aa r X i v : . [ m a t h . AG ] F e b REPRESENTABILITY OF DERIVED STACKS
J.P.PRIDHAM
Abstract.
Lurie’s representability theorem gives necessary and sufficient conditionsfor a functor to be an almost finitely presented derived geometric stack. We establishseveral variants of Lurie’s theorem, making the hypotheses easier to verify for manyapplications. Provided a derived analogue of Schlessinger’s condition holds, the theoremreduces to verifying conditions on the underived part and on cohomology groups. Anothersimplification is that functors need only be defined on nilpotent extensions of discreterings. Finally, there is a pre-representability theorem, which can be applied to associateexplicit geometric stacks to dg-manifolds and related objects.
Contents
Introduction 11. Representability of derived stacks 31.1. Background 31.2. Tangent spaces and homogeneity 41.3. Finite presentation 71.4. Sheaves 81.5. Representability 101.6. Strong quasi-compactness 132. Complete simplicial and chain algebras 142.1. Nilpotent algebras 172.2. A nilpotent representability theorem 182.3. Covers 213. Pre-representability 213.1. Simplicial structures 213.2. Deriving functors 233.3. Representability 26References 27
Introduction
Artin’s representability theorem ([Art]) gives necessary and sufficient conditions for afunctor from R -algebras to groupoids to be representable by an algebraic Artin stack,locally of finite presentation. In his thesis, Lurie established a similar result not justfor derived Artin 1-stacks, but for derived geometric Artin n -stacks. Explicitly, given afunctor F : s Alg R → S from simplicial R -algebras to simplicial sets, [Lur] Theorems 7.1.6and 7.5.1 give necessary and sufficient conditions for F to be representable by a derivedgeometric Artin n -stack, almost of finite presentation over R .Lurie’s Representability Theorem is more natural than Artin’s in one important respect:in the derived setting, existence of a functorial obstruction theory is an automatic conse-quence of left-exactness. However, Lurie’s theorem can be difficult to verify for problems The author was supported during this research by the Engineering and Physical Sciences ResearchCouncil [grant number EP/F043570/1]. not explicitly coming from topology. The most basic difficulty can be showing that afunctor is homotopy-preserving, or finding a suitable functor which is. It tends to be evenmore difficult to show that a functor is almost of finite presentation, or to verify that it isa hypersheaf. The purpose of this paper is to adapt the representability theorems in [Lur]and [TV], simplifying these criteria for a functor F : s Alg R → S to be a geometric n -stack.In [Lur], the key exactness properties used were cohesiveness and infinitesimal cohesive-ness. These are said to hold for a functor F : s Alg R → S if the maps θ : F ( A × B C ) → F ( A ) × hF ( B ) F ( C )to the homotopy fibre product are weak equivalences for all surjections (resp. nilpotentsurjections) A ։ B and C ։ B . The key idea of this paper is to introduce a notion morein line with Schlessinger’s conditions ([Sch]). We say that F is homotopy-homogeneous if θ is a weak equivalence for all nilpotent surjections A ։ B and arbitrary maps C → B .The first major consequence is Theorem 1.23, showing that if F is homotopy-homogeneous, then it is almost finitely presented whenever the restriction π ( F ) :Alg H R → S and the cohomology theories D ix ( F, − ) of the tangent spaces of F at dis-crete points x are all finitely presented. This reduces the question to familiar invariants,since the cohomology groups are usually naturally associated to the moduli problem. Like-wise, Proposition 1.32 shows that to ensure that a homotopy-homogeneous functor F is ahypersheaf, it suffices to check that π F is a hypersheaf and that the modules D ix ( F, − )are quasi-coherent.These results are applied to Proposition 1.33, which shows that with certain additionalfiniteness hypotheses on D ix ( F ), a cotangent complex and obstruction theory exist for F .This leads to Theorem 1.34, which replaces Lurie’s almost finite presentation conditionwith those of Theorem 1.23. We then obtain Corollary 1.36, which incorporates the furthersimplifications of Proposition 1.32.A key principle in derived algebraic geometry is that the derived structure is no morethan an infinitesimal thickening of the underived objects. For instance, every simplicialring can be expressed as a composite of homotopy square-zero extensions of a discrete ring.Proposition 2.7 strictifies this result, showing that we can work with extensions which arenilpotent (rather than just homotopy nilpotent). This approach leads to Theorem 2.17,which shows how the earlier representability results can be reformulated for functors ondg or simplicial rings A for which A → H A is nilpotent, thereby removing the need forLurie’s nilcompleteness hypothesis.The last major result is Theorem 3.16, which shows how to construct representablefunctors from functors which are not even homotopy-preserving. The key motivation isExample 3.17, which constructs explicit derived geometric stacks from Kontsevich’s dgmanifolds.The structure of the paper is as follows.In Section 1, we recall Lurie’s Representability Theorem, introduce homotopy-homogeneity, and establish the variants Theorem 1.34 and Corollary 1.36 of Lurie’s the-orem. We also establish Proposition 1.38, which identifies weak equivalences betweengeometric derived n -stacks, and Proposition 1.40, which gives a functorial criterion forstrong quasi-compactness.Section 2 then introduces simplicial or dg algebras A for which A → H A is a nilpotentextension, showing in Theorem 2.17 how to re-interpret representability in terms of functoron such algebras.Finally, Section 3 introduces the notion of homotopy-surjecting functors; these mapsquare-zero acyclic extensions to surjections. For any such functor F , we construct anotherfunctor ¯ W F , and Proposition 3.10 shows that this is homotopy-preserving whenever F ishomotopy-homogeneous and homotopy-surjecting. This leads to Theorem 3.16, whichgives sufficient conditions on F for ¯ W F to be a derived geometric n -stack. EPRESENTABILITY OF DERIVED STACKS 3 Representability of derived stacks
We denote the category of simplicial sets by S , the category of simplicial rings by s Ring,and the category of simplicial R -algebras by s Alg R . We let dg + Alg R be the category ofdifferential graded-commutative R -algebras in non-negative chain degrees. The homotopycategory Ho( C ) of a category C is obtained by formally inverting weak equivalences.1.1. Background.
Given a simplicial ring R , a derived geometric n -stack over R is afunctor F : s Alg R → S satisfying many additional conditions. These are detailed in [TV] Chapter 2.2 or [Lur] § Theorem 1.1.
A homotopy-preserving functor F : s Alg R → S is a geometric derived n -stack which is almost of finite presentation if and only if (1) The functor F commutes with filtered colimits when restricted to k -truncated objectsof s Alg R , for each k ≥ . (2) For any discrete commutative ring A , the space F ( A ) is n -truncated. (3) The functor F is a hypersheaf for the ´etale topology. (4) The functor F is cohesive: for any pair A → C , B → C of surjective morphismsin s Alg R , the induced map F ( A × C B ) → F ( A ) × hF ( C ) F ( B ) is a weak equivalence. (5) The functor F is nilcomplete: for any A ∈ s Alg R , the natural map F ( A ) → lim ←− hk F ( P k A ) is an equivalence, where { P k A } k denotes the Moore-Postnikov towerof A . (6) Let B be a complete, discrete, local, Noetherian R -algebra, and m ⊂ B the maximalideal. Then the natural map F ( B ) → lim ←− hn F ( B/m n ) is a weak equivalence. (7) Let x ∈ F ( C ) , where C is a (discrete) integral domain which is finitely generatedas a π R -algebra. For each i, n , the tangent module D n − ix ( F, C ) := π i ( F ( C ⊕ C [ − n ]) × hF ( C ) { x } ) is a finitely generated C -module. (8) R is a derived G-ring: (a) π R is Noetherian, (b) for each prime ideal p ⊂ π R , the p ( π R ) p -adic completion of ( π R ) p is ageometrically regular π R -algebra, and (c) for all n , π n R is a finite π R -module. (9) R admits a dualising module in the sense of [Lur] Definition 3.6.1. [For discreterings, this is equivalent to a dualising complex. In particular, Z and Gorensteinlocal rings are all derived G-rings with dualising modules.]Proof. [Lur] Theorem 7.5.1. (cid:3) Readers unfamiliar with the conditions of this theorem should not despair, since theconditions will be explained and considerably simplified over the course of this paper.
Remark . Note that there are slight differences in terminology between [TV] and [Lur].In the former, only disjoint unions of affine schemes are 0-representable, so arbitraryschemes are 2-geometric stacks, and Artin stacks are 1-geometric stacks if and only if they
J.P.PRIDHAM have affine diagonal. In the latter, algebraic spaces are 0-stacks. A geometric n -stack iscalled n -truncated in [TV], and it follows easily that every n -geometric stack in [TV] is n -truncated. Conversely, every geometric n -stack is ( n + 2)-geometric.We can summarise this by saying that for a derived geometric stack X to be n -truncatedmeans that X → X S n +1 is an equivalence, or equivalently that X → X S n − is representableby derived algebraic spaces. For X to be n -geometric means that X → X S n − is repre-sentable by disjoint unions of derived affine schemes.Theorem 1.34 takes the convention from [Lur], so “geometric derived n -stack” means“ n -truncated derived geometric stack”.1.2. Tangent spaces and homogeneity.Definition 1.3.
