline{Hom}_{D(C)}(t, t′),

the first isomorphism following from homogeneity, and the second because \underline{Hom}_{D(A)}(t, t′) → \underline{Hom}_{D(B)}(t, t′) is a fibration, by quasi-smoothness. This completes the proof that

D(A ×_{B} C) ≃ D(A) ×_{D(B)}^{h} D(C)

in sCat. □

Definition 2.28. Say that D : dN^{♭} → sCat has formally quasi-presmooth (resp. formally quasi-smooth) \underline{Hom}-spaces if \underline{Hom}_{D}(x, y) : A↓dN^{♭} → 𝕊 is formally quasi-presmooth (resp. formally quasi-smooth) for all A ∈ dN^{♭} and x, y ∈ D(A).

Definition 2.29. Given a simplicial category C, define W(C) to be the full simplicial subcategory in which morphisms are maps whose images in π_{0}C are invertible. In particular, this means that π_{0}W(C) is the core (maximal subgroupoid) of π_{0}C.

Proposition 2.30. If D : dN^{♭} → sCat is 2-homogeneous with formally quasi-smooth \underline{Hom}-spaces, then W(D) → D is formally étale, in the sense that for any square-zero extension A → B, the map

W(D)(A) → D(A) ×_{D(B)} W(D)(B)

is an isomorphism.

Proof. Since f ∈ \underline{Hom}_{C}(x, y)_{n} lies in W(C) if and only if (∂_{0})^{n}f is a homotopy equivalence, it suffices to show that

W(D)_{0}(A) → D_{0}(A) ×_{D_{0}(B)} W(D)_{0}(B)

is an isomorphism.

Take a morphism f ∈ \underline{Hom}_{D(B)}(x, y)_{0}, with objects x, y lifting to x, y ∈ D(A). If we set D_{f}^{n}(M) := D_{f}^{n}(\underline{Hom}_{D}(x, y)/A, M), then Proposition 2.8 shows that the obstruction o(f) to lifting f̄ to \underline{Hom}_{D(A)}(x, y)_{0} lies in D1 ; f̄(I), where I = ker(A → B).

Assume that f is a homotopy equivalence, so there exist g ∈ \underline{Hom}_{D(B)}(y, x)_{0} and homotopies h^{x} ∈ \underline{Hom}_{D(B)}(x, x)_{1}, h^{y} ∈ \underline{Hom}_{D(B)}(y, y)_{1} such that

∂_{1}h^{x} = id_{x}, ∂_{1}h^{y} = id_{y}, ∂_{0}h^{x} = g ∘ f̄, ∂_{0}h^{y} = f̄∘ g.

We now need to study the construction of obstruction maps.

For the acyclic extension B̃ → B as in the Proof of Proposition 2.8, we may lift f to f̃ in D(B̃), and do the same for g. Then the obstruction o(f ∘ g) to lifting f ∘ g is the class of the image of f̃∘ g̃ in π_{0}\underline{Hom}D(B ⊕ I[−1])(y, y), which is just

o(f ∘ g) = f ∘ g + f_{∗}o(g) + g^{∗}o(f) ∈ D_{f∘g}^{1}(I).

Since \underline{Hom}_{D(B̃)}(x, x) → \underline{Hom}_{D(B)}(x, x) is a trivial fibration (by quasi-smoothness), we may also lift the homotopies h^{x}, h^{y} to give homotopies h̃^{x} ∈ \underline{Hom}_{D(B̃)}(x, x)_{1}, h̃^{y} ∈ \underline{Hom}_{D(B̃)}(y, y)_{1} making f̃and g̃ homotopy inverses. Then h̃^{y} ensures that o(f∘g) = 0 (since the image of f̃∘ g̃ is homotopic to id_{y}), and therefore f_{∗}o(g) = −g^{∗}o(f) ∈ D_{y}^{1}(I). Likewise g_{∗}o(f) = −f^{∗}o(g) ∈ D_{x}^{1}(I). Since o(g) = (gf)_{∗}(fg)^{∗}o(g), this gives o(g) = −g_{∗}g^{∗}o(f)

If f lifts to f′ ∈ \underline{Hom}_{D(A)}(x, y)_{0}, then o(f) = 0, and the formula above ensures that o(g) = 0, so g also lifts to g′ in D(A). Now, f′ ∘ g′ is in the fibre of \underline{End}_{D(A)}(y)_{0} → \underline{End}_{D(B)}(y)_{0} over id_{y}, so [f′ ∘ g′] ∈ π_{0}\underline{End}_{D(A)}(y) lies in the image of

π_{0}T_{id_{y}}(\underline{End}_{D}(y)/A, I) = D_{y}^{0}(I).

Another choice for g′ is of the form g′+a for a ∈ D_{g}^{0}(I), and then [f′∘(g′+1)] = [f′∘g′]+f_{∗}a. Since f is a homotopy equivalence, f_{∗} : D_{g}^{0}(I) → D_{y}^{0}(I) is an isomorphism, so we may set a = −(f_{∗})^{−1}[f′ ∘ g′]. Without loss of generality, we may therefore assume that f′ and g′ are homotopy inverses.

Finally, write [g′ ∘ f′] = id_{x} + b, for b ∈ D_{x}^{0}(I). Since f′ ∘ g′ is homotopic to id_{y}, applying f_{∗} to b gives

[f′] + f_{∗}b = [f′ ∘ g′ ∘ f′] = [f′],

so f_{∗}b = 0 ∈ D_{f}^{0}(I). Thus b = 0, as f_{∗} : D_{x}^{0}(I) → D_{f}^{0}(I) is an isomorphism. □

Corollary 2.31. If D : dN^{♭} → sCat is formally 2-quasi-smooth and 2-homogeneous, then so is W(D).

Proof. For any square-zero extension A → B in dN^{♭}, we have

W(D)(A) = D(A) ×_{D(B)} W(D)(B),

by Proposition 2.30. Thus

W(D)(A) ×_{W(D)(B)}^{(2)} W(D)(B) = D(A) ×_{D(B)}^{(2)} W(D)(B),

so θ : W(D)(A) ×_{W(D)(B)}^{(2)} W(D)(B) → W(D)(B) is the pullback of D(A) ×_{D(B)}^{(2)} D(B) → D(B) along W(D)(B) → D(B). By 2-quasi-smoothness of D, this means that θ is a fibration in sCat, and a trivial fibration whenever A → B is a tiny acyclic extension. Thus W(D) is formally 2-quasi-smooth.

To see that W(D) is 2-homogeneous, first observe that D(A) ×_{D(B)}^{(2)} D(C) and

W(D)(A) ×_{W(D)(B)}^{(2)} W(D)(C) have the same objects. For a square-zero extension A → B and a map C → B, 2-homogeneity of D then gives that

W(D)_{0}(A ×_{B} C) → W(D)_{0}(A) ×_{W(D)_{0}(B)}^{(2)} W(D)_{0}(C)

is essentially surjective on objects. Proposition 2.30 combines with 2-homogeneity of D and Lemma 2.18 to give that \underline{Hom}_{W(D)}(x, y) is homogeneous, since \underline{Hom}_{W(D)}(x, y) → \underline{Hom}_{D}(x, y) is formally étale. □

Proposition 2.32. Take a ring R, a functor F: Alg_{R} → sCat for which W̄ W(F): Alg_{R} → 𝕊 is an étale hypersheaf, and a subfunctor E ⊂ F for which each E(A) is a full simplicial subcategory of F(A).

Then W̄ W(E) is an étale hypersheaf if and only if for any A ∈ Alg_{R} and any étale cover {f_{i}: A → B_{i}}_{i∈I}, the map

cπ_{0}E(A) → cπ_{0}F(A) ×_{(∏_{i} cπ_{0}F(B_{i}))} (∏_{i} cπ_{0}E(B_{i}))

is surjective, where cC denotes the set of isomorphism classes in a category C.

Proof. The “only if” implication is almost automatic, so we will just prove the “if” implication. First, take a finite set {A_{i}}_{i∈I} of R-algebras. Since F := W̄ W(F) is a hypersheaf, the map

F (∏_{i} A_{i}) → ∏_{i} F (A_{i})

is a weak equivalence.

Setting E := W̄ W(E), the fact that E ⊂ F is a full simplicial subcategory implies that the map E → F is injective on π_{0} and an isomorphism on all other homotopy groups. Writing A := ∏_{i} A_{i}, it follows that

π_{n}(E(A), x) ≅ ∏_{i} π_{n}(E(A_{i}), x)

for all n > 0 and all x, and that π_{0}E(A) ↪ ∏_{i} π_{0}E(A_{i}).

Now π_{0}E = cπ_{0}E and the set {A → A_{i}}_{i∈I} is an étale cover, so we also have

π_{0}E(A) ↠ π_{0}F (A) ×_{(∏_{i} π_{0}F (A_{i}))} (∏_{i} π_{0}E(A_{i})) = ∏_{i} π_{0}E(A_{i}),

and hence

E(A) ≃ ∏_{i} E(A_{i}).

Now take an affine étale hypercover B → B^{•}; we need to show that the map

ϕ: E(B) → ho\underleftarrow{lim}_{n} E(B^{n})

is a weak equivalence. Because E ≃ π_{0}E ×_{π_{0}F} F, we have

ho\underleftarrow{lim}_{n} E(B^{n}) ≃ (ho\underleftarrow{lim}_{n} π_{0}E(B^{n})) ×_{(ho⟵^{lim}_{n} π_{0}F (B^{n}))} (ho\underleftarrow{lim}_{n} F (B^{n}))
≃ (\underleftarrow{lim}_{n} π_{0}E(B^{n})) ×_{(⟵^{lim}_{n} π_{0}F (B^{n}))} (ho\underleftarrow{lim}_{n} F (B^{n}))
≃ (\underleftarrow{lim}_{n} π_{0}E(B^{n})) ×_{(⟵^{lim}_{n} π_{0}F (B^{n}))} F (B).

Since π_{0}E ⊂ π_{0}F and \underleftarrow{lim}_{n} π_{0}E(B^{n}) ⊂ π_{0}E(B^{0}), we can rewrite this as

π_{0}E(B^{0}) ×_{π_{0}F (B^{0})} F (B),

so ϕ induces an isomorphism on all higher homotopy groups and is injective on π_{0}. Because B → B^{0} is an étale cover, the map

π_{0}E(B) → π_{0}F (B) ×_{π_{0}F (B^{0})} π_{0}E(B^{0})

is surjective, so ϕ is also surjective on π_{0}, and hence a weak equivalence. □

3. Derived moduli of schemes

3.1. Derived moduli of affine schemes.

Definition 3.1. Given A ∈ dg_{+}Alg_{R}, say that B ∈ dg_{+}Alg_{A} is quasi-free if the underlying map A_{∗} → B_{∗} of graded algebras is freely generated. Say that B ∈ dg_{+}Alg_{A} is cofibrant if it is a retract of a quasi-free dg algebra.

This is equivalent to satisfying the condition that for all acyclic surjections C′ → C in dg_{+}Alg_{A}, the map Hom_{dg_{+}Alg_{A}}(B, C′) → Hom_{dg_{+}Alg_{A}}(B, C) is surjective.