We say that a map A → B in s Ring is a square-zero extension if it issurjective, and the kernel I is square-zero, i.e. satisfies I = 0. Lemma 1.4. In Ho( s Alg R ) , square-zero extensions A → B with kernel I correspond upto weak equivalence to the small extensions A of B by I in the sense of [Lur] Definition3.3.1.Proof.
Given a square-zero extension A → B , observe that the kernel I is a simplicial B -module. Choose an inclusion i : I ֒ → N of simplicial B -modules, with N acyclic, andset ˜ B to be the simplicial algebra A ⊕ I N . Then ˜ B → B is a trivial fibration, and if welet C = coker i , then A = ˜ B × B ⊕ C B. Now we need only observe that Ω C ≃ I in the notation of [TV], so ˜ B → B ⊕ C gives ahomotopy derivation s : B → I [ − A = B ⊕ s I := B × h id+ s,B ⊕ I [ − , id+0 B, so A → B is a small extension in Lurie’s sense.Conversely, given a homotopy derivation s : B → M [ − B iscofibrant, so lift this to a morphism B → B ⊕ M [ −
1] of simplicial R -algebras. Taking asurjection f : N ։ M [ −
1] of simplicial B -modules, with N acyclic, we see that B × h id+ s,B ⊕ M [ − , id+0 B ≃ B × id+ s,B ⊕ M [ − , id ( B ⊕ N ) , since the right-hand map is a fibration. But this maps surjectively to B , with kernel I := ker f , which is a B -module, so square-zero. Moreover M ≃ I , so the respectivesquare-zero extensions are by the same module. (cid:3) Remark . Given a simplicial ring B and a simplicial B -module M , we may define aderivation t : B ⊕ M [ − → M [ −
1] given by 0 on B , and by the identity on M [ − B ⊕ M [ − ⊕ t M is equivalent to B . In particular,this means that B → B ⊕ M [ −
1] is weakly equivalent to a square-zero extension.
Definition 1.6.
Say that a functor between model categories is homotopy-preserving ifit maps weak equivalences to weak equivalences.
Definition 1.7.
We say that a functor F : s Alg R → S is homotopy-homogeneous if for all square-zero extensions A → B and all maps C → B in s Alg R , the natural map F ( A × B C ) → F ( A ) × hF ( B ) F ( C )to the homotopy fibre product is a weak equivalence. EPRESENTABILITY OF DERIVED STACKS 5
Definition 1.8.
Given a homotopy-preserving homotopy-homogeneous functor F : s Alg R → S , a simplicial R -algebra A , and a point x ∈ F ( A ), define the tangent func-tor T x ( F/R ) : s Mod A → S by T x ( F/R )( M ) := F ( A ⊕ M ) × hF ( A ) { x } . Lemma 1.9. If F satisfies the conditions of Definition 1.8, then up to non-canonical weakequivalence, T x ( F/R )( M ) is an invariant of the class [ x ] ∈ π F ( A ) .Proof. Given a path γ : ∆ → F ( A ), we have equivalences T γ (0) ( F/R )( M ) ≃ ∆ × hγ,F ( A ) F ( A ⊕ M ) ≃ T γ (1) ( F/R )( M ) , so paths in F ( A ) give equivalences between stalks. Considering maps ∆ → F ( A ), wesee that these equivalences satisfy the cocycle condition up to homotopy, with the maps∆ n → F ( A ) giving higher homotopies. Thus T ( − ) ( F/R )( M ) forms a weak local coefficientsystem on F ( A ). (cid:3) Definition 1.10.
Given a simplicial abelian group A • , we denote the associated nor-malised chain complex by N A . Recall that this is given by N ( A ) n := T i> ker( ∂ i : A n → A n − ), with differential ∂ . Then H ∗ ( N A ) ∼ = π ∗ ( A ).Using the Eilenberg-Zilber shuffle product, normalisation N extends to a functor N : s Ring → dg + Ringfrom simplicial rings to differential graded rings in non-negative chain degrees.By the Dold-Kan correspondence, normalisation gives an equivalence of categories be-tween simplicial abelian groups and chain complexes in non-negative degrees. For any R ∈ s Ring, this extends to an equivalence s Mod R → dg + Mod NR between simplicial R -modules and N R -modules in non-negatively graded chain complexes.
Definition 1.11.
Given a chain complex V , let V [ r ] be the chain complex V [ r ] i := V r + i .Given a simplicial abelian group M and n ≥
0, let M [ − n ] := N − ( N M [ − n ]), where N − is inverse to the normalisation functor N .For R ∈ s Ring, observe that this extends to a functor [ − n ] : s Mod R → s Mod R . Notethat π i M [ − n ] = π i − n M . Lemma 1.12.
For all
F, A, M, x as in Definition 1.8, there is a natural abelian structureon π i T x F ( M ) . Moreover, there are natural isomorphisms π i T x ( F/R )( M ) ∼ = π i +1 T x F ( M [ − , where homotopy groups are defined relative to the basepoint given by the image of T x ( F/R )(0) → T x ( F/R )( M ) .Proof. Addition in M gives a morphism( A ⊕ M ) × A ( A ⊕ M ) ∼ = A ⊕ ( M ⊕ M ) → A ⊕ M, so the corresponding map F ( A ⊕ M ) × hF ( A ) F ( A ⊕ M ) → F ( A ⊕ M ) . induces an abelian structure on π i T x F ( M ).For the second part, observe that M = 0 × hM [ −
0, and that 0 → M [ −
1] is surjective(in the sense that it is surjective on π ), so F ( A ⊕ M ) ≃ F ( A ) × hF ( A ⊕ M [ − F ( A ) J.P.PRIDHAM by homotopy-homogeneity, giving T x ( F/R )( M ) ≃ × hT x ( F/R )( M [ − . Thus π i T x ( F/R )( M ) ∼ = π i +1 T x ( F/R )( M [ − (cid:3) Definition 1.13.
For all
F, A, x as above, and all simplicial A -modules M , defineD n − ix ( F, M ) := π i ( T x ( F/R )( M [ − n ])) , observing that this is well-defined, by Lemma 1.12. Remark . Observe that if F is a derived geometric n -stack, and x : Spec A → F overSpec R , then D jx ( F, M ) = Ext jA ( x ∗ L F/ Spec R • , M ), for L F/R • the cotangent complex of F over R . Lemma 1.15.
For
F, A, x as above, with f : A → B a morphism of simplicial R -algebras,and M a simplicial B -module, there are natural isomorphisms T x ( F/R )( f ∗ M ) ≃ T f ∗ x ( F/R )( M ) , and hence D jx ( F, f ∗ M ) ∼ = D jf ∗ x ( F, M ) .Proof. This is just the observation that A ⊕ f ∗ M = A × B ( B ⊕ M ), so F ( A ⊕ f ∗ M ) ≃ F ( A ) × hF ( B ) F ( B ⊕ M ). (cid:3) Lemma 1.16. If X : s Alg R → S is homotopy-preserving and homotopy-homogeneous,take an object A ∈ s Alg R and an A -module M . Then there is a local coefficient system D ∗ ( X, M ) on the simplicial set X ( A ) , whose stalk at x ∈ X ( A ) is D ∗ x ( X, M ) . In particular, D ∗ x ( X, M ) depends (up to non-canonical isomorphism) only on the image of x in π X ( A ) .Proof. This follows straightforwardly from the proof of Lemma 1.9. (cid:3)
Proposition 1.17. If F : s Alg R → S is homotopy-preserving and homotopy-homogeneous,then for any square-zero extension e : I → A f −→ B in s Alg R , there is a sequence of sets π ( F A ) f ∗ −→ π ( F B ) o e −→ Γ( F B, D ( F, I )) , where Γ( − ) denotes the global section functor. This is exact in the sense that the fibre of o e over is the image of f ∗ . Moreover, there is a group action of D x ( F, I ) on the fibre of π ( F A ) → π ( F B ) over x , whose orbits are precisely the fibres of f ∗ .For any y ∈ F 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 · · · e ∗ / / π n ( F A, y ) f ∗ / / π n ( F B, x ) o e / / D − ny ( F, I ) e ∗ / / π n − ( F A, y ) f ∗ / / · · ·· · · f ∗ / / π ( F B, x ) o e / / D y ( F, I ) −∗ y / / π ( F A ) . Proof.
The proof of [Pri1] Theorem 1.45 carries over to this context. The main idea is thatas in the proof of Lemma 1.4, there is a trivial fibration ˜ B → B , and A = ˜ B × B ⊕ I [ − B ,with ˜ B → B ⊕ I [ −
1] a square-zero extension. By homotopy-homogeneity, F ( A ) ≃ F ( ˜ B ) × hF ( B ⊕ I [ − F ( B ) , and F ( ˜ B ) ≃ F ( B ) since F is homotopy-preserving.The rest of the proof then follows by studying the long exact sequence of homotopygroups associated to the homotopy fibres of F ( ˜ B ) → F ( B ⊕ I [ − F A → F B , noting that F ( A × B A ) ≃ F ( A ) × hF ( B ) F ( B ⊕ I ). (cid:3) EPRESENTABILITY OF DERIVED STACKS 7
Finite presentation.Definition 1.18.