Definition 3.2. Given A ∈ sAlg_{R}, say that B ∈ sAlg_{A} is quasi-free if there are sets Σ_{q} ⊂ B_{q} with B_{q} = A_{q}[Σ_{q}], such that ⋃_{q} Σ_{q} is closed under the degeneracy operations. Say that B ∈ sAlg_{A} is cofibrant if it is a retract of a quasi-free simplicial algebra.

This is equivalent to satisfying the condition that for all acyclic surjections C′ → C in sAlg_{A}, the map Hom_{sAlg_{A}}(B, C′) → Hom_{sAlg_{A}}(B, C) is surjective.

Definition 3.3. Let Δ^{n} ∈ 𝕊 be the standard n-simplex, and ∂Δ^{n} ∈ 𝕊 its boundary. Given 0 ≤ k ≤ n, define the kth horn Λ_{k}^{n} of Δ^{n} to be the simplicial set obtained from Δ^{n} by removing the interior and the kth face. See [GJ] §I.1 for explicit descriptions.

Definition 3.4. There is a simplicial structure on sAlg_{A} given as follows. For B ∈ sAlg_{A} and K ∈ 𝕊, B^{K} is defined by

(B^{K})_{n} := Hom_{𝕊}(K × Δ^{n}, B).

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

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

Definition 3.5. As in [Pri1] Lemma 3.4, put a simplicial structure on dg_{+}Alg_{A} as follows. First set Ω_{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 B^{Δ^{n}} as the good truncation B^{Δ^{n}} := τ_{≥0}(B ⊗ Ω_{n}). Spaces \underline{Hom}_{dg_{+}Alg_{A}}(C, B) ∈ 𝕊 of morphisms are then given by

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

Definition 3.6. For A ∈ dg_{+}Alg_{R} (resp. A ∈ sAlg_{R}), define ALG(A) to be the full simplicial subcategory of the category dg_{+}Alg_{A} (resp. sAlg_{A}) on cofibrant objects.

Proposition 3.7. The functor ALG : dN^{♭} → sCat defined above is 2-homogeneous and formally 2-quasi-smooth.

Proof. We first prove that ALG is 2-homogeneous, taking a square-zero extension A → B and a morphism C → B. Homogeneity of \underline{Hom}_{ALG} is almost immediate, because S ∈ ALG(A ×_{B} C) is flat over A ×_{B} C as a graded algebra (resp. levelwise flat over A ×_{B} C), giving the pullback square

\underline{Hom}_{ALG(A×_{B}C)}(S, S′)  ⟶ \underline{Hom}_{ALG(A)}(S ⊗_{A×_{B}C} A, S′ ⊗_{A×_{B}C} A)
↓  ↓
\underline{Hom}_{ALG(C)}(S ⊗_{A×_{B}C} C, S′ ⊗_{A×_{B}C} C) ⟶ \underline{Hom}_{ALG(B)}(S ⊗_{A×_{B}C} B, S′ ⊗_{A×_{B}C} B).

Given objects S_{A} ∈ ALG(A) and S_{C} ∈ ALG(C), and an isomorphism α : S_{A} ⊗_{A} B → S_{C} ⊗_{C} B, we can define S ∈ ALG(A ×_{B} C) by S := S_{A} ×_{α,S_{C}⊗_{C}B} S_{C}, noting that this is automatically cofibrant, with S ⊗_{A×_{B}C} A ≅ S_{A} and S ⊗_{A×_{B}C} C ≅ S_{C}. This completes the proof that ALG is 2-homogeneous.

Now, since any S ∈ ALG(A) is cofibrant over A, the map f : \underline{Hom}_{ALG(A)}(S, S′) → \underline{Hom}_{ALG(B)}(S ⊗_{A} B, S′ ⊗_{A} B) is a fibration for all square-zero extensions A → B, because S′ → S′ ⊗_{A} B is also a square-zero extension. If A → B is also a weak equivalence, then so is S′ → S′ ⊗_{A} B (as S′ is levelwise flat over A, so the kernel is S′ ⊗_{A} I), and f is therefore a trivial fibration. We have therefore shown that \underline{Hom}_{ALG}(S, S′) is formally quasi-smooth

It only remains to show that ALG(A) → ALG(B) satisfies Condition (F2) from Definition 2.22 for all square-zero extensions A → B, and satisfies condition (TF2) whenever the extension is acyclic. [Pri3] Proposition 8.6 shows that the obstruction (in the simplicial case) to lifting S ∈ ALG(B) to ALG(A) lies in

𝔼xt_{B}^{2}(Ω^{S/B}, S ⊗_{B} I) = D_{id_{S}}^{2}(\underline{Hom}_{ALG}(S, S), I),

where I = ker(A → B). The idea is that as an almost simplicial algebra (i.e. discarding ∂_{0}), S has a unique levelwise flat lift to S̃ over A. We can then choose a lifting δ of ∂_{0} with ∂_{i}δ = δ∂_{i−1} for all i > 1, and the obstruction comes from comparing δ^{2} and δ∂_{1}. The same proof adapts to the dg case, where we lift S as a flat graded algebra, lift d to δ, and get δ^{2} as the obstruction. In particular, this ensures that (TF2) is satisfied, since if H_{∗}(I) = 0, then 𝔼xt_{B}^{2}(Ω^{S/B}, S ⊗_{B} I) = 0.

We now establish (F2): take S ∈ ALG(A), write S̄ := S ⊗_{A} B, and take a homotopy equivalence θ : S̄ → T. We now proceed with the proof in the dg case — the simplicial case is entirely similar, replacing graded algebras with almost simplicial algebras. As a graded algebra, T has a unique lift T̃ over A, and we can then lift θ to θ̃ : S → T̃ (as a graded morphism), since S is smooth. We also lift the differential d on T to δ on T̃. The obstruction to these giving a morphism of dg algebras is then

(u, v) := (δ^{2}, δ ∘ θ̃ − θ̃ ∘ d) ∈ HOM_{B}^{2}(Ω^{T/B}, T ⊗_{B} I) × HOM_{B}^{1}(Ω^{S̄/B}, T ⊗_{B} I),

which satisfies du = 0, dv = u ∘ θ. Here, HOM_{B}^{i} denotes B-linear graded morphisms of chain degree −i. Alternative choices for (δ, θ̃) would be of the form (δ + a, θ̃ + b), for

(a, b) ∈ HOM_{B}^{1}(Ω^{T/B}, T ⊗_{B} I) × HOM_{B}^{0}(Ω^{S̄/B}, T ⊗_{B} I);

these send (u, v) to (u + da, v + db + a ∘ θ). Thus the obstruction to lifting θ and T compatibly lies in H^{2} of the cone complex

HOM_{B}^{•}(Ω^{T/B}, T ⊗_{B} I) ⟶^{θ^{∗}} HOM_{B}^{•}(Ω^{S̄/B}, T ⊗_{B} I).

Since θ is a weak equivalence, θ^{∗} is a quasi-isomorphism, so cohomology of this complex is 0, and suitable liftings thus exist. □

Definition 3.8. Given a functor C : Alg_{H_{0}R} → sCat and a functorial simplicial subcategory M ⊂ C, say that M is an open simplicial subcategory of C if

(1) M is a full simplicial subcategory, i.e. for all X, Y ∈ M(A), the map \underline{Hom}_{M(A)}(X, Y) → \underline{Hom}_{C(A)}(X, Y) is a weak equivalence;

(2) the map M → C is homotopy formally étale, in the sense that for any square-zero extension A → B, the map

π_{0}M(A) → π_{0}C(A) ×_{π_{0}C(B)}^{(2)} π_{0}M(B)

is essentially surjective on objects.

Definition 3.9. Given a simplicial category C for which π_{0}C is a groupoid, say that C is n-truncated if

π_{i}\underline{Hom}_{C(A)}(X, Y) = 0

for all i ≥ n. Thus C is n-truncated if and only if π_{i}(W̄ C) = 0 for all i > n.

Corollary 3.10. Take an n-truncated presheaf M : Alg_{H_{0}R} → sCat, open in π^{0}W(ALG), such that

(0) if {f_{i}: A → B_{i}}_{i∈I} is an étale cover in Alg_{H_{0}R}, then an object X ∈ π_{0}W(ALG)(A) lies in the essential image of π_{0}M(A) whenever f_{i}(X) lies in the essential image of π_{0}M(B_{i}) for all i.

(1) for all finitely generated A ∈ Alg_{H_{0}R} and all X ∈ M(A), the functors

𝔼xt_{X}^{i}(𝕃^{X/A}, X ⊗_{A}^{L} −) : Mod_{A} → Ab

preserve filtered colimits for all i > 1.

(2) for all finitely generated integral domains A ∈ Alg_{H_{0}R} and all X ∈ M(A), the groups 𝔼xt_{X}^{i}(𝕃^{X/A}, X) are all finitely generated A-modules.

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

the map

M(A) → \underleftarrow{lim}^{h}M(A/𝔪^{n})

is a weak equivalence (see Remark 2.11 for a reformulation).

(4) The functor c(π_{0}M) : Alg_{π_{0}R} → Set of components of the groupoid π_{0}M preserves filtered colimits.

(5) For all A ∈ Alg_{π_{0}R} and all X ∈ M(A), the functors π_{i}\underline{Hom}_{M}(X, X) : Alg_{A} → Set preserve filtered colimits for all i ≥ 0.

Let M̃ : dN_{R}^{♭} → sCat be the full simplicial subcategory of W(ALG)(A) consisting of objects C for which C ⊗_{A} H_{0}A is weakly equivalent (in W(ALG)(H_{0}A)) to an object of M(H_{0}A). Then the nerve W̄ M̃ is a geometric derived n-stack.

Proof. The proof of [Pri3, Lemma 5.23] adapts from modules to algebras, showing that A ↦ dg_{+}Alg_{A} and A ↦ sAlg_{A} are left Quillen hypersheaves, and hence that W̄ WALG is a hypersheaf by [Pri3, Proposition 5.9]. Then Proposition 2.32 ensures that W̄ WM is an étale hypersheaf.

Since π_{0}W(ALG) is a groupoid-valued functor, and M ↪ π^{0}W(ALG) is open, the map

M → π^{0}W(ALG)

is homotopy formally étale. This also implies that W̄ M̃ is a model for

A ↦ W̄ M(H_{0}A) ×_{W̄ W(ALG)(H_{0}A)}^{h} W̄ W(ALG)(A).

Now observe that this satisfies the conditions of Theorem 2.10, using Lemmas 2.24 and 2.27, Propositions 2.30 and 3.7, and Corollary 2.31. The proof of Proposition 3.7 shows that

D_{X}^{i}(W̄ M̃/R, M) = 𝔼xt_{X}^{i+1}(𝕃^{X/A}, X ⊗_{A}^{L} M).
□

We now give a simple motivating example. This is already well-known; indeed, [TV2] Proposition 2.2.6.8 generalises it to algebras for any operad.

Example 3.11 (Moduli of finite schemes). Define M : Alg_{H_{0}R} → sCat by setting M(A) ⊂ π^{0}W(ALG)(A) to consist of dg (resp. simplicial) A-algebras S for which H_{0}S is a finite flat A-module, and H_{i}S = 0 for all i > 0. Then we get

M̃ ≃ M ×_{π^{0}ALG}^{(2)} ALG.