Recall (e.g. from [GJ] Definition VI.3.4) that the Moore-Postnikovtower { P n X } of a fibrant simplicial set X is given by P n X q := Im ( X q → Hom(sk n ∆ q , X )) , with the obvious simplicial structure. Here, sk n K denotes the n -skeleton of K , the sim-plicial set generated by K ≤ n .The spaces P n X form an inverse system X → . . . → P n X → P n − X → . . . , with X = lim ←− P n X , and π q P n X = (cid:26) π q X q ≤ n q > n. The maps P n X → P n − X are fibrations. If X is reduced, then so is P n X . Definition 1.19.
Define τ ≤ k ( s Alg R ) to be the full subcategory of s Alg R consisting ofobjects A with A = P k A , the k th Moore-Postnikov space. Definition 1.20.
Define the category τ ≤ k Ho( s Alg R ) to be the full subcategory ofHo( s Alg R ) consisting of objects A with π i A = 0 for i > k . Note that τ ≤ k Ho( s Alg R )is equivalent to the category Ho( τ ≤ k ( s Alg R )) obtained by localising τ ≤ k ( s Alg R ) at weakequivalences. Definition 1.21.
Recall from [Lur] Proposition 5.3.10 that a homotopy-preserving functor F : s Alg R → S is said to be almost of finite presentation if for all k and all filtered directsystems { A α } α ∈ I in τ ≤ k ( s Alg R ), the maplim −→ F ( A α ) → F (lim −→ A α )is a weak equivalence. Definition 1.22.
Given a functor F : s Alg R → S , define π F : Alg π R → S by π F ( A ) = F ( A ). Theorem 1.23.
If a homotopy-preserving functor F : s Alg R → S is homotopy-homogeneous, then it is almost of finite presentation if and only if the following hold: (1) the functor π F : Alg π R → S preserves filtered colimits; (2) for all finitely generated A ∈ Alg π R and all x ∈ F ( A ) , the functors D ix ( F, − ) :Mod A → Ab preserve filtered colimits for all i > .Proof. Note that since π F preserves filtered colimits, Lemma 1.12 implies that the func-tors D ix ( F, − ) : Mod A → Ab preserve filtered colimits for all i ≤ F preserves filtered homotopy colimits in the categories τ ≤ k ( s Alg R ). We prove this by induction on k , the case k = 0 following by hypothesis.Take a filtered direct system { A α } in τ ≤ k ( s Alg R ), with homotopy colimit A . Let B α = P k − A α , B = P k − A . Let M α := π k A α , M := π k A , and observe that these are π A α - and π A -modules respectively.Now, A α → B α and A → B are square-zero extensions up to homotopy (see for instance[TV] Lemma 2.2.1.1), coming from essentially unique homotopy derivations δ : B α → M α [ − k − A α ≃ B α × h id+ δ,B α ⊕ M α [ − k − , id+0 B α = B α × h id+ δ,π ( A α ) ⊕ M α [ − k − , id+0 π ( A α ) . Now, by Remark 1.5, the map π ( A α ) → π ( A α ) ⊕ M α [ − k −
1] is weakly equivalent toa square-zero extension. Thus, since F is homotopy-homogeneous, F ( A α ) ≃ F ( B α ) × h id+ δ,F ( π ( A α ) ⊕ M α [ − k − F ( π ( A α ))and similarly for A . J.P.PRIDHAM
We wish to show that θ : lim −→ F ( A α ) → F ( A ) is a weak equivalence, and our inductivehypothesis gives lim −→ F ( B α ) ≃ F ( B ). It therefore suffices to consider the homotopy fibre of θ over y ∈ F ( B ), which lifts to some ˜ y ∈ F ( B β ). If we let ˜ y α be the image of ˜ y in F ( B α ),this gives θ y : lim −→{ ˜ y α } × h id+ δ,F ( π ( A α ) ⊕ M α [ − k − F ( π ( A α )) → { y } × h id+ δ,F ( π ( A ) ⊕ M [ − k − F ( π ( A )) . Since we know that F ( π ( A )) = lim −→ F ( π ( A α )), it suffices to show that for the images˜ x α ∈ F ( π A α ) , x ∈ F ( π A ) of ˜ y α , y , the mapslim −→ F ( π ( A α ) ⊕ M α [ − k − × F ( π ( A α )) { ˜ x α } → F ( π ( A ) ⊕ M [ − k − × F ( π ( A )) { x } are equivalences. Taking homotopy groups, this becomeslim −→ D k +1 − i ˜ x α ( F, M α ) → D k +1 − ix ( F, M ) , which by Lemma 1.15 is lim −→ D k +1 − i ˜ x ( F, M α ) → D k +1 − i ˜ x ( F, M ) , for ˜ x ∈ F ( π A β ) the image of ˜ y .It will therefore suffice to show that the functors D i ˜ x ( F, − ) : Mod π A β → Ab preservefiltered colimits. If we express π A β as a filtered colimit of finitely generated π R -algebras,then the condition that π F preserves filtered colimits allows us to write [˜ x ] = [ f ∗ z ] ∈ π F ( A ), for z ∈ F ( C ) , with C a finitely-generated π R -algebra. ThenD i ˜ x ( F, − ) ∼ = D if ∗ z ( F, − ) ∼ = D iz ( F, f ∗ − ) , which preserves filtered colimits by hypothesis. (cid:3) Sheaves.Definition 1.24.
Let R Tot S : c S → S be the derived total space functor from cosimplicialsimplicial sets to simplicial sets, given by R Tot S X • = holim ←− n ∈ ∆ X n , as in [GJ] § VIII.1. Explicitly, R Tot S X • = { x ∈ Y n ( X n ) ∆ n : ∂ iX x n = ( ∂ i ∆ ) ∗ x n +1 , σ iX x n = ( σ i ∆ ) ∗ x n − } , whenever X is Reedy fibrant. Homotopy groups of the total space are related to a spectralsequence given in [GJ] § VIII.1.
Definition 1.25.
A morphism f : A → B in s Ring is said to be ´etale if π f is ´etale and themaps π n ( A ) ⊗ π ( A ) π ( B ) → π n ( B ) are isomorphisms for all n . An ´etale morphism is saidto be an ´etale covering if the morphism Spec π f : Spec π B → Spec π A is a surjection ofschemes. Definition 1.26.
Given A ∈ s Ring and B • ∈ ( s Alg A ) ∆ , we may regard B as a cocontin-uous functor B : S → s Alg A , determined by B n = B (∆ n ). Then B • is said to be Reedycofibrant if the latching morphisms f n : B ( ∂ ∆ n ) → B n are cofibrations for all n ≥ B ( ∂ ∆ ) = B ( ∅ ) = A ). Definition 1.27.
A Reedy cofibrant object B • ∈ ( s Alg A ) ∆ is an ´etale hypercover if thelatching morphisms are ´etale coverings. An arbitrary object C • ∈ ( s Alg A ) ∆ is an ´etalehypercover if there exists a levelwise weak equivalence f : B • → C • , for B • a Reedycofibrant ´etale hypercover. Definition 1.28.
Given a simplicial hypercover A → B • , and a presheaf P over A , definethe cosimplicial complex ˇ C • ( B • /A, P ) by ˇ C n ( B • /A, P ) = P ( B n ). EPRESENTABILITY OF DERIVED STACKS 9
Definition 1.29.
A homotopy-preserving functor F : s Alg → S is said to be a hypersheaffor the ´etale topology if it satisfies the following conditions.(1) It preserves finite products up to homotopy; this means that for any finite (possiblyempty) subset { A i } of s Alg R , the map F ( Y A i ) → Y F ( A i )is a weak equivalence.(2) For all ´etale hypercovers A → B • , the natural map F ( A ) → R Tot ˇ C ( B • /A, F )is a weak equivalence, for ˇ C as in Definition 1.28. Remark . The same definition applies for functors Alg π R → S . Given a groupoid-valued functor Γ : Alg π R → Gpd, the nerve B Γ : Alg π R → S is a hypersheaf if and onlyif Γ is a stack (in the sense of [LMB]). Definition 1.31.
Say that a functor F : s Alg → B is nilcomplete if for any A ∈ s Alg R ,the natural map F ( A ) → lim ←− hk F ( P k A ) to the homotopy limit is an equivalence. Proposition 1.32.