Now, H_{0} gives a functor from M(A) to the groupoid W(FALG(A)) of finite flat A-algebras, and this is a weak equivalence in sCat, so

M ≃ W(FALG),

which is a hypersheaf by [Pri1] Remark 1.30. Thus the corollary above implies that the nerve W̄ M̃ is a geometric derived 1-stack, which we can now regard as the derived moduli stack of finite flat algebras.

We now check that this example is consistent with the flatness criteria used in [TV2] and [Lur1].

Definition 3.12. Given A ∈ dg_{+}Alg_{R} (resp. A ∈ sAlg_{R}) say that M ∈ dg_{+}Mod_{A} (resp. M ∈ sMod_{A}) is homotopy-flat over A if H_{0}M is flat over H_{0}A, and the maps

H_{n}(A) ⊗_{H_{0}A} H_{0}M → H_{n}M

are isomorphisms for all n.

Note that [TV2] Lemma 2.2.2.2 implies that this is the same as flatness in the sense of [TV2] Definition 1.2.4.1, which corresponds to the functor ⊗_{A}^{L}M : Ho(dg_{+}Mod_{A}) → Ho(dg_{+}Mod_{A}) (resp. ⊗_{A}^{L}M : Ho(sMod_{A}) → Ho(sMod_{A})) preserving homotopy pullbacks.

Lemma 3.13. A morphism f : A → B in A ∈ dg_{+}Alg_{R} (resp. sAlg_{R}) is homotopy-flat if and only if B ⊗_{A}^{L} H_{0}A is weakly equivalent to a discrete flat H_{0}A-algebra.

Proof. If f is flat, then the proof of [TV2] Lemma 2.2.2.2(2) shows that for any A-module N, we have H_{∗}(Lf^{∗}N) ≅ H_{0}B ⊗_{H_{0}A} H_{∗}N. Setting N = H_{0}A, this gives Lf^{∗}H_{0}A ≃ H_{0}B. This is flat over H_{0}A by hypothesis, completing the proof of one implication.

For the converse, assume that Lf^{∗}H_{0}A ≃ H_{0}B, flat over H_{0}A. By considering free simplicial resolutions, this implies that Lf^{∗}M ≃ H_{0}B ⊗_{H_{0}A} M for all H_{0}A-modules M. We have a spectral sequence

H_{i}(Lf^{∗}H_{j}A) ⟹ H_{i+j}(B),

giving H_{j}B ≅ H_{0}B ⊗_{H_{0}A} H_{j}A, which completes the proof. □

In particular, this means that B is homotopy-flat over A if and only if for all H_{0}Aalgebras C and all i > 0, we have H_{i}(B ⊗_{A}^{L} C) = 0.

Proposition 3.14. For T ∈ dg_{+}Alg_{R} (resp. T ∈ sAlg_{R}) flat as a graded algebra over R, the functor ALG_{T} := ALG(T ⊗ −) : dN^{♭} → sCat is 2-homogeneous and formally 2-quasi-smooth.

Proof. The proof of Proposition 3.7 carries over to this generality, noting that for any (acyclic) square-zero extension A → B, we get another (acyclic) square-zero extension T ⊗ A → T ⊗ B. □

Corollary 3.15. Take an n-truncated presheaf M : Alg_{H_{0}R} → sCat, open in π^{0}W(ALG_{T}), such that

(0) if {f_{i}: A → B_{i}}_{i∈I} is an étale cover in Alg_{H_{0}R}, then an object X ∈ π_{0}W(ALG_{T})(A) lies in the essential image of π_{0}M(A) whenever f_{i}(X) lies in the essential image of π_{0}M(B_{i}) for all i.

(1) for all finitely generated A ∈ Alg_{H_{0}R} and all X ∈ M(A), the functors

𝔼xt_{X}^{i}(𝕃^{X/A⊗T} , X ⊗_{A}^{L} −) : Mod_{A} → Ab

preserve filtered colimits for all i > 0.

(2) for all finitely generated integral domains A ∈ Alg_{H_{0}R} and all X ∈ M(A), the groups 𝔼xt_{X}^{i}(𝕃^{X/A⊗T} , X) are all finitely generated A-modules.

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

the map

M(A) → \underleftarrow{lim}^{h}M(A/𝔪^{n})

is a weak equivalence (see Remark 2.11 for a reformulation).

(4) The functor c(π_{0}M) : Alg_{π_{0}R} → Set of components of the groupoid π_{0}M preserves filtered colimits.

(5) For all A ∈ Alg_{π_{0}R} and all X ∈ M(A), the functors π_{i}\underline{Hom}_{M}(X, X) : Alg_{A} → Set preserve filtered colimits for all i ≥ 0.

Let M̃ : dN_{R}^{♭} → sCat be the full simplicial subcategory of W(ALG_{T})(A) consisting of objects C for which C ⊗_{A} H_{0}A is weakly equivalent (in W(ALG_{T})(H_{0}A)) to an object of M(H_{0}A). Then the nerve W̄ M is a geometric derived n-stack.

Proof. The proof of Corollary 3.10 carries over. □

Example 3.16 (Moduli of finite schemes over an affine base). Fix T ∈ dg_{+}Alg_{R} (resp. T ∈ sAlg_{R}) flat as a graded algebra over R, define M : Alg_{H_{0}R} → sCat by setting M(A) ⊂ π^{0}WALG(T ⊗_{R} A) to consist of dg (resp. simplicial) T ⊗_{R} A-algebras S for which H_{0}S is a finite flat A-module, and H_{i}S = 0 for all i > 0. Then we get

M̃ ≃ M ×_{π^{0}ALG_{T}}^{(2)} ALG_{T} .

Now, H_{0} gives a functor from M(A) to the groupoid W(FALG_{H_{0}T} (A)) of (H_{0}T ⊗_{H_{0}R} A)- algebras, finite and flat over A, and this is a weak equivalence in sCat, so

M ≃ WFALG_{H_{0}T} ,

which is a hypersheaf by [Pri1] Remark 1.30. Thus the corollary above implies that the nerve W̄ M̃ is a geometric derived 1-stack, which we can now regard as the derived moduli stack of finite flat algebras equipped with a morphism from T.

3.2. Derived moduli of schemes. Recall that we write dAlg_{R} for either dg_{+}Alg_{R} or sAlg_{R}.

3.2.1. Cosimplicial derived rings.

Definition 3.17. Define Δ to be the ordinal number category. For any category C, this means that C^{Δ} (resp. C^{Δ^{opp}}) is the category of cosimplicial (resp. simplicial) diagrams in C.

Definition 3.18. Define a simplicial structure on (dAlg_{A})^{Δ} as follows. For C ∈ (dAlg_{A})^{Δ}, first define h(Δ^{n}, C) ∈ (dAlg_{A})^{Δ} by h(Δ^{n}, C)^{i} := (C^{i})^{Δ_{i}^{n}}, and then define the Hom-spaces by

\underline{Hom}_{(dAlg_{A})^{Δ}}(B, C)_{n} = Hom_{(dAlg_{A})^{Δ}}(B, h(Δ^{n}, C)).

Definition 3.19. Given A ∈ (dAlg_{R})^{Δ}, define L^{n}A to represent the functor M_{n}Spec A given by

(M_{n}Spec A)(B) = {x ∈ ∏_{i=0}^{n} Hom_{dAlg_{R}}(A^{n−1}, B) : (∂^{i})^{∗}x_{j} = (∂^{j−1})^{∗}x_{i} if 0 ≤ i < j ≤ n}.

Definition 3.20. Say that a morphism A → B in (dAlg_{R})^{Δ} is a Reedy cofibration if the latching maps

A^{n} ⊗_{L^{n}A} L^{n}B → B^{n}

in dAlg_{A} are cofibrations (in the sense of Definitions 3.1 and 3.2) for all n ≥ 0.

This is equivalent to satisfying the conditions that for all acyclic surjections in C′ → C in dAlg_{R}, the map

Hom_{dAlg_{R}}(B, C′) → Hom_{dAlg_{R}}(B, C) ×_{Hom_{dAlg_{R}}(A,C)} Hom_{dAlg_{R}}(A, C′)

is a trivial fibration in 𝕊, where we set Hom_{dAlg_{R}}(A, C)_{n} := Hom_{dAlg_{R}}(A^{n}, C).

Definition 3.21. For A ∈ dAlg_{R}, define cALG_{T} (A) to be the full simplicial subcategory of the category (dAlg_{(T ⊗_{R}A)})^{Δ} on cofibrant objects (over T ⊗_{R} A).

Proposition 3.22. For T levelwise flat as a graded algebra over R, the functor cALG_{T} : dN^{♭} → sCat defined above is 2-homogeneous and formally 2-quasi-smooth.

Proof. This is a fairly direct consequence of the proof of Proposition 3.14. Again the only part which is not immediate is showing that cALG_{T} (A) → cALG_{T} (B) satisfies Condition (F2) from Definition 2.22 for all square-zero extensions A → B, and satisfies condition (TF2) whenever the extension is acyclic.

Given a square-zero extension A → B and an object S in cALG_{T} (B), Proposition 3.14 gives criteria for lifting S^{0} to S̃^{0} ∈ ALG_{T^{0}}(A). Similar criteria apply for lifting the L^{0}S⊗_{L^{0}T} T^{1}-algebra S^{1} to an L^{0}S̃⊗_{L^{0}T} T^{1}-algebra S̃^{1}, with an augmentation S̃^{1} → M^{1}(S̃) extending the augmentation S^{1} → M^{1}S. We then proceed by induction on the levels of the cosimplicial diagram. A similar argument applies to lift morphisms. □

Definition 3.23. Say that a map f : A → B in dAlg_{R} is homotopy-flat (resp. homotopysmooth, resp. homotopy-étale) if H_{0}A → H_{0}B is flat (resp. smooth, resp. étale) and the maps

H_{n}A ⊗_{H_{0}A} H_{0}B → H_{n}B

are isomorphisms for all n.

Say that a homotopy-flat map A → B is a cover if H_{0}A → H_{0}B is faithfully flat.

Definition 3.24. Say that a Reedy cofibration A → B in (dAlg_{R})^{Δ} is a homotopy-étale hypercover if the latching maps

A^{n} ⊗_{L^{n}A} L^{n}B → B^{n}

in dAlg_{A} are homotopy-étale covers for all n ≥ 0.

By [Pri3] Proposition 3.24, there is a functor Q from cAlg_{T} (A) to the category ind(cAlg_{T} (A)) of direct systems, equipped with a natural transformation id → Q. This has the properties that QC = {(QC)_{α}}_{α} is a system of Reedy cofibrant homotopy-étale hypercovers of C, and that any such hypercover of QC has a retraction.

Definition 3.25. Define cALG_{T,hyp} : dN^{♭} → sCat by setting cALG_{T,hyp}(A) to have the same objects as cAlg_{T} (A), and morphisms

\underline{Hom}_{cALG_{T,hyp}(A)}(B, C) = \underline{Hom}_{ind(cAlg_{T} (A))}(QB, QC)
= \underleftarrow{lim}_{β} \underrightarrow{lim}_{α} \underline{Hom}_{ind(cAlg_{T} (A))}((QB)_{β}, (QC)_{α})
≃ \underrightarrow{lim}_{α} \underline{Hom}_{ind(cAlg_{T} (A))}(B, (QC)_{α}).

Proposition 3.26. The functor cALG_{T,hyp} : dN^{♭} → sCat defined above is 2-homogeneous and formally 2-quasi-smooth.

Proof. This follows immediately from Proposition 3.22, making use of the fact that filtered colimits commute with finite limits. □

3.2.2. Artin hypergroupoids.

Definition 3.27. Define the category dAff_{R} of derived affine R-schemes to be opposite to dAlg_{R} (i.e. dg_{+}Alg_{R} or sAlg_{R}). Write sdAff_{R} for the category of simplicial objects in dAff_{R}; this is opposite to (dAlg_{R})^{Δ}. Say that a morphism in dAff_{R} is a fibration if the corresponding morphism in dAlg_{R} is a cofibration.

The following definition is adapted from [Pri3]. For the definitions of ∂Δ^{m} and Λ_{k}^{m}, see Definition 3.3.

Definition 3.28. Given Y_{•} ∈ sdAff_{R}, define a derived Artin (resp. Deligne–Mumford) n-hypergroupoid over Y_{•} to be a morphism X_{•} → Y_{•} in sdAff_{R}, satisfying the following conditions :

(1) for each m ≥ 0, the matching map

X_{m} → Hom_{𝕊}(∂Δ^{m}, X) ×_{Hom_{𝕊}(∂Δ^{m},Y)} Y_{m}

is a fibration (this is equivalent to saying that X → Y is opposite to a Reedy cofibration in (dAlg_{R})^{Δ});

(2) for each k, m, the partial matching map

X_{m} → Hom_{𝕊}(Λ_{k}^{m}, X) ×_{Hom_{𝕊}(Λ_{k}^{m},Y)} Y_{m}

is a homotopy-smooth (resp. homotopy-étale) cover (in the sense of Definition 3.23);

(3) for all m > n and all k, the partial matching maps

X_{m} → Hom_{𝕊}(Λ_{k}^{m}, X) ×_{Hom_{𝕊}(Λ_{k}^{m},Y)} Y_{m}

are weak equivalences.

Proposition 3.29. The simplicial category of strongly quasi-compact n-geometric derived Artin (resp. Deligne–Mumford) stacks over the hypersheafification Spec (T ⊗_{R}^{L}A)^{♯} is weakly equivalent to the opposite of the full subcategory HYP_{T}^{n} (A) of cALG_{T,hyp}(A) whose objects are dual to derived Artin (resp. Deligne–Mumford) n-hypergroupoids.

Proof. This is [Pri3] Theorem 4.15. The functor is given by sending C ∈ HYP^{n}(A) to the hypersheafification of the presheaf Spec C : dAff_{R}^{opp} → 𝕊. □

Definition 3.30. Given a flat ∞-geometric Artin stack 𝔜 over H_{0}R, and A ∈ Alg_{H_{0}R}, let F Stack_{𝔜}^{n}(A) be the simplicial category of n-geometric Artin stacks over 𝔜 ⊗_{H_{0}R} A, flat over A, and set F Stack_{𝔜}(A) = \underrightarrow{lim}_{n} F Stack_{𝔜}^{n}(A). Likewise, given an ∞-geometric derived Artin stack 𝔜 over R and A ∈ dAlg_{R}, let dStack_{𝔜}^{n}(A) be the simplicial category of n-geometric derived Artin stacks over 𝔜 ⊗_{R}^{L} A, and set dStack_{𝔜}(A) = \underrightarrow{lim}_{n} dStack_{𝔜}^{n}(A).

Note that if we set 𝔜 := Spec (T)^{♯}, then Proposition 3.29 shows that sheafification gives an ∞-equivalence HYP_{T}^{n} (A) → dStack_{𝔜}^{n}(A). Also note that the maps dStack_{𝔜}^{n}(A) → dStack_{𝔜}(A) are homotopy formally étale, as are the maps F Stack_{𝔜}^{n}(A) → dStack_{𝔜}^{n}(A), F Stack_{𝔜}(A) → dStack_{𝔜}(A) and the embedding F Sch_{𝔜}(A) → F Stack_{𝔜}(A) of algebraic spaces into ∞-geometric stacks.

Lemma 3.31. Given a strongly quasi-compact ∞-geometric derived Artin stack 𝔜 over R, the functor dStack_{𝔜} : dN_{R}^{♭} → sCat is homotopy-preserving and homotopy-homogeneous.

Proof. Choose a representative T ∈ cALG(R) with (Spec T)^{♯} ≃ 𝔜 (as in Proposition 3.29). If we write HYP_{T} (A) := \underrightarrow{lim}_{n} HYP_{T}^{n} (A), then Proposition 3.29 allows us to replace dStack_{𝔜}(A) with HYP_{T} (A). The map HYP_{T} → cALG_{T,hyp}(A) is homotopy formally étale, so it suffices to show that cALG_{T,hyp}(A) is homotopy-preserving and homotopyhomogeneous. This follows from combining Lemmas 2.24 and 2.27 with Proposition 3.26.

□

Theorem 3.32. Take an ∞-geometric derived Artin stack 𝔜, homotopy-flat over R.

Assume that we have an n-truncated presheaf M : Alg_{H_{0}R} → sCat, open in W(F Stack_{𝔜}) (in the sense of Definition 3.8), noting that homotopy-flatness ensures that Y_{A} := 𝔜 ⊗_{R}^{L} A is a flat ∞-geometric (underived) Artin stack over A for all A ∈ Alg_{H_{0}R}.

Also assume that this satisfies the following conditions

(0) if {f_{i}: Spec B_{i} → Spec A}_{i∈I} is an étale cover in affine H_{0}R-schemes, then an object X ∈ π_{0}W(F Stack_{𝔜})(A) lies in the essential image of π_{0}M(A) whenever f_{i}^{∗}(X) lies in the essential image of π_{0}M(B_{i}) for all i.

(1) for all finitely generated A ∈ Alg_{H_{0}R} and all X ∈ M(A), the functors

𝔼xt_{X}^{i}(𝕃^{X/Y_{A}}, 𝒪_{X} ⊗_{A}^{L} −) : Mod_{A} → Ab

preserve filtered colimits for all i > 1.

(2) for all finitely generated integral domains A ∈ Alg_{H_{0}R} and all X ∈ M(A), the groups 𝔼xt_{X}^{i}(𝕃^{X/Y_{A}}, 𝒪_{X}) are all finitely generated A-modules.

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

the map

M(A) → \underleftarrow{lim}^{h}M(A/𝔪^{n})

is a weak equivalence (see Remark 2.11 for a reformulation).

(4) The functor c(π_{0}M) : Alg_{H_{0}R} → Set of components of the groupoid π_{0}M preserves filtered colimits.

(5) For all A ∈ Alg_{H_{0}R} and all X ∈ M(A), the functors π_{i}\underline{Hom}_{M}(X, X) : Alg_{A} → Set preserve filtered colimits for all i ≥ 0.

Let M̃ : dN_{R}^{♭} → sCat be the full simplicial subcategory of W(cALG_{T,hyp}(A)) consisting of objects 𝔛 for which 𝔛 ⊗_{A} H_{0}A is weakly equivalent in dStack_{𝔜}(H_{0}A) to an object of M(H_{0}A). Then the nerve W̄ M̃ is a geometric derived n-stack.

Proof. As in Lemma 3.31, we may replace cALG_{T,hyp}(A) with dStack_{𝔜}(A), so

M̃(A) ≃ M(H_{0}A) ×_{W(dStack_{𝔜}(H_{0}A))}^{h} W(dStack_{𝔜}(A)),

because M is open in F Stack_{𝔜}, which is open in π^{0}dStack_{𝔜}.

The proof of Corollary 3.10 now carries over to this context, using Lemma 3.31 in place of Proposition 3.7. The only non-trivial part is the calculation

D_{X}^{i}(W̄ M̃/R, M) = 𝔼xt_{X}^{i+1}(𝕃^{X/Y_{A}}, 𝒪_{X} ⊗_{A}^{L} M),

but this follows from the calculations of [Pri3] §7.1, for instance. □

Remark 3.33. Since W̄ M̃ is a geometric derived stack, it extends canonically to a functor

W̄ M̃ : dAlg_{R} → 𝕊

where dAlg_{R} = sAlg_{R} or dg_{+}Alg_{R}. The obvious candidate for this functor is given by adapting the description above; M̃ should be the full simplicial subcategory of W(cALG_{T,hyp}) (or equivalently of W(dStack_{𝔜})) consisting of objects 𝔛 for which 𝔛⊗_{A}^{L}H_{0}A is weakly equivalent in dStack_{𝔜}(H_{0}A) to an object of M(H_{0}A).

To check that this is indeed the case, it would suffice to verify the conditions of [Pri1] Corollary 1.36 for this functor. These reduce to showing that the functor

W̄ W(dStack_{𝔜}) : dAlg_{R} → 𝕊

is a homotopy-preserving functor (which is certainly true), and nilcomplete in the sense that for all A ∈ dAlg_{R}, the map

W̄ W(dStack_{𝔜})(A) → \underleftarrow{lim}^{h}W̄ W(dStack_{𝔜})(P_{k}A)

is an equivalence, for {P_{k}A} the Postnikov tower of A. Nilcompleteness certainly seems plausible, as similar statements in [Lur1] §8 are asserted to be automatic. However, broadly similar comparisons in [TV2] involve weighty strictification theorems (such as [TV2] Theorem B.0.7 or [TV1] Theorem 4.2.1).

We now give a generalisation which will be used for moduli of pointed schemes.

Definition 3.34. Given a morphism f : ℨ → 𝔜 of ∞-geometric derived Artin stacks over R, define dStack_{𝔜}^{ℨ} : dN_{R}^{♭} → sCat as follows. First set ℨ_{A} := ℨ ⊗_{R}^{L} A, 𝔜_{A} := 𝔜 ⊗_{R}^{L} A; objects of dStack_{𝔜}^{ℨ}(A) are factorisations

ℨ_{A} ⟶^{i} X ⟶^{p} 𝔜_{A}

of f in the category of ∞-geometric derived Artin stacks over A. Morphism spaces are given by homotopy fibre products

\underline{Hom}_{dStack_{𝔜}^{ℨ}(A)}(X, X′) := \underline{Hom}_{𝔜_{A}}(X, X′) ×_{i^{∗},\underline{Hom}_{𝔜_{A}}(ℨ_{A},X′)}^{h} \{i′\}.

Theorem 3.35. Take a morphism ℨ → 𝔜 of homotopy-flat ∞-geometric derived Artin stacks over R. Assume that we have an n-truncated presheaf M : Alg_{H_{0}R} → sCat, open in the functor A ↦ W((ℨ ⊗_{R}^{L} A) ↓ F Stack_{𝔜}) (in the sense of Definition 3.8), noting that Y_{A} := 𝔜 ⊗_{R}^{L} A and Z_{A} := ℨ ⊗_{R}^{L} A are flat ∞-geometric (underived) Artin stacks over A for all A ∈ Alg_{H_{0}R}.