Take a homotopy-homogeneous nilcomplete homotopy-preserving func-tor F : s Alg → S . If (1) π F : Alg π R → S is a hypersheaf, and (2) for all A ∈ Alg π R , all x ∈ F ( A ) , all A -modules M and all ´etale morphisms f : A → A ′ , the maps D ∗ x ( F, M ) ⊗ A A ′ → D ∗ fx ( F, M ⊗ A A ′ ) (induced by Lemma 1.15) are isomorphisms,then F is a hypersheaf.Proof. Take an ´etale hypercover f : A → B • . The first observation to make is that P k A → P k B • is also an ´etale hypercover. Assume inductively that F ( P k − A ) → R Tot ˇ C ( P k − B • /P k − A, F )is an equivalence (the case k = 1 following because π F is a hypersheaf). Now P k A → P k − A is a square-zero extension up to homotopy (see for instance [TV] Lemma 2.2.1.1),coming from an essentially unique homotopy derivation δ : P k − A → ( π k A )[ − k − P k A ≃ P k − A × h id+ δ,π A ⊕ ( π k A )[ − k − π A. Since F is homotopy-homogeneous and homotopy-preserving, this means that F ( P k A ) ≃ F ( P k − A ) × hF ( π A ⊕ ( π k A )[ − k − F ( π A ) . For the inductive step, it suffices to show that for any point x ∈ π F ( A ), the homotopyfibres of F ( P k A ) and of R Tot ˇ C ( P k B • /P k A, F ) over x are weakly equivalent. From theexpression above, we see that F ( P k A ) x ≃ F ( P k − A ) x × hT x ( F/R, ( π k A )[ − k − { } , and the corresponding statement for the hypercover is R Tot ˇ C ( P k B • /P k A, F ) ≃ R Tot ( ˇ C ( P k − B • /P k − A, F ) fx × hT fx ( F/R, ( π k B • )[ − k − { } ) ≃ F ( P k − A ) x × h R Tot T fx ( F/R, ( π k B • )[ − k − { } , using the inductive hypothesis and the fact the R Tot commutes with homotopy fibreproducts.
This reduces the problem to showing that the map T x ( F/R, ( π k A )[ − k − → R Tot T fx ( F/R, ( π k B • )[ − k − ∗ commute with ´etale base change, it follows that the map R Tot T x ( F/R, ( π k A )[ − k − ⊗ A B • → R Tot T fx ( F/R, ( π k B • )[ − k − A → B • is an ´etale hypercover (and hence an fpqc hypercover),the map T x ( F/R, ( π k A )[ − k − → R Tot T x ( F/R, ( π k A )[ − k − ⊗ A B • is also a weak equivalence, completing the inductive step.Finally, since F is nilcomplete, we get F ( A ) ≃ lim ←− k h F ( P k A ) R Tot ˇ C ( B • /A, F ) ≃ lim ←− k h R Tot ˇ C ( P k B • /P k A, F ) , which completes the proof. (cid:3) Representability.Proposition 1.33.
Take a Noetherian simplicial ring R , and a homotopy-preserving func-tor F : s Alg R → S , satisfying the following conditions: (1) For all discrete rings A , F ( A ) is n -truncated, i.e. π i F ( A ) = 0 for all i > n . (2) F is homotopy-homogeneous, i.e. for all square-zero extensions A ։ C and allmaps B → C , the map F ( A × C B ) → F ( A ) × hF ( C ) F ( B ) is an equivalence. (3) F is nilcomplete, i.e. for all A , the map F ( A ) → lim ←− h F ( P k A ) is an equivalence, for { P k A } the Postnikov tower of A . (4) F is a hypersheaf for the ´etale topology. (5) π F : Alg π R → S preserves filtered colimits. (6) R admits a dualising module, in the sense of [Lur] Definition 3.6.1. Examples areanything admitting a dualising complex in the sense of [Har]
Ch. V, such as Z orGorenstein local rings, and any simplicial ring almost of finite presentation over aNoetherian ring with a dualising module. (7) for all finitely generated A ∈ Alg π R and all x ∈ F ( A ) , the functors D ix ( F, − ) :Mod A → Ab preserve filtered colimits for all i > . (8) for all finitely generated integral domains A ∈ Alg π R and all x ∈ F ( A ) , thegroups D ix ( F, A ) are all finitely generated A -modules.Then there is an almost perfect cotangent complex L F/R in the sense of [Lur] .Proof.
This is an adaptation of [Lur] Theorem 7.4.1. After applying Theorem 1.23 to showthat F is almost of finite presentation, the only difference is in condition (2), where weonly consider square-zero extensions A → C (rather than all surjections), but also allowarbitrary maps B → C (rather than just surjections). The key observation is that we stillsatisfy the conditions of [Lur] Theorem 3.6.9, guaranteeing local existence of the cotangentcomplex, while Lemma 1.15 provides the required compatibility. (cid:3) Theorem 1.34.
Let R be a derived G-ring admitting a dualising module, and F : s Alg R → S a homotopy-preserving functor. Then F is a geometric derived n -stack which is almost EPRESENTABILITY OF DERIVED STACKS 11 of finite presentation if and only if the conditions of Proposition 1.33 hold, and for allcomplete discrete local Noetherian π R -algebras A , with maximal ideal m , the map F ( A ) → lim ←− h F ( A/ m r ) is a weak equivalence.Proof. This is essentially the same as [Lur] Theorem 7.5.1, by combining ibid. Theorem7.1.6 with Proposition 1.33 (rather than ibid. Theorem 7.4.1).Note that our revised condition (2) implies infinitesimal cohesiveness, since, for anysquare-zero extensions 0 → M → ˜ A → A →
0, we may set B to be the mapping cone (so B ≃ A ), and consider the fibre product ˜ A ≃ B × hA ⊕ M [ − A .To see that the revised condition (2) is necessary, we adapt [Lur] Proposition 5.3.7. Itsuffices to show that for any smooth surjective map U → F of n -stacks, the map U ( A ) × hU ( C ) U ( B ) → F ( A ) × hF ( C ) F ( B )is surjective, for all square-zero extensions A ։ C . Moreover, the argument of [Lur]Proposition 5.3.7 allows us to replace A × C B with an ´etale algebra over it, giving a locallift of a point x ∈ F ( B ) to u ∈ U ( B ). The problem then reduces to showing that U ( A ) × hU ( C ) U ( B ) → F ( A ) × hF ( C ) U ( B )is surjective, but this follows from pulling back the surjection U ( A ) → U ( C ) × hF ( C ) F ( A )given by the smoothness of U → F . (cid:3) Remark . The Milnor exact sequence ([GJ] Proposition 2.15) gives a sequence • → lim ←− r π i +1 F ( A/ m r ) → π i (lim ←− h F ( A/ m r )) → lim ←− r π i F ( A/ m r ) → • , which is exact as groups for i ≥ i = 0. Thus the condition ofTheorem 1.34 can be rephrased to say that the map f : π F ( A ) → lim ←− r π F ( A/ m r )is surjective, that for all x ∈ F ( A ) the maps f i,x : π i ( F A, x ) → lim ←− r π i ( F ( A/ m r ) , x )are surjective for all i ≥ f i,x → lim ←− r π i +1 ( F ( A/ m r ) , x )are surjective for all i ≥ { G r } r ∈ N of groups satisfies the Mittag-Lefflercondition 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 thatlim ←− { G r } r = 1.Hence, if each system { Im ( π i ( F ( A/ m s ) , x ) → π i ( F ( A/ m r ) , x )) } s ≥ r satisfies the Mittag-Leffler condition for i ≥
1, then the condition of Theorem 1.34 reduces to requiring thatthe maps π i F ( A ) → lim ←− r π i F ( A/ m r )be isomorphisms for all i . Corollary 1.36.
Let R be a derived G-ring admitting a dualising module (in the sense of [Lur] Definition 3.6.1) and F : s Alg R → S a homotopy-preserving functor. Then F is ageometric derived n -stack which is almost of finite presentation if and only if the followingconditions hold (1) For all discrete rings A , F ( A ) is n -truncated, i.e. π i F ( A ) = 0 for all i > n . (2) F is homotopy-homogeneous, i.e. for all square-zero extensions A ։ C and allmaps B → C , the map F ( A × C B ) → F ( A ) × hF ( C ) F ( B ) is an equivalence. (3) F is nilcomplete, i.e. for all A , the map F ( A ) → lim ←− h F ( P k A ) is an equivalence, for { P k A } the Postnikov tower of A . (4) π F : Alg π R → S is a hypersheaf for the ´etale topology. (5) π π F : Alg π R → Set preserves filtered colimits. (6)
For all A ∈ Alg π R and all x ∈ F ( A ) , the functors π i ( π F, x ) : Alg A → Set preserve filtered colimits for all i > . (7) for all finitely generated integral domains A ∈ Alg π R , all x ∈ F ( A ) and all ´etalemorphisms 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 R and all x ∈ F ( A ) , the functors D ix ( F, − ) :Mod A → Ab preserve filtered colimits for all i > . (9) for all finitely generated integral domains A ∈ Alg π R and all x ∈ F ( A ) , thegroups D ix ( F, A ) are all finitely generated A -modules. (10) for all complete discrete local Noetherian π R -algebras A , with maximal ideal m ,the map F ( A ) → lim ←− n h F ( A/ m r ) is a weak equivalence (see Remark 1.35 for a reformulation).Proof. If F is a derived n -stack of almost finite presentation, then the ´etale sheaf A ′ D ifx ( F, A ′ ) on Y := Spec A is just E xt i O Y ( x ∗ L F/R , O Y ) , which is necessarily quasi-coherent, as x ∗ L F/R is equivalent to a complex of finitely gener-ated locally free sheaves (for instance by the results of [Pri3] § π F : Alg π R → S preserving filtered colimits.For the converse, we just need to show that F is a hypersheaf in order to ensure that itsatisfies the conditions of Theorem 1.34. This follows almost immediately from Proposition1.32, first noting that condition (7) above combines with almost finite presentation andexactness of the tangent complex to ensure that for all A ∈ Alg π R , all x ∈ F ( A ) , all A -modules M and all ´etale morphisms f : A → A ′ , the mapsD ∗ x ( F, M ) ⊗ A A ′ → D ∗ fx ( F, M ⊗ A A ′ )are isomorphisms. (cid:3) Remark . Although Corollary 1.36 seems more complicated than Theorem 1.34, sinceit has an extra condition, it is much easier to verify in practice. This is because F ( A ) is EPRESENTABILITY OF DERIVED STACKS 13 only n -truncated when A is discrete, so it is much easier to check that π F is a hypersheafthan to do the same for F . Proposition 1.38.
Take a morphism α : F → G of almost finitely presented geometricderived n -stacks a over R . Then α is a weak equivalence if and only if (1) π α : π F → π G is a weak equivalence of functors Alg π R → S , and (2) the maps D ix ( F, A ) → D iαx ( G, A ) are isomorphisms for all finitely generated integraldomains A ∈ Alg π R , all x ∈ F ( A ) , and all i > .Proof. It suffices to show that L F/G • ≃
0. For if this is the case, then [TV] Corollary 2.2.5.6implies that α is ´etale. By applying [TV] Theorem 2.2.2.6 locally, it follows that an ´etalemorphism α must be a weak equivalence whenever π α is so.Now, L F/G • is the cone of α ∗ L G/R → L F/R , so we wish to show that this map is anequivalence locally. This is equivalent to saying that for all integral domains A ∈ π R , all π R -modules M , all x ∈ F ( A ) and all i , the mapsD ix ( F, M ) → D iαx ( G, M )are isomorphisms.For i ≤
0, these isomorphisms follow immediately from the hypothesis that π α be anequivalence. For i >
0, we first note that finite presentation of π F means that we mayassume that A is finitely generated. We then have an almost perfect complex x ∗ L F/G • withthe property that Ext iA ( x ∗ L F/G • , A ) = 0for all i , so Ext iA ( x ∗ L F/G • , P ) = 0 for all almost perfect A -complexes P (using nilcomplete-ness of F and G ). In particular,D ix ( F/G, M ) = Ext iA ( x ∗ L F/G • , M ) = 0for all finite A -modules. Almost finite presentation of F and G now gives thatD ix ( F/G, M ) = 0 for all A -modules M , completing the proof. (cid:3) Strong quasi-compactness.Lemma 1.39. If S is a set of separably closed fields, and X = Spec ( Q k ∈ S k ) , then everysurjective ´etale morphism f : Y → X of affine schemes has a section.Proof. Since f is surjective, the canonical maps Spec k → X admit lifts to Y , for all k ∈ S ,combining to give a map ` k ∈ S Spec k → Y . Since Y is affine, this is equivalent to givinga map X → Y , and this is automatically a section of f . (cid:3) Proposition 1.40.
A morphism F → G of geometric m -stacks is strongly quasi-compactif and only if for all sets S of separably closed fields, the map F ( Y k ∈ S k ) → ( Y k ∈ S F ( k )) × h ( Q k ∈ S G ( k )) G ( Y k ∈ S k ) is a weak equivalence in S .Proof. Let Z = Spec ( Q k ∈ S k ), and fix an element g ∈ G ( Z ) . If F → G is stronglyquasi-compact, then F × hG,g Z is strongly quasi-compact, so by [Pri3] Theorem 4.7, thereexists a simplicial affine scheme X whose sheafification X ♯ is F × hG Z . Now, Lemma 1.39implies that Z is weakly initial in the category of ´etale hypercovers of Z , so (for instanceby [Pri3] Corollary 4.10) X ♯ ( Z ) ≃ X ( Z ). Now, since X is simplicial affine, it preservesarbitrary limits of rings, so X ( Z ) ∼ = Y k ∈ S X ( k ) ∼ = Y k ∈ S X ♯ ( Z ) , which proves that the condition is necessary.To prove that the condition is sufficient, we need to show that for any affine scheme U and any morphism U → G , the homotopy fibre product F × hG U is strongly quasi-compact.Since U is affine, it satisfies the condition, so F × hG U will also, and so we may assumethat G = U or even Spec Z .Now, it follows (for instance from the proof of [Pri3] Theorem 4.7) that if an n -geometricstack F admits an n -atlas U → F , with U quasi-compact, and the diagonal F → F × F is strongly quasi-compact, then F is strongly quasi-compact.We will proceed by induction on n (noting that we use n -geometric, as in Remark 1.2,rather than n -truncated). A 0-geometric stack F is a disjoint union of affine schemes, sois separated, and in particular its diagonal is strongly quasi-compact.Assume that an n -geometric stack F has strongly quasi-compact diagonal and satisfiesthe condition above, and take an n -atlas V → F for V S be a set of representatives of equivalence classes ofgeometric points of V , and set Z = Spec ( Q k ∈ S k ). Since F satisfies the condition above, F ( Z ) ∼ = Y k ∈ S F ( k ) , so the points Spec k → V → F combine to define a map f : Z → F .As V → F is an atlas, for some ´etale cover Z ′ → Z , f lifts to a map ˜ f : Z ′ → V . ButLemma 1.39 implies that Z ′ → Z has a section, so we have a lifting ˜ f : Z → V of f .Now, V = ` α ∈ I V α is a disjoint union of affine schemes, and since Z is quasi-compact,there is some finite subset J ⊂ I with U := ` α ∈ J V α containing the image of Z . But U is then quasi-compact, and U → F is surjective, hence an n -atlas, which completes theinduction. (cid:3) Corollary 1.41.
A morphism F → G of geometric derived stacks is strongly quasi-compact if and only if for all sets S of separably closed fields, the map F ( Y k ∈ S k ) → ( Y k ∈ S F ( k )) × h ( Q k ∈ S G ( k )) G ( Y k ∈ S k ) is a weak equivalence in S .Proof. The morphism F → G is strongly quasi-compact if and only if π F → π G is astrongly quasi-compact morphism of geometric stacks, so we apply Proposition 1.40. (cid:3) Complete simplicial and chain algebras
Proposition 2.1.
Take a cofibrantly generated model category C . Assume that D is acomplete and cocomplete category, equipped with an adjunction D U ⊤ / / C F o o , with U preserving filtered colimits. If U F maps generating trivial cofibrations to weakequivalences, then D has a cofibrantly generated model structure in with a morphism f isa fibration or a weak equivalence whenever U f is so.This adjunction is a pair of Quillen equivalences if and only if the unit morphism A → U F A is a weak equivalence for all cofibrant objects A ∈ C .Proof. To see that this defines a model structure on D , note that since U preserves filteredcolimits, for any small object I ∈ C , the object F I is small in D , so we may apply [Hir]Theorem 11.3.2 to obtain the model structure on D .Since U reflects weak equivalences, by [Hov] Corollary 1.3.16, the functors F ⊢ U form a pair of Quillen equivalences if and only if the morphisms R η : A → R U F A are
EPRESENTABILITY OF DERIVED STACKS 15 weak equivalences for all cofibrant A ∈ C . Since U preserves weak equivalences, the map U B → R U B is a weak equivalence for all B ∈ D . Thus the unit η : A → U F A is a weakequivalence if and only if R η is so. (cid:3) Fix a Noetherian ring R . Definition 2.2.
Say that a simplicial R -algebra A is finitely generated if there are finitesets Σ q ⊂ A q of generators, closed under the degeneracy operations, with only finitelymany elements of S q Σ q being non-degenerate.Define F Gs
Alg R to be the category of finitely generated simplicial R -algebras. De-fine F Gdg + Alg R to be the category of finitely generated non-negatively graded chain R -algebras (if R is a Q -algebra). Definition 2.3.
Given A ∈ s Alg R , define ˆ A := lim ←− n A/I nA , for I A = ker( A → π A ). Given A ∈ dg + Alg R , define ˆ A := lim ←− n A/I nA , for I A = ker( A → H A ). Definition 2.4.
Define \ F Gs
Alg R to be the full subcategory of s Alg R consisting of objectsof the form ˆ A , for A ∈ F Gs
Alg R . Define \ F Gdg + Alg R to be the full subcategory of dg + Alg R consisting of objects of the form ˆ A , for A ∈ F Gdg + Alg R Lemma 2.5.
The categories \ F Gs
Alg R and \ F Gdg + Alg R contain all finite colimits.Proof. The initial object is ˆ R (which equals R whenever R is discrete), and the cofibrecoproduct of A ← B → C is given by A ˆ ⊗ B C := \ A ⊗ B C. (cid:3) Proposition 2.6.