Also assume that these satisfy the following conditions

(0) if {f_{i}: Spec B_{i} → Spec A}_{i∈I} is an étale cover in affine H_{0}R-schemes, then an object X ∈ π_{0}W((ℨ ⊗_{R}^{L} A) ↓ F Stack_{𝔜})(A) lies in the essential image of π_{0}M(A) whenever f_{i}^{∗}(X) lies in the essential image of π_{0}M(B_{i}) for all i.

(1) for all finitely generated A ∈ Alg_{H_{0}R} and all Z_{A} ⟶^{i} X in M(A), the functors

𝔼xt_{X}^{j}(𝕃^{X/Y_{A}}, (𝒪_{X} → Ri_{∗}𝒪_{Z_{A}}) ⊗_{A}^{L} −) : Mod_{A} → Ab

preserve filtered colimits for all j > 1.

(2) for all finitely generated integral domains A ∈ Alg_{H_{0}R} and all X ∈ M(A), the groups 𝔼xt_{X}^{j}(𝕃^{X/Y_{A}}, 𝒪_{X} → Ri_{∗}𝒪_{Z_{A}}) are all finitely generated A-modules.

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

the map

M(A) → \underleftarrow{lim}^{h}M(A/𝔪^{n})

is a weak equivalence (see Remark 2.11 for a reformulation).

(4) The functor c(π_{0}M) : Alg_{π_{0}R} → Set of components of the groupoid π_{0}M preserves filtered colimits.

(5) For all A ∈ Alg_{π_{0}R} and all X ∈ M(A), the functors π_{j}\underline{Hom}_{M}(X, X) : Alg_{A} → Set preserve filtered colimits for all j ≥ 0.

Let M̃ : dN_{R}^{♭} → sCat be the full simplicial subcategory of W(dStack_{𝔜}^{ℨ}) consisting of morphisms i : Z_{A} → 𝔛 for which 𝔛 ⊗_{A} H_{0}A is weakly equivalent in dStack_{𝔜}^{ℨ}(H_{0}A) to an object of M(H_{0}A). Then the nerve W̄ M̃ is a geometric derived n-stack.

Proof. Lemma 3.31 ensures that dStack_{𝔜} is homotopy-preserving and homotopyhomogeneous, so it follows immediately that dStack_{𝔜}^{ℨ} inherits these properties. Lemma 2.30 ensures that the same is true of W(dStack_{𝔜}^{ℨ}).

In order to verify the conditions of Theorem 2.10, we now just need to establish the isomorphisms

D_{(Z_{A}⟶^{i}X)}^{j}(W̄ M̃/R, M) = 𝔼xt_{X}^{j+1}(𝕃^{X/Y_{A}}, 𝒪_{X} ⊗_{A}^{L} M → Ri_{∗}𝒪_{Z_{A}} ⊗_{R}^{L} M).

As in the proof of Theorem 3.32, we know that the loop space of T_{(Z_{A}⟶^{i}X)}(W̄ M̃/R)(M)

is the homotopy fibre of

R\underline{Hom}_{X}(𝕃^{X/Y_{A}}, 𝒪_{X} ⊗_{A}^{L} M) → R\underline{Hom}_{Z_{A}}(Li^{∗}𝕃^{X/Y_{A}}, 𝒪_{Z_{A}} ⊗_{R}^{L} M)

over 0. We can rewrite the second space as R\underline{Hom}_{X}(𝕃^{X/Y_{A}}, Ri_{∗}𝒪_{Z_{A}} ⊗_{R}^{L} M), and the result follows by noting that homotopy fibres correspond to mapping cones of complexes. □

3.3. Examples. In the examples which follow, we will always choose a functorial full subgroupoid M of the core W(F Stack_{Y}^{n}) on objects which are fibred in geometric 0-stacks (i.e. algebraic spaces) over Y ⊗_{H_{0}R} A — this has the effect that M is a bona fide groupoid, not just an ∞-groupoid. We get a derived 1-stack by taking the nerve W̄ M̃ of M̃ as constructed in Theorem 3.32.

Our first application generalises the examples of Hilbert schemes and moduli of curves in [Lur1] §8, but beware that the derived schemes featuring in [Lur1] form a larger category than derived geometric 0-stacks.

Example 3.36 (Moduli of schemes). Let 𝔜 be a homotopy-flat derived geometric 0-stack over R, so Y := 𝔜 ⊗_{R}^{L} H_{0}R is a flat algebraic space over H_{0}R. Choose a stack M equipped with a functorial fully faithful embedding of M into the core of F Sch_{Y}, with the property that M is closed under infinitesimal deformations. This ensures that W̄ M → W̄ F Sch_{Y} is open.

An example satisfying these conditions is when 𝔜 = R and M is some moduli stack of schemes with fixed dimension. Another example is to let M(A) be the set of closed subschemes of Y ⊗_{H_{0}R} A.

Then we can define M̃ as above, and we get a derived 1-stack W̄ M, with π^{0}W̄ M̃ = BM.

If M is known to be an algebraic stack, then the last two conditions of Theorem 3.32 are automatically satisfied, so for geometricity of W̄ M̃ we need only check properties of the cohomology groups

D_{f}^{i}(W̄ M̃/R, M) = 𝔼xt_{X}^{i+1}(𝕃^{X/Y ⊗_{H_{0}R}A}, 𝒪_{X} ⊗_{A}^{L} M),

for A ∈ Alg_{H_{0}R} and X ∈ M(A).

Our next application gives a generalisation of [TV2] Theorem 2.2.6.11. In particular,

[TV2] Corollary 2.2.6.14 addresses the case when X is a flat projective scheme and Y a smooth projective scheme.

Example 3.37 (Derived Hom-stacks). Another example is to fix n-geometric derived stacks 𝔛, ℨ, homotopy-flat over R. Set X := π^{0}𝔛 = 𝔛 ⊗_{R}^{L} H_{0}R, Z := π^{0}ℨ; these are flat ngeometric stacks over H_{0}R.

Now, let 𝔜 := 𝔛 ×_{Spec R} ℨ with Y := π^{0}𝔜, and set M(A) ⊂ Sch_{Y} (A) to consist of maps

U → X ×_{Spec H_{0}R} Z ×_{Spec H_{0}R} Spec A

for which the projection U → X ×_{Spec H_{0}R} Spec A is an equivalence. Then W̄ M(A) is isomorphic to the simplicial set \underline{Hom}(X ×_{Spec H_{0}R} Spec A, Z). The derived moduli stack W̄ M̃(A) is weakly equivalent to the nerve of

dStack_{𝔜}(A) ×_{dStack(A)}^{h} {ℨ}

— W̄ M̃ is the derived Hom-stack Hom(𝔛, ℨ).

By considering the arrow category of dStack(A), we see that this in turn is equivalent to \underline{Hom}(𝔛 ×_{Spec R} Spec A, ℨ). Thus a far simpler characterisation of this moduli functor is given by choosing B, C ∈ HYP_{R}^{n} with 𝔛 = (Spec B)^{♯}, 𝔛 = (Spec C)^{♯}. Then

M̃(A) ≃ \underline{Hom}_{HYP_{R}^{n}(A)}(C ⊗_{R} A, B ⊗_{R} A).

Of course, if 𝔛, ℨ are affine schemes, then we can take n = 0. If they are semi-separated schemes, we can take n = 1.

For f : X ⊗ A → Z in M(A), Hom(𝔛, ℨ) has cohomology groups

D_{f}^{i}(Hom(𝔛, ℨ)/R, M) = 𝔼xt_{X⊗A}^{i+1}(𝕃^{X⊗A/(X×Z)⊗A}, 𝒪_{X} ⊗_{H_{0}R} M)
= 𝔼xt_{X⊗A}^{i+1}(f^{∗}𝕃^{Z/H_{0}R}[1], 𝒪_{X} ⊗_{H_{0}R} M)
= 𝔼xt_{Z}^{i}(𝕃^{Z/H_{0}R}, Rf_{∗}𝒪_{X} ⊗_{H_{0}R} M).

Theorem 3.32 (or a direct application of Theorem 2.10 to \underline{Hom}(𝔛 ×_{Spec R} Spec A, ℨ)) then shows that Hom(𝔛, ℨ) is a derived n-geometric stack provided X is strongly quasicompact (i.e. quasi-compact, quasi-separated …), that Z is almost of finite presentation, and that the groups 𝔼xt_{X}^{i}(f^{∗}𝕃^{Z/H_{0}R}, 𝒪_{X} ⊗_{H_{0}R}−) have suitable finiteness properties (which are satisfied if X is proper).

When 𝔛 is a smooth projective scheme and G = GL_{r}, the following appears as [TV2] Corollary 2.2.6.15:

Example 3.38 (Moduli of torsors). A special case of Example 3.37 is to take a smooth group scheme G over R, and to set ℨ to be the classifying stack BG. Then Hom(𝔛, BG) is the derived stack of G-torsors on 𝔛 — when G = GL_{r}, this gives derived moduli of vector bundles, and when G = SL_{r} it gives derived moduli of vector bundles with trivial determinant.

To calculate the cohomology groups D_{f}^{i}(M̃/R, M) for f : 𝔛 × Spec A → BG, write e : Spec R → G for the unit. Then e^{∗}Ω(G/R) has a G-action, so can be regarded as a quasi-coherent sheaf 𝒞 on BG. We then have 𝕃^{BG/R} ≃ 𝒞 [1], so

D_{f}^{i}(Hom(𝔛, BG)/R, M) = Ext_{X⊗A}^{i+1}(f^{∗}𝒞 , 𝒪_{X} ⊗_{R} M).

If f corresponds to a G-torsor P, then note that f^{∗}𝒞 = P ×_{G} (e^{∗}Ω(G/R)).

If G = GL(V), then 𝒞 ≅ End(V) with the adjoint action, and for f : 𝔛 × Spec A → BGL_{n} corresponding to a vector bundle ℰ, we have f^{∗}𝒞 ≅ ℰnd(ℰ), so

D_{f}^{i}(Hom(𝔛, BG)/R, M) = Ext_{X⊗A}^{i+1}(ℰnd(ℰ), 𝒪_{X} ⊗_{R} M)
= Ext_{X⊗A}^{i+1}(ℰ , ℰ ⊗_{R} M).

Example 3.39 (Moduli of polarised schemes). For an example combining aspects of all the problems so far, consider moduli of polarised schemes. In Theorem 3.32, take 𝔜 = B𝔾_{m}, and let M(A) be some groupoid of flat schemes X over A, equipped with polarisations f : X → B𝔾_{m}, with morphisms required to respect the polarisations. Then we have M̃ as defined at the the beginning of the section. Again, W̄ M̃ will be a geometric derived 1-stack whenever M is an algebraic 1-stack and the cohomology groups D^{i}(W̄ M̃/R, −) satisfy the requisite finiteness conditions.

Explicitly,

D_{X}^{i}(W̄ M̃/R, M) = 𝔼xt_{X}^{i+1}(𝕃^{X/B𝔾_{m}}, 𝒪_{X} ⊗_{H_{0}R} M).

The calculations of the previous example show that f^{∗}𝕃^{B𝔾_{m}} = 𝒪_{X}[1], so the cotangent complex 𝕃^{X/B𝔾_{m}} is an extension of 𝒪_{X} by 𝕃^{X/H_{0}R}. This gives a long exact sequence

… H^{i+1}(X, 𝒪_{X} ⊗_{H_{0}R} M) → D_{X}^{i}(W̄ M̃/R, M) → 𝔼xt_{X}^{i+1}(𝕃^{X/H_{0}R}, 𝒪_{X} ⊗_{H_{0}R} M) → …,

so properness of X gives finiteness.

Remark 3.40. When considering moduli of polarised varieties, it is common only to consider the polarisation up to equivalence, so (X, L^{m}) ∼ (X, L^{n}). The corresponding derived moduli stack can be constructed by taking the filtered colimit

\underrightarrow{lim}_{n∈ℕ^{×}} W̄ M̃,

for M̃ as in Example 3.39, with the transition maps M̃ → M̃ (corresponding to m → mn in ℕ^{×}) coming from the multiplication map [n] : 𝔾_{m} → 𝔾_{m}.

Alternatively, we could simply replace 𝔾_{m} with the (non-quasi-separated) algebraic group space \underrightarrow{lim}_{n∈ℕ^{×}} 𝔾_{m} = 𝔾_{m}/μ_{∞}, where μ_{∞} = ⋃_{n∈ℕ^{×}} μ_{n}.

Example 3.41 (Moduli of group schemes). Given a smooth group scheme G over A, the nerve BG is a 1-truncated geometric Artin stack over A. However, beware that \underline{Hom}_{A}(BG, BG′) is the nerve of the groupoid [Hom_{GpSch}(G, G′)/G′(A)], where Hom_{Gp} denotes group scheme homomorphisms and G′ acts by conjugation. If M(A) is the nerve of some open 2-subcategory of the 2-category of group schemes over A, then B gives a map M → F Stack(A), so we may apply Theorem 3.32 to give a derived moduli functor M̃ whenever M satisfies the conditions of that theorem.

Since we usually wish to kill the 2-morphisms (given by inner automorphisms), an alternative approach is to apply Theorem 3.35, taking ℨ = 𝔜 = Spec R. Making use of the canonical basepoint i : Spec A → BG, we then have

\underline{Hom}_{A}(BG, BG′) ×_{i^{∗},\underline{Hom}_{A}(Spec A,BG′)} \{i′\} ≃ Hom_{GpSch}(G, G′),

so the nerve functor allows us to regard smooth group schemes over A as an open subcategory of dStack_{Spec R}^{Spec R}(A).

As in Example 3.38, 𝕃^{BG/A} ≃ 𝒞 [1], where 𝒞 = e^{∗}Ω(G/A) with its canonical G-action. Thus

D_{G}^{i}(W̄ M̃/R, M) ≅ Ext_{BG}^{i+2}(𝒞 , 𝒪_{BG} ⊗_{A}^{L} M)

for i ≥ 0 (or, in the unpointed case, for all i). For i < −1, the groups D_{G}^{i}(W̄ M̃/R, M) are 0 in the pointed case, with i = −1 giving an exact sequence

0 → Hom_{BG}(𝒞 , 𝒪_{BG} ⊗_{A}^{L} M) → Hom_{A}(e^{∗}𝒞 , M) →
D_{G}^{−1}(W̄ M̃/R, M) → Ext_{BG}^{1}(𝒞 , 𝒪_{BG} ⊗_{A}^{L} M) → 0.

Remark 3.42. Given a flat finitely presented group scheme G over A, [Art] Theorem 6.1 implies that BG is a 1-truncated geometric Artin stack over A. The analysis of the example above then carries over, subject to replacing Ω_{G} with the complex χ_{G} defined in [Ill2] §2.5.1. The same substitution will also work in the following example, with [Toë2] Theorem 0.1 ensuring that K(G, n) be an n-truncated geometric Artin stack over A.

Example 3.43 (Moduli of commutative group schemes). For smooth commutative group schemes G over A, we can no longer study moduli via BG, since this has non-commutative deformations. However, for all n ≥ 2, the construction K(G, n) gives an n-truncated geometric Artin stack over A, whose deformations all correspond to commutative group schemes. Rather than fixing one value of n, the most natural approach is to use them all simultaneously.

The loop space functor (X, x) ↦ X^{S^{1}} ×_{X} {x} gives a endofunctor Ω of dStack_{Spec R}^{Spec R}. This maps K(G, n) to K(G, n − 1) (up to weak equivalence). We define dSpStack_{R}(A) to be the homotopy inverse limit of the diagram

… ⟶^{Ω} dStack_{Spec R}^{Spec R} ⟶^{Ω} dStack_{Spec R}^{Spec R}(A);

this is the ∞-category of spectral derived stacks. Defining F SpStack_{R}(A) from F Stack_{R}^{R}(A) analogously, we have a functor from the category SGpSch(A) smooth commutative group schemes G over A to F SpStack_{R}(A), given by G ↦ {K(G, n)}_{n}. Explicitly, K(G, 0) = G, and we form K(G, n) from the group n − 1-stack K(G, n − 1) as the nerve K(G, n) := W̄ K(G, n − 1). Then SGpSch is an open subcategory of F SpStack_{R}(A), each of the projections K(−, n) : SGpSch → dStack_{Spec R}^{Spec R}(A) also being open for n ≥ 2.

We can now adapt Theorem 3.35, replacing dStack_{𝔜}^{ℨ} with dSpStack, noting that the latter is homotopy-preserving and homotopy-homogeneous. In order to determine which open subcategories M of SGpSch yield derived geometric stacks

M̃(A) := M(H_{0}A) ×_{F SpStack(H_{0}A)}^{h} dSpStack(A),

we need to calculate the cohomology groups D_{G}^{i}(W̄ M̃/R, M). The theorem implies that these groups are given by cohomology of

\underleftarrow{lim}_{n} RHOM_{K(G,n)}(𝕃^{K(G,n)/A}, 𝒪_{K(G,n)} ⊗_{A}^{L} M → Ri_{∗}M)[1]

= \underleftarrow{lim}_{n} RHOM_{K(G,n)}(𝒪_{K(G,n)} ⊗_{A} e^{∗}Ω(G/A)[n + 1], 𝒪_{K(G,n)} ⊗_{A}^{L} M → Ri_{∗}M)
= RHOM_{A}(e^{∗}Ω(G/A), \underleftarrow{lim}_{n} RΓ̃(K(G, n), 𝒪_{K(G,n)}) ⊗_{A}^{L} M)[n + 1],

where RΓ̃ = RΓ/H^{0}. We can rewrite this as

D_{G}^{i}(W̄ M̃/R, M) = 𝔼xt_{ℤ}^{i+1}(\underrightarrow{lim}_{n} NℤK(G, n)[n], Hom_{A}(e^{∗}Ω(G/A), M)),

where N denotes Dold–Kan normalisation, and 𝔼xt is taken in the category of abelian fppf sheaves over Spec A. This ties in with the approach of [Ill1] §VII.4 (as sketched in [Ill2] Theorem 5.3).

Now, [Car] Theorem 9.7 (adapted to abelian sheaves as in [Bre] Theorem 3) shows that

\underrightarrow{lim}_{n} ℋ_{n}(NℤK(G, n)) = G,

and that for fixed q, the groups ℋ_{n+q}(NℤK(G, n)) are constant for n > q, and can be expressed in terms of G/pG and _{p}G for primes p. [Explicit descriptions up to q=5 are given at the end of [Bre] §1.] In particular, ℋ_{n+1}(\underrightarrow{lim}_{n} NℤK(G, n)) = 0 for n > 1, so

D_{G}^{i}(W̄ M̃/R, M) = Ext_{ℤ}^{i+1}(G, Hom_{A}(e^{∗}Ω(G/A), M))

for all i ≤ 0.

Remark 3.44. This approach means that we can think of (derived) commutative group schemes as functors from simplicial (or dg) algebras to spectra. By contrast, derived stacks are functors from simplicial algebras to simplicial sets. This might help to explain why Shimura varieties are one of the rare examples of moduli spaces which extend naturally to brave new stacks (i.e. functors from ring spectra to simplicial sets), as observed in [Lur2].

Example 3.45 (Moduli of abelian schemes). A special case of the previous example is when M is an open subcategory of the category of abelian schemes. The description of the cohomology groups in this case simplifies greatly. The key observation is that

RΓ(G, 𝒪_{G}) ≃ Symm_{A}(H^{1}(G, 𝒪_{G})[−1]),

with H^{1}(G, 𝒪_{G}) projective over A.

If we have H^{1}(G, 𝒪_{G}) ≅ A^{g}, then the equivalence becomes

RΓ(G, 𝒪_{G}) ≃ C^{•}(K(ℤ^{g}, 1), ℤ) ⊗_{ℤ}^{L} A.

Thus

\underleftarrow{lim}_{n} RΓ(K(G, n), 𝒪_{K(G,n)})[n] ≃ \underleftarrow{lim}_{n} C^{•}(K(ℤ^{g}, n + 1), ℤ)[n] ⊗_{ℤ}^{L} A
= C_{st}^{•−1}(Hℤ^{g}, ℤ) ⊗_{ℤ}^{L} A
= C_{st}^{•−1}(Hℤ, ℤ) ⊗_{ℤ}^{L} A^{g},

the complex of stable cochains of the integral Eilenberg–Mac Lane spectrum Hℤ.

In general, \underleftarrow{lim}_{n} RΓ(K(G, n), 𝒪_{K(G,n)})[n] can be rewritten A-linearly in terms of the cocommutative A-coalgebra RΓ(G, 𝒪_{G}) (a model for which is the normalised Čech complex with Eilenberg-Zilber comultiplication). This gives us a morphism

C_{st}^{•}(Hℤ, ℤ) ⊗_{ℤ}^{L} \underleftarrow{lim}_{n} RΓ(K(G, n), 𝒪_{K(G,n)})[n] → \underleftarrow{lim}_{n} RΓ(K(G, n), 𝒪_{K(G,n)})[n],

inducing a map

C_{st}^{•−1}(Hℤ, ℤ) ⊗_{ℤ}^{L} H^{1}(G, 𝒪_{G}) → \underleftarrow{lim}_{n} RΓ(K(G, n), 𝒪_{K(G,n)})[n].

We know that this is an isomorphism locally on A (as H^{1}(G, 𝒪_{G}) is locally free), so it is an isomorphism in general.

Thus

D_{G}^{i}(W̄ M̃/R, M) ≅ Hom_{A}(e^{∗}Ω(G/A), H^{1}(G, 𝒪_{G})) ⊗_{A} H_{st}^{i}(Hℤ, M).

These stable cohomology groups can be calculated by applying the universal coefficient theorem to [Car] Theorem 7; alternatively we could apply the universal coefficient theorem for Tor to the integral Steenrod algebra H_{st}^{i}(Hℤ, ℤ). The important features are that H_{st}^{<0}(Hℤ, M) = 0, while H_{st}^{0}(Hℤ, M) = M and H_{st}^{>0}(Hℤ, M) is an invariant of the torsion and cotorsion in M. In particular, if M is a ℚ-module, then H_{st}^{>0}(Hℤ, M) = 0. We always have H_{st}^{1}(Hℤ, M) = 0, which is why moduli of abelian varieties are unobstructed.