For C = \ F Gs
Alg R or \ F Gdg + Alg R , the category ind( C ) is equivalent tothe category of left-exact functors F : C opp → Set , i.e. functors for which (1) F ( ˆ R ) is the one-point set, and (2) the map F ( A ˆ ⊗ B C ) → F ( A ) × F ( B ) F ( C ) is an isomorphism for all diagrams A ← B → C .The equivalence is given by sending a direct system { A α } α to the functor F ( B ) =lim −→ α Hom C ( B, A α ) .Proof. For A ∈ C , a subobject of A opp ∈ C opp is just a surjective map A → B in C , orequivalently a simplicial (resp. dg) ideal of A . Since A is Noetherian, it satisfies ACCon such ideals, and hence A opp satisfies DCC on strict subobjects. Therefore C opp isan Artinian category containing all finite limits, so the required result is given by [Gro],Corollary to Proposition 3.1. (cid:3) Proposition 2.7.
There are cofibrantly generated model structures on the categories ind( \ F Gs
Alg R ) and ind( \ F Gdg + Alg R ) in which a morphism f : { A α } α → { B β } β is afibration or a weak equivalence whenever the corresponding map lim −→ f : lim −→ α A α → lim −→ β B β in s Alg R or dg + Alg R is so.For these model structures, the functors U : ind( \ F Gs
Alg R ) → s Alg R U : ind( \ F Gdg + Alg R ) → dg + Alg R given by U ( { A α } α ) = lim −→ α A α are right Quillen equivalences. Proof.
We begin by showing that ind( \ F Gs
Alg R ) and ind( \ F Gdg + Alg R ) are complete andcocomplete. By Lemma 2.5, they contain finite colimits, and the proof of [Isa] Proposition11.1 then ensures that they contain arbitrary coproducts, and hence arbitrary colimits.It follows immediately from Proposition 2.6 that the categories contain arbitrary limits,since any limit of left-exact functors is left-exact.We need to establish that the functors U have left adjoints. Since R is Noetherian,finitely generated objects over R are finitely presented, so the functorslim −→ : ind( F Gs
Alg R ) → s Alg R lim −→ : ind( F Gdg + Alg R ) → dg + Alg R are equivalences of categories. The left adjoints F : ind( F Gs
Alg R ) → ind( \ F Gs
Alg R ) F : ind( F Gdg + Alg R ) → ind( \ F Gdg + Alg R )to U are thus given by { A α } α
7→ { ˆ A α } α .It is immediate that U preserves filtered colimits, so we may apply Proposition 2.1 toconstruct the model structures. It only remains to show that U is a Quillen equivalence.By Proposition 2.1, we need only show that, for any cofibrant A ∈ s Alg R or A ∈ dg + Alg R ,the map A → U F A is a weak equivalence. If we write A = lim −→ α A α , for A α ∈ F Gs
Alg R (or A α ∈ F Gdg + Alg R ),then U F A = lim −→ α ˆ A α . Thus it suffices to show that for A ∈ F Gs
Alg R (or A ∈ F Gdg + Alg R ), the map A → ˆ A is a weak equivalence. If A ∈ F Gs
Alg R , then each A n is Noetherian, so [Pri3] Theorem6.6 gives the required equivalence. If A ∈ F Gdg + Alg R , then A is Noetherian and each A n is a finite A -module, so [Pri3] Lemma 6.37 gives the required equivalence. (cid:3) Lemma 2.8.
The category ind( \ F Gs
Alg R ) (resp. ind( \ F Gdg + Alg R ) ) is equivalent to a fullsubcategory C of s Alg R (resp. dg + Alg R ). If I A = ker( A → H A ) , then A is an object of C if and only if it contains the I A -adic completions of all its finitely generated subalgebras.Proof. It is immediate that A satisfies the condition above if and only if A = U F A forthe functors U and F from the proof of Proposition 2.7. Thus we need only show thatthe functor U : ind( \ F Gs
Alg R ) → F s
Alg given by { A α } 7→ lim −→ α A α is full and faithful.It suffices to show that for A ∈ \ F Gs
Alg R and B ∈ ind( \ F Gs
Alg R ), Hom( A, lim −→ B β ) =lim −→ β Hom(
A, B β ).To do this, recall that A = c A ′ for some finitely generated A ′ , and express A as lim −→ A α ,for A ′ ⊂ A α ∈ F Gs
Alg R . ThenHom( A, lim −→ B β ) = lim ←− α Hom( A α , lim −→ B β )= lim ←− α lim −→ β Hom( A α , B β )= lim ←− α lim −→ β Hom( ˆ A α , B β ) , but ˆ A α = A , giving the required result. (cid:3) EPRESENTABILITY OF DERIVED STACKS 17
Nilpotent algebras.Definition 2.9.
Say that a surjection A → B in dg + Alg R (resp. s Alg R ) is a little extension if the kernel K satisfies I A · K = 0. Say that an acyclic little extension is tiny if K (resp. N K ) is of the form cone( M )[ − r ] for some H A -module (resp. π A -module) M .Note that acyclic little extensions are necessarily square-zero, but that arbitrary littleextensions need not be. Definition 2.10.
Define dg + N R (resp. s N R ) to be the full subcategory of dg + Alg R (resp. s Alg R ) consisting of objects A for which the map A → H A (resp. A → π A ) hasnilpotent kernel. Define dg + N ♭R (resp. s N ♭R ) to be the full subcategory of dg + N R (resp. s N R ) consisting of objects A for which A i = 0 (resp. N i A = 0) for all i ≫ Lemma 2.11.
Every surjective weak equivalence f : A → B in dg + N ♭R (resp. s N ♭R )factors as a composition of tiny acyclic extensions.Proof. We first prove this for dg + N ♭R . Let K = ker( f ), and observe that the good trunca-tions ( τ ≥ r K ) i = K i i > r Z r K i = r i < r are also dg ideals in A . Since A is concentrated in degrees [0 , d ] for some d , we get afactorisation of f into acyclic surjections A = A/ ( τ ≥ d K ) → A/ ( τ ≥ ( d − K ) → . . . → A/ ( τ ≥ K ) = B. We therefore reduce to the case where K is concentrated in degrees r, r + 1.Let s be least such that K r · I sA = 0; if s = 1 then f is already a tiny acyclic extension.We will proceed by induction on s . Since K ։ ( K/I A K ), we have H r ( K/I A K ) = 0. Thismeans that the inclusion τ >r ( K/I A K ) → ( K/I A K ) is a quasi-isomorphism of ideals in A . If we set B ′ := ( A/I A K ) / ( τ >r K/I A K ) and K ′′ := ker( A → B ′ ), then I A · K ′′ = 0so f ′′ : B ′ → B is an acyclic little extension. In fact, for M := ( K/I A · K ) r , we have K ′′ = cone( M )[ − r ], so f ′′ is a tiny acyclic extension.Now, for K ′ := ker( f ′ : A → B ′ ) we have K ′ r = ( I A K ) r , so K ′ r · I s − A = 0, so by induction f ′ factors as a composition of tiny acyclic extensions. This completes the inductive step.Finally, for f : A → B in s N ♭R , normalisation gives an equivalence of categories betweensimplicial A -modules and non-negatively graded dg N A -modules. In particular, it givesan equivalence between the categories of ideals, and hence quotients of A correspond toquotients of N A . If
N f is a tiny acyclic extension, then so is f , since N K is automati-cally an H N A -module, and H N A = π A . The proof above expresses N A → N B as acomposition of tiny acyclic extensions, which thus yields such an expression for A → B . (cid:3) Definition 2.12.
Define \ F Gs
Alg R♭ (resp. \ F Gdg + Alg R♭ ) to be the full subcategory of \ F Gs
Alg R (resp. \ F Gdg + Alg R ) consisting of objects A for which A i = 0 (resp. N i A = 0)for all i ≫ Lemma 2.13.
For any surjective weak equivalence f : A → B in \ F Gs
Alg R♭ (resp. \ F Gdg + Alg R♭ ), the associated morphism { A/I nA } → { B/I nB } in pro( dg + N ♭R ) (resp. pro( s N ♭R ) ) is isomorphic to an inverse limit of surjective weakequivalences in dg + N R (resp. s N R ). Proof.