Beware that the example above only gives a derived Artin stack when a suitable moduli stack of abelian varieties exists.

Remark 3.46. Most approaches to moduli of abelian varieties involve polarisations, and we now sketch a possible approach. A (principal) polarisation is a symmetric bilinear map G × G → 𝔾_{m}, such that the restriction to the diagonal G → G × G gives an ample line bundle on G, and the the induced map G → G^{∨} is an isogeny (resp. an isomorphism).

For an associated derived moduli problem, we could take spectral derived geometric stacks 𝔛 over A equipped with morphisms

α : [𝔛 ∧ 𝔛/S_{2}] → Ω^{−1}H(𝔾_{m} ⊗ A),

where Ω^{−1}Hπ = {K(π, n + 1)}_{n}, ∧ is the smash product on spectra, and S_{2} acts by transposition.

Since isogenies and isomorphisms are open in the space of group scheme homomorphisms, we could then take M(A) to be the category of polarised (or principally polarised) abelian schemes over A. For G ∈ M(A), the associated spectral derived geometric stack 𝔛 would then be HG, with α given by the composition

[𝔛 ∧ 𝔛/S_{2}] → [H(G ⊗_{ℤ} G)/S_{2}] → Ω^{−1}H(𝔾_{m} ⊗ A),

the right-hand map coming from the polarisation.

In order to verify that this defines a geometric derived stack, the difficult part would be to calculate the cohomology groups in order to check that they satisfy the requisite finiteness conditions.

4. Derived moduli of quasi-coherent sheaves

4.1. Derived quasi-coherent sheaves. Given a simplicial ring A_{•}, the simplicial normalisation functor N^{s} of Definition 2.1 induces an equivalence between the categories sMod(A) of simplicial A_{•}-modules, and dg_{+}Mod(N^{s}A) of N^{s}A-modules in non-negatively graded chain complexes, where the Eilenberg–Zilber shuffle product makes N^{s}A into a graded-commutative algebra. As observed in [TV2] §2.2.1, this extends to give a weak equivalence between the ∞-category of stable A-modules, and the ∞-category dgMod(N^{s}A) of N^{s}A-modules in ℤ-graded chain complexes, and hence an equivalence between the corresponding homotopy categories. More precisely, the ∞-structure on dgMod(N^{s}A) comes from a model structure:

Definition 4.1. In dgMod(N^{s}A), fibrations are surjections, weak equivalences are quasiisomorphisms, and cofibrations satisfy LLP with respect to trivial fibrations. Explicitly, f : L → M is a a cofibration if it is a retract of a transfinite composition of monomorphisms f′ : L′ → M′ with coker f′ ≅ (N^{s}A)[n] for some n.

In particular, if A is discrete and f is a monomorphism with coker f bounded below and levelwise projective, then f is a cofibration.

Definition 4.2. Given L, M ∈ dgMod(N^{s}A), we can then define HOM_{A}(L, M) to be the chain complex given in level n by A-linear maps L → M[n] of graded objects. The differential on HOM_{A}(L, M) is given by d(f) = d ∘ f ± f ∘ d. The simplicial structure on dgMod(N^{s}A) is then given by setting

\underline{Hom}_{A}(L, M) := (N^{s})^{−1}(τ_{≥0}HOM_{A}(L, M)),

where τ_{≥0} is good truncation.

The associated ∞-category is given by the full simplicial subcategory of dgMod(N^{s}A) on cofibrant objects. Equivalently, we have derived function complexes on dgMod(N^{s}A) given by R\underline{Hom}(L, M) := \underline{Hom}(L̃, M), for L̃ → L a cofibrant replacement.

Definition 4.3. An object X_{•} in sdAff_{R} is said to be a homotopy derived Artin nhypergroupoid if it admits a map X_{•} → X̃_{•} to a derived Artin n-hypergroupoid with the property that the level maps X_{n} → X̃_{n} are all weak equivalences in dAff_{R}.

Definition 4.4. If X is a homotopy derived Artin l-hypergroupoid, define the simplicial cosimplicial algebra O(X)_{•}^{•} by O(X)_{m}^{n} = Γ(X_{n}^{m}, 𝒪_{X_{n}^{m}}). Define the category dgMod(X) to have objects M consisting of complexes M_{•}^{n} ∈ dgMod(N^{s}O(X)_{•}^{n}) for all n, together with morphisms

∂^{i} : M_{•}^{n} ⊗_{N^{s}O(X)_{•}^{n},∂_{i}^{∗}} N^{s}O(X)_{•}^{n+1} → M_{•}^{n+1}
σ^{i} : M_{•}^{n} ⊗_{N^{s}O(X)_{•}^{n},σ_{i}^{∗}} N^{s}O(X)_{•}^{n−1} → M_{•}^{n−1},

satisfying the usual cosimplicial identities.

Definition 4.5. A morphism f : M → N in dgMod(X) is said to be a weak equivalence if each map f^{n} : M^{n} → N^{n} is a weak equivalence in dgMod(N^{s}O(X)_{•}^{n}). Say that f is a fibration if each map f^{n} : M^{n} → N^{n} is surjective.

Finally, f is said to be a cofibration if it satisfies the LLP with respect to trivial fibrations. For an explicit characterisation, first form the nth latching object L^{n}M ∈ dgMod(N^{s}O(X)_{•}^{n}) as the cokernel of

(α − β) : ⨁_{i=0}^{n} ⨁_{j=0}^{i−1} ∂_{i}^{∗}∂_{j}^{∗}M^{n−2} → ⨁_{i=0}^{n−1} ∂_{i}^{∗}M^{n−1},

where for x ∈ ∂_{i}^{∗}∂_{j}^{∗}M^{n−2} = ∂_{j}^{∗}∂_{i−1}^{∗}M^{n−2}, we define α(x) = ∂_{i}^{∗}(∂^{j})x ∈ ∂_{i}^{∗}M^{n−1}, and β(x) = ∂_{j}^{∗}(∂^{i−1})x ∈ ∂_{j}^{∗}M^{n−1}. Then f is a cofibration if and only if the maps

L^{n}N ⊕_{L^{m}M} M^{n} → N^{n}

are cofibrations in dgMod(N^{s}O(X)_{•}^{n}) for all n.

We can then form the simplicial structure by analogy with dgMod(N^{s}A): define the chain complex

HOM_{X}(M, N)_{n} := Hom_{O(X)}(M, N[n]),

where Hom respects cosimplicial, but not chain, structures. Then \underline{Hom}(M, N) := (N^{s})^{−1}(τ_{≥0}HOM_{X}(M, N)), with R\underline{Hom} given by taking a cofibrant replacement for M.

Definition 4.6. Define dgMod(X)_{cart} ⊂ dgMod(X) to be the full subcategory consisting of those M for which the morphisms ∂^{i} are all weak equivalences. Let Ho(dgMod(X))_{cart}) be the category obtained by localising at weak equivalences. Note that dgMod(X)_{cart} also inherits a simplicial structure from dgMod(X). Objects of dgMod(X)_{cart} are called derived quasi-coherent sheaves on X.

Proposition 4.7. Fix a homotopy derived Artin n-hypergroupoid X. The simplicial category of derived quasi-coherent sheaves on X is equivalent to the ∞-category dgMod_{cart}(X^{♯}) of homotopy-Cartesian modules on X^{♯} (as in [TV2] Definition 1.2.12.1) or equivalently, of quasi-coherent complexes on X^{♯} (in the sense of [Lur1] §5.2).

Proof. This is [Pri3] Corollary 5.12. □

In particular, this means that dgMod(X)_{cart} is essentially independent of the atlas X chosen for X^{♯}.

Definition 4.8. Define an Artin (resp. Deligne–Mumford) n-hypergroupoid over a ring R to be an object X_{•} ∈ sAff_{R}, such that the partial matching maps

X_{m} → M_{Λ_{k}^{m}}X

are smooth (resp. étale) covers for all k, m, and isomorphisms for all m > n and all k.

In particular, note that every n-hypergroupoid is a homotopy derived n-hypergroupoid. Specialising to underived n-stacks, Proposition 3.29 shows that every strongly quasicompact n-geometric Artin (resp. Deligne–Mumford) stack is of the form X^{♯}, for some Artin (resp. Deligne–Mumford) n-hypergroupoid X.

For our purposes, the main applications of derived quasi-coherent sheaves stem from the following:

Proposition 4.9. (1) Take an Artin n-hypergroupoid X. Then the category of quasicoherent sheaves on the geometric stack X^{♯} is equivalent to the full subcategory of Ho(dgMod(X)_{cart}) consisting of complexes concentrated in chain degree 0.

(2) If Y is a semi-separated quasi-compact scheme, take an open affine cover {U_{i}}_{i=1}^{n}. Let U = ∐_{i} U_{i}, and set X to be the Deligne-Mumford 1-hypergroupoid given by

X_{n} = \overbrace{U ×_{Y} U ×_{Y} … ×_{Y} U}^{n+1} .

Then Ho(dgMod(X)_{cart}) is equivalent to the derived category of complexes of quasicoherent sheaves on Y.

Proof. The first statement is [Pri3] Corollary 5.12. Once we note that X^{♯} = Y, the second statement follows from the observation in the introduction of [Hüt] that the proof of [Hüt] Theorem 4.5.1 works for any quasi-compact scheme equipped with a finite semi-separating affine covering. □

4.2. Moduli.

Definition 4.10. Take A ∈ dg_{+}Alg_{R} (resp. A ∈ sAlg_{R}), and a homotopy derived Artin n-hypergroupoid X over R, chosen so that each X_{n} is levelwise flat over R (which holds for instance if X is a derived Artin n-hypergroupoid). Then observe that X ⊗_{R} A is a homotopy derived Artin n-hypergroupoid over A, and define

dCART_{X}(A)

to be the full simplicial subcategory of the category of dgMod(X ⊗_{R} A)_{cart} on cofibrant objects.

Proposition 4.11. The functor dCART_{X} : dN^{♭} → sCat defined above is 2-homogeneous and formally 2-quasi-smooth.

Proof. The proof of Proposition 3.22 carries over to this context, with cofibrancy once again ensuring the existence of suitable lifts. □

Theorem 4.12. Take a strongly quasi-compact m-geometric derived Artin stack 𝔜 over R, and choose a homotopy derived Artin n-hypergroupoid 𝔜̌ with 𝔜̌^{♯} = 𝔜 and each 𝔜̌_{n} flat over R.

Assume that we have an n-truncated presheaf M : Alg_{H_{0}R} → sCat, open in the functor A ↦ W(dgMod_{cart}(𝔜 ⊗_{R}^{L} A)).

Also assume that this satisfies the following conditions:

(0) if {f_{i}: Spec B_{i} → Spec A}_{i∈I} is an étale cover in affine H_{0}R-schemes, then an object ℰ ∈ π_{0}W(dgMod_{cart}(𝔜⊗_{R}^{L}A)) lies in the essential image of π_{0}M(A) whenever f_{i}^{∗}ℰ lies in the essential image of π_{0}M(B_{i}) for all i.

(1) for all finitely generated A ∈ Alg_{H_{0}R} and all ℰ ∈ M(A), the functors

𝔼xt_{𝔜⊗_{R}^{L}A}^{i}(ℰ , ℰ ⊗_{A}^{L} −) : Mod_{A} → Ab

preserve filtered colimits for all i ≠ 1.

(2) for all finitely generated integral domains A ∈ Alg_{H_{0}R} and all ℰ ∈ M(A), the groups 𝔼xt_{𝔜⊗_{R}^{L}A}^{i}(ℰ , ℰ) are all finitely generated A-modules.

(3) The functor c(π_{0}M) : Alg_{π_{0}R} → Set of components of the groupoid π_{0}M preserves filtered colimits.

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

the map

c(π_{0}M(A)) → \underleftarrow{lim}_{r} c(π_{0}M(A/𝔪^{r}))

is an isomorphism, as are the maps

𝔼xt_{𝔜⊗_{R}^{L}A}^{i}(ℰ , ℰ) → \underleftarrow{lim}_{r} 𝔼xt_{𝔜⊗_{R}^{L}A}^{i}(ℰ , ℰ /𝔪^{r})

for all ℰ ∈ M(A) and all i ≤ 0.

Let M̃ : dN_{R}^{♭} → sCat be the full simplicial subcategory of W(dCART𝔜̌(A)) consisting of objects ℱ for which ℱ ⊗_{A} H_{0}A is weakly equivalent in dgMod_{cart}(𝔜 ⊗_{R}^{L} H_{0}A) to an object of M(H_{0}A). Then the nerve W̄ M is an geometric derived n-stack.

Proof. First note that Proposition 4.7 allows us to replace dCART𝔜̌(A) with dgMod_{cart}(𝔜 ⊗_{R}^{L} A), so

M̃(A) ≃ M(H_{0}A) ×_{W(dgMod_{cart}(𝔜⊗_{R}^{L}H_{0}A))}^{h} W(dgMod_{cart}(𝔜 ⊗_{R}^{L} A)),

because M is open in π^{0}W(dgMod_{cart}(𝔜 ⊗_{R}^{L} −)).

Now, we have

D_{ℰ}^{i} (W̄ M̃/R, M) = 𝔼xt_{𝔜⊗_{R}^{L}A}^{i+1}(ℰ , ℰ ⊗_{A}^{L} M),

for instance from the calculations of [Pri3] §7, where 𝔼xt groups are also characterised as

𝔼xt_{𝔜⊗_{R}^{L}A}^{i}(ℰ , ℱ) ≅ H_{−i}HOM_{X⊗_{R}A}(ℰ , ℱ),

provided ℰ is chosen to be cofibrant.

Note that π_{i}\underline{Hom}_{M(B)}(ℰ , ℰ) = 𝔼xt^{−i}(ℰ , ℰ ⊗_{A}^{L} B) for i ≥ 1, and that π_{0}\underline{Hom}_{M(B)}(ℰ , ℰ) consists of invertible elements in 𝔼xt^{0}(ℰ , ℰ ⊗_{A}^{L} B). Thus Condition (1) ensures that for all A ∈ Alg_{π_{0}R} and all ℰ ∈ M(A), the functors π_{i}\underline{Hom}_{M}(ℰ , ℰ) : Alg_{A} → Set preserve filtered colimits for all i ≥ 0.

Next, we need to establish that for all complete discrete local Noetherian H_{0}R-algebras A, with maximal ideal 𝔪, the map

M(A) → \underleftarrow{lim}^{h}M(A/𝔪^{n})

is a weak equivalence.

Observe that

Aut_{π_{0}M(A)}(ℰ) = {(g, h) ∈ 𝔼xt_{𝔜⊗_{R}^{L}A}^{0}(ℰ , ℰ) × 𝔼xt_{𝔜⊗_{R}^{L}A}^{0}(ℰ , ℰ) : gh = hg = id},

so Condition 4 implies that Aut_{π_{0}M(A)}(ℰ) ≅ \underleftarrow{lim}_{r} Aut_{π_{0}M(A/𝔪^{r})}(ℰ /𝔪^{r}). Since

π_{1}(W̄ M(A), ℰ)) = Aut_{π_{0}M(A)}(ℰ)
π_{i}(W̄ M(A), ℰ)) = 𝔼xt_{𝔜⊗_{R}^{L}A}^{1−i}(ℰ , ℰ)

for i ≥ 2, by Remark 2.11 it suffices to show that each system

{Im (π_{i}(W̄ M(A/𝔪^{s}), x = ℰ) → π_{i}(W̄ M(A/𝔪^{r}), ℰ))}_{s≥r}

satisfies the Mittag-Leffler condition for i ≥ 1 and ℰ ∈ M(A), and that Condition (4) holds.

Since 𝔼xt_{𝔜⊗_{R}^{L}A}^{1−i}(ℰ , ℰ /𝔪^{r}) is a finitely generated A/𝔪^{r}-module, it is Artinian, so the Mittag-Leffler condition is automatically satisfied for i ≥ 2. For i = 1, we can study the obstruction maps of Proposition 2.8 to deduce that

Aut_{π_{0}M(A/𝔪^{r})}(ℰ) = 𝔼xt_{𝔜⊗_{R}^{L}A}^{0}(ℰ , ℰ /𝔪^{r}) ×_{𝔼xt_{𝔜⊗^{L}_{R}A}^{0}(ℰ ,ℰ /𝔪)} Aut_{π_{0}M(A/𝔪)}(ℰ /𝔪).

Since 𝔼xt_{𝔜⊗_{R}^{L}A}^{0}(ℰ , ℰ /𝔪^{r}) is Artinian, we deduce that the Mittag-Leffler condition holds for i = 1, so we have satisfied the conditions of Remark 2.11.

The proof of Corollary 3.10 now carries over to this context, using Proposition 4.11 in place of Proposition 3.7. For the hypersheaf condition, observe that the argument of Corollary 3.10 shows that W̄ W applied to the full simplicial subcategory of dgMod(X⊗_{R}−) on cofibrant objects gives an étale hypersheaf, so W̄ WdCART_{X} and hence W̄ WM are hypersheaves by Proposition 2.32. □

Remarks 4.13. If R = H_{0}R, we can let 𝔜 = Y be a semi-separated quasi-compact scheme, then form the Čech nerve Y̌ as in part 2 of Proposition 4.9, giving an explicit description of M̃.

In order to satisfy the finiteness hypotheses for 𝔼xt^{∗}, a likely candidate for a derived geometric n-stack would be to let M(A) consist of perfect complexes ℰ_{•} of 𝒪_{Y} ⊗_{R}^{L} Amodules of Tor-dimension (n − 1) over A, with ℋ_{i}(ℰ) quasi-coherent of proper support on Y.

Definition 4.14. Given an m-geometric (underived) Artin stack Y over H_{0}R, and A ∈ Alg_{H_{0}R}, let F Mod_{Y} (A) be the category of quasi-coherent sheaves (not complexes) on Y ⊗_{H_{0}R} A, flat over A.

Example 4.15. For 𝔜 as in Theorem 4.12, set Y := π^{0}𝔜 (an m-geometric (underived) Artin stack over H_{0}R). Of course, if R = H_{0}R, then we will have Y = 𝔜. Choose a stack M equipped with functorial fully faithful embeddings of M into the core of F Mod_{Y}, with the property that M is closed under infinitesimal deformations. This ensures that W̄ M → W̄ F Mod_{Y} is open.

Note that flatness ensures that for any morphism A → B in Alg_{H_{0}R} and ℱ ∈ F Mod_{Y} (A), the complex ℱ ⊗_{A}^{L} B is weakly equivalent to ℱ ⊗_{A} B, so F Mod_{Y} (A) embeds into Mod_{cart}(𝔜⊗_{R}^{L} A)). In fact, we can characterise F Mod_{Y} (A) as being weakly equivalent to the subcategory of Mod_{cart}(𝔜 ⊗_{R}^{L} A) consisting of complexes ℱ for which the homology sheaves ℋ_{i}(ℱ ⊗_{A}^{L}B) = 0 for all i ≠ 0 and all B ∈ Alg_{A}. Thus F Mod → π^{0}Mod_{cart}(𝔜⊗_{R}^{L}−) is an open embedding.

Provided that the objects of M satisfy the conditions of Theorem 4.12, this gives us a geometric derived 1-stack of quasi-coherent sheaves of 𝔜. An example for which the conditions are satisfied arises when Y is a projective scheme over R and M(A) consists of coherent sheaves on Y ⊗_{R} A, flat over A.

4.2.1. Comparisons. If we wish to study derived moduli of rank r vector bundles on a homotopy-flat derived geometric ∞-stack 𝔛 over R, we now have two possible approaches, which we now show are consistent. Let X = 𝔛 ⊗_{R} H_{0}R, and define M : Alg_{H_{0}R} → Gpd by letting M(A) be the groupoid of rank r vector bundles on X ⊗_{H_{0}R} A.

Example 3.38 gives one extension Hom(𝔛, BGL_{r}) : dN_{R}^{♭} → 𝕊 of BM. Theorem 4.12 gives another candidate W̄ M̃ for an extension, where the objects of M̃ : dN_{R}^{♭} → sCat are quasi-coherent complexes ℰ on 𝔛 ⊗_{R}^{L} A for which ℰ ⊗_{A}^{L} H_{0}A is quasi-isomorphic to a rank r vector bundle on 𝔛 ⊗_{R}^{L} H_{0}A.

The following lemma is essentially the same as [TV2] Proposition 1.3.7.10 and its subsequent remarks.

Lemma 4.16. The derived geometric 1-stacks Hom(𝔛, BGL_{r}) and W̄ M̃ defined above are weakly equivalent.

Proof. The argument of Example 3.38 gives a morphism from Hom(𝔛, BGL_{n}) to W̄ W(T ), where T (A) is the full simplicial subcategory of dStack_{𝔛×BGL_{r}}(A) on objects ℨ for which the projection f_{1} : ℨ → 𝔛 ⊗_{R}^{L} A is a weak equivalence. We can then define a morphism W(T ) → M̃ as follows: let 𝒰 be the universal rank r vector bundle on BGL_{r}, and send

(ℨ ⟶^{(f_{1},f_{2})} (𝔛 × BGL_{r}) ⊗_{R}^{L} A)

in T (A) to the quasi-coherent complex Rf_{1∗}f^{∗}𝒰 on 𝔛.

Thus we have constructed a map Hom(𝔛, BGL_{n}) → W̄ M̃. It is immediate that this gives an equivalence on π^{0}, and straightforward to check that it gives isomorphisms on cohomology groups D^{i}, so it is a weak equivalence, by [Pri1] Proposition 1.38. □

Remark 4.17. We could adapt Theorem 4.12 to moduli of polarised varieties, by considering pairs (T, M), for T ∈ cALG(A) and M ∈ dCART_{T} (A). The resulting derived stack would parametrise pairs (𝔛, ℰ), for 𝔛 a derived geometric m-stack and ℰ a quasi-coherent complex on 𝔛. The argument of Lemma 4.16 then adapts to show that an open substack of the resulting moduli stack is equivalent to the derived stack from Example 3.39.

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