With reasoning as at the end of Lemma 2.11, it suffices to prove this for \ F Gdg + Alg R♭ . The first observation to make is that if f and g are composable mor-phisms satisfying the conclusions of this lemma, then f g also satisfies the conclusions. Let K = ker( f ); since A is concentrated in degrees [0 , d ] for some d , we get a factorisation of f into acyclic surjections A = A/ ( τ ≥ d K ) → A/ ( τ ≥ ( d − K ) → . . . → A/ ( τ ≥ K ) = B, and therefore reduce to the case where K is concentrated in degrees r, r + 1.Set I := I A and J := I B ; we now define a dg ideal I ( n ) ′ ✁ A to be generated by I n and K r +1 ∩ d − ( I n ), and set A ( n ) ′ := A/I ( n ) ′ . There is a surjection A ( n ) ′ → B/J n , withkernel K/ ( K ∩ I ( n ) ′ ). This is given by( K/K ∩ I ( n ) ′ ) i = K r / ( K ∩ I n ) r i = rK r +1 / ( K r +1 ∩ d − ( I n )) i = r + 10 i = r, r + 1 . Since d : K r +1 → K r is an isomorphism, so is d : K r +1 ∩ d − I n → ( K ∩ I n ) r , which meansthat H ∗ ( K/K ∩ I ( n ) ′ ) = 0, so A ( n ′ ) → B/J n is a weak equivalence.Thus it only remains to show that the pro-objects { A/I n } n and { A/I ( n ) ′ } n are isomor-phic. Since I n ⊂ I ( n ) ′ , there is an obvious morphism A/I n → A/I ( n ) ′ , and it remains toconstruct an inverse in the pro-category, Observe that A is a Noetherian ring, and that( I r ) r and K r are finitely generated A -modules.Now, ( K ∩ I n ) r = K r ∩ I n − r ( I r ) r for all n ≥ r . By the Artin–Rees Lemma ([Mat]Theorem 8.5), there exists some c ≥ r such that for all n ≥ c , this is I n − c ( K r ∩ I c − r ( I r ) r ) = I n − c ( K r ∩ ( I c ) r ) . Thus K r +1 ∩ d − ( I n ) is just I n − c K r +1 ∩ d − ( I c ). Therefore I ( n ) ′ ⊂ I n − c , so giving maps A/I ( n ) ′ → A/I n − c , and hence the required inverse in the pro-category. (cid:3) A nilpotent representability theorem.
Let d N ♭R (or simply d N ♭ ) be either of thecategories s N ♭R or dg + N ♭R . Remark . Note that the constructions of § d N ♭R , sincethey are closed under fibre products. Lemma 2.15.
Given a weak equivalence f : A → B between fibrant objects in a rightproper model category C , there exists a diagram BA i / / C g > > ⑦⑦⑦⑦⑦⑦⑦⑦ g ❅❅❅❅❅❅❅❅ A , such that g , g are trivial fibrations, g ◦ i = f and g ◦ i = id .Proof. Let C := A × f,B, ev B I , for B I the path object of B , and let g be given byprojection onto A . The projection C → B I is the pullback of A → B along the fibration B I → B , so is a weak equivalence by right properness. Define g to be the compositionof this with the trivial fibration ev : B I → B . The projection g is the pullback of thetrivial fibration ev : B I → B along f , so is a trivial fibration.It only remains to show that g is a fibration. Since B I → B × B is a fibration, pullingback along f shows that ( g , g ) : C → A × B is a fibration, and since A is fibrant, wededuce that A × B → B is a fibration, so g must be a fibration. (cid:3) EPRESENTABILITY OF DERIVED STACKS 19
Lemma 2.16.
If a homotopy-preserving functor F : d N ♭R → S is homotopy-homogeneous,then it is almost of finite presentation if and only if the following hold: (1) the functor π F : Alg π R → S preserves filtered colimits; (2) for all finitely generated A ∈ Alg π R and all x ∈ F ( A ) , the functors D ix ( F, − ) :Mod A → Ab preserve filtered colimits for all i > .Proof. This is essentially the same as Theorem 1.23 — we need only show that any square-zero extension A → B in s N ♭R (resp. dg + N ♭R ) is of the form A = B × B ⊕ M ˜ B , for ˜ B → B a weak equivalence, and some derivation B → M . Now just note that such an expressionis constructed in the proof of Lemma 1.4. (cid:3) Theorem 2.17.
Let R be a derived G-ring admitting a dualising module (in the senseof [Lur] Definition 3.6.1) and take a functor F : d N ♭R → S . Then F is the restriction ofan almost finitely presented geometric derived n -stack F ′ : d Alg R → S if and only if thefollowing conditions hold (1) F maps tiny acyclic extensions to weak equivalences. (2) For all discrete rings A , F ( A ) is n -truncated, i.e. π i F ( A ) = 0 for all i > n . (3) F is homotopy-homogeneous, i.e. for all square-zero extensions A ։ C and allmaps B → C , the map F ( A × C B ) → F ( A ) × hF ( C ) F ( B ) is an equivalence. (4) π F : Alg π R → S is a hypersheaf for the ´etale topology. (5) π π F : Alg π R → Set preserves filtered colimits. (6)
For all A ∈ Alg π R and all x ∈ F ( A ) , the functors π i ( π F, x ) : Alg A → Set preserve filtered colimits for all i > . (7) for all finitely generated integral domains A ∈ Alg π R , all x ∈ F ( A ) and all ´etalemorphisms 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 R and all x ∈ F ( A ) , the functors D ix ( F, − ) :Mod A → Ab preserve filtered colimits for all i > . (9) for all finitely generated integral domains A ∈ Alg π R and all x ∈ F ( A ) , thegroups D ix ( F, A ) are all finitely generated A -modules. (10) for all complete discrete local Noetherian π R -algebras A , with maximal ideal m ,the map π F ( A ) → lim ←− h F ( A/ m r ) is a weak equivalence (see Remark 1.35 for a reformulation).Moreover, F ′ is uniquely determined by F (up to weak equivalence).Proof. We will deal with the simplicial case. Since normalisation gives an equivalence N : s N ♭R → dg + N ♭R when R is a Q -algebra, the dg case is entirely similar.First observe that F extends to a functor ˆ F : pro( s N ♭R ) → S , given by ˆ F ( { A ( i ) } i ∈ I ) =lim ←− hi ∈ I F ( A ( i ) ).Define F ′ as follows. For any A ∈ s Alg R , write A = lim −→ A α , for A α ∈ F Gs
Alg R , andset F ′ ( A ) := lim ←− k h lim −→ α ˆ F ( { P k A α /I nA α } n ∈ N ) . We first show that F ′ is homotopy-preserving; it follows from Lemma 2.11 and the proofof Proposition 2.7 that F is homotopy-preserving. Note that the formula for F ′ definesa functor F ′′ on ind( \ F Gs
Alg R ), and that F ′ is the composition of F ′′ with the derived left Quillen functor of Proposition 2.7. By the proof of Lemma 2.15, it suffices to showthat F ′′ maps trivial fibrations to weak equivalences. Any such morphism is isomorphicto one of the form { A α } α → { B α } α , where each A α → B α is a surjective weak equivalencein \ F Gs
Alg R . Note that P k A α → P k B α is also a surjective weak equivalence, so we mayapply Lemma 2.13, which implies thatˆ F ( { P k A α /I nA α } n ∈ N ) → ˆ F ( { P k B α /I nB α } n ∈ N )is a weak equivalence, since F is homotopy-preserving. Thus F ′′ (and hence F ′ ) ishomotopy-preserving.If A ∈ s N ♭R , note that F ′ ( A ) = lim −→ α F ( A α ) ≃ F ( A ) , by nilpotence and almost finite presentation, respectively, noting that as in the proof ofTheorem 1.34, conditions (5), (6) and (8) ensure almost finite presentation of F . Thus F ≃ F ′ | s N ♭R ; in particular, this ensures that D ix (( F ′ ) , M ) ∼ = D ix ( F, M ).Since P k A = lim −→ P k A α (for A α as above), it follows immediately that F is nilcomplete.Likewise, π F automatically preserves filtered colimits, as do the functors D ix ( F, − ) :Mod A → Ab. Therefore F ′ satisfies the conditions of Corollary 1.36.Finally, it remains to show that F ′ is uniquely determined by F . Assume that wehave some geometric derived stack G : s Alg R → S , almost of finite presentation, with G | s N ♭R ≃ F . Then, since G is nilcomplete and almost of finite presentation, we must have G ( A ) ≃ lim ←− k h G ( P k A ) ≃ lim ←− k h lim −→ α G ( P k A α ) ≃ lim ←− k h lim −→ α G ( P k ˆ A α ) , where we write A = lim −→ α A α as a filtered colimit of finitely generated subalgebras, and thefinal isomorphism comes from the weak equivalence A α → ˆ A α of [Pri3] Theorem 6.6.Now, if we take an inverse system { B i } i in s Alg in which the morphisms B i → B j induce isomorphisms π B i → π B j , then G (lim ←− h B i ) ≃ lim ←− h G ( B i ) (as G is a geometricderived stack, so has an atlas as in [Pri3] Theorem 4.7). In particular, G ( P k ˆ A α ) = G (lim ←− n P k ˆ A α / ( I nA α )) ≃ lim ←− n h G ( P k ˆ A α / ( I nA α ))= lim ←− n h F ( P k ˆ A α / ( I nA α ))= ˆ F ( P k ˆ A α ) . Thus G ( A ) ≃ lim ←− k h lim −→ α ˆ F ( P k ˆ A α ) , as required. (cid:3) Remark . Note that if we replace d N ♭R with s N R or dg + N R , then the theorem remainstrue, provided we impose the additional condition that F be nilcomplete, in the sense thatfor all A , the map F ( A ) → lim ←− hk F ( P k A ) is a weak equivalence. EPRESENTABILITY OF DERIVED STACKS 21
Covers.
We end this section with a criterion which allows us to verify the key rep-resentability properties on formally ´etale covers.
Definition 2.19.
A transformation α : F → G of functors F, G : d N ♭ → S is said to behomotopy formally ´etale if for all square-zero extensions A → B , the map F ( A ) → F ( B ) × hG ( B ) G ( A )is an equivalence. Proposition 2.20.
Let α : F → G be a homotopy formally ´etale morphism of functors F, G : d N ♭ → S . If G is homotopy-homogeneous (resp. homotopy-preserving), then so is F . Conversely, if α is surjective (in the sense that π F ( A ) ։ π G ( A ) for all A ) and F ishomotopy-homogeneous (resp. homotopy-preserving), then so is G .Proof. Take a square-zero extension A → B , and a morphism C → B , noting that A × B C → C is then another square-zero extension. Since α is homotopy formally ´etale, F ( A × B C ) ≃ G ( A × B C ) × hGC F CF A × hF B F C ≃ [ G ( A ) × hG ( B ) F ( B )] × hF B F C ≃ G ( A ) × hG ( B ) F C ≃ ( GA × hGB GC ) × hGC F C.
Thus homogeneity of G implies homogeneity of F , and if π F C → π GC is surjective forall C , then homogeneity of F implies homogeneity of G .Now take a tiny acyclic extension A → B in d N ♭ . Since α is homotopy formally ´etale, F ( A ) ≃ G ( A ) × hG ( B ) F ( B ) , so if G is homotopy-preserving, then F maps tiny acyclic extensions to weak equivalences.By Lemma 2.11 and the proof of Lemma 2.15, this implies that F is homotopy-preserving.If π F ( B ) → π G ( B ) is surjective for all B , then the converse holds. (cid:3) Pre-representability
Simplicial structures.Definition 3.1.
Define simplicial structures (in the sense of [GJ] Definition II.2.1) on s Alg R and ind( \ F Gs
Alg R ) as follows. For A ∈ s Alg R and K ∈ S , A K is defined by( A K ) n := Hom S ( K × ∆ n , A ) . Then for A ∈ ind( \ F Gs
Alg R ), A K is uniquely determined via Lemma 2.8 by the propertythat U ( A K ) = ( U A ) K .Spaces Hom( A, B ) ∈ S of morphisms are then given byHom( A, B ) n := Hom( A, B ∆ n ) . We need to check that this is well-defined:
Lemma 3.2.
For A ∈ ind( \ F Gs
Alg R ) and K ∈ S , we have A K ∈ ind( \ F Gs
Alg R ) . More-over, if A → π A is a nilpotent extension, then so is A K → π ( A K ) .Proof. A K can be expressed as the limitlim ←− (∆ n f −→ K ) ∈ ∆ ↓ K A ∆ n ;since the inclusion functor U : ind( \ F Gs
Alg R ) → s Alg R is a right adjoint, it preservesarbitrary limits, so it suffices to show that A ∆ n ∈ ind( \ F Gs
Alg R ). Write A := lim −→ α A α , for A α ∈ \ F Gs
Alg R . Since ∆ n is finite, we have A ∆ n = lim −→ α A ∆ n α ,so we may assume that A ∈ \ F Gs
Alg R .The exact sequence 0 → I A → A → π A → → I ∆ n A → A ∆ n → π A → π A ) ∆ n = π A , since ∆ n is connected). Since ∆ n is contractible, π ( I ∆ n A ) = π ( I A ) = 0, so I A ∆ n = I ∆ n A . Hencelim ←− m ( A ∆ n /I mA ∆ n ) = lim ←− m ( A ∆ n / ( I ∆ n A ) m ) = lim ←− ( A/I mA ) ∆ n = A ∆ n , so A ∆ n ∈ \ F Gs
Alg R .Finally, if I mA = 0, then ( I ∆ n A ) m = 0, so I mA ∆ n = 0 for all n , and hence I mA K = 0 for all K ∈ S . (cid:3) In fact, this makes ind( \ F Gs
Alg R ) into a simplicial model category in the sense of [GJ]Ch. II (with U : ind( \ F Gs
Alg R ) → s Alg R becoming a simplicial right Quillen equivalence).Although the same is not true for dg + Alg R or ind( \ F Gdg + Alg R ), we now show that theycarry compatible weak simplicial structures. Definition 3.3.
Explicitly, we say that a model category C has a weak simplicial structureif we have the following data:(1) a functor Hom C : C opp × C → S such that Hom C ( A, B ) = Hom C ( A, B ).(2) a functor ( f S ) opp × C → C (where f S is the category of finite simplicial sets),denoted by ( K, B ) B K , with natural isomorphismsHom C ( A, B K ) ∼ = Hom S ( K, Hom C ( A, B )) . These must satisfy the property (known as SM7) that if i : A → B is a cofibration in C ,and p : X → Y a fibration, thenHom C ( B, X ) → Hom C ( A, X ) × Hom C ( A,Y ) Hom C ( B, Y )is a fibration in S which is trivial whenever either i or p is a weak equivalence.This means that C satisfies all of the axioms of a simplicial model category from [GJ]Ch. II except for conditions (2) and (3) of Definition II.2.1 (which require that for allobjects A ∈ C , the functors Hom C ( A, − ) : C → S and Hom C ( − , A ) : C opp → S have leftadjoints).Note that this is enough to ensure that C is still a simplicial model category in the senseof [Qui]. Lemma 3.4.
The model categories dg + Alg R and ind( \ F Gdg + Alg R ) carry weak simplicialstructures.Proof. First set Ω n = Ω(∆ n ) to be the cochain algebra Q [ t , t , . . . , t n , dt , dt , . . . , dt n ] / ( X t i − , X dt i )of rational differential forms on the n -simplex ∆ n . These fit together to form a simplicialcomplex Ω • of DG-algebras, and we define A ∆ n as the good truncation A ∆ n := τ ≥ ( A ⊗ Ω n ).Note that this construction only commutes with finite limits, so only extends to define A K for finite simplicial sets K , and does not have a left adjoint.For A ∈ \ F Gdg + Alg R , we replace A K with its completion over H ( A K ), and extend thisconstruction to ind( \ F Gdg + Alg R ) in the obvious way.That these have the required properties follows because the matching maps Ω n → M n Ω = Ω( ∂ ∆ n ) are surjective. Explicitly, M n Ω ∼ = Ω n / ( t · · · t n , X i t · · · t i − ( dt i ) t i +1 · · · t n ) . EPRESENTABILITY OF DERIVED STACKS 23 (cid:3)
Definition 3.5.
Although the categories s N ♭R and dg + N ♭R are not model categories, weendow them with weak simplicial structures inherited from s Alg R and dg + Alg R , respec-tively. The key observation is that for K ∈ f S and A ∈ d N ♭ , the object A K lies in d N ♭ .3.2. Deriving functors.Definition 3.6.
Given a functor F : d N ♭ → S , we define a functor F : d N ♭ → s S to thecategory of bisimplicial sets by F ( A ) n := F ( A ∆ n ) . For a functor F : C →
Set, we will abuse notation by also writing F : C → S for thecomposition C F −→ Set → S . Proposition 3.7. If F : d N ♭ → S is homotopy-homogeneous, then for A → B an acycliclittle extension in d N ♭ and K ∈ S finite, the map F ( A K ) → ( M hK F ( A )) × h ( M hK F ( B )) F ( B K ) is a weak equivalence in S , where M hK denotes the Reedy homotopy K -matching object.Proof. We prove this by induction on the dimension of K . If K is dimension 0 (i.e.discrete), then the map is automatically an equivalence, as M hK F ( A ) = F ( A ) K = F ( A K ) . Now assume the statement holds for all finite simplicial sets of dimension < n , take K of dimension n , and let K ′ := sk n − K , the ( n − ∂ ∆ n × N n K ) ⊔ (∆ n × L n K ) −−−−→ ∆ n × K n y y K ′ −−−−→ K, where L n K is the n th latching object and N n K = K n − L n K . Hence we have a pullbacksquare A K −−−−→ B K × B K ′ A K ′ y y B K × B (∆ n × Kn ) A (∆ n × K n ) −−−−→ B K × [ B ( ∂ ∆ n × NnK ) × B (∆ n × LnK ) ] [ A ( ∂ ∆ n × N n K ) × A (∆ n × L n K ) ] . Now, since A → B is an acyclic little extension, the map A ∆ n → A ∂ ∆ n × B ∂ ∆ n B ∆ n is asquare-zero extension, so the bottom map in the diagram above is a square-zero extension,giving a homotopy pullback square F ( A K ) −−−−→ F ( B K ) × hF ( B K ′ ) F ( A K ′ ) y y F ( B K ) × hF ( B ∆ n ) Kn F ( A ∆ n ) K n −−−−→ F ( B K ) × h [ F ( B ∂ ∆ n ) NnK × F ( B ∆ n ) LnK ] [ F ( A ∂ ∆ n ) N n K × F ( A ∆ n ) L n K ] . Here, the top right isomorphism comes from A K ′ ։ B K ′ , the bottom left from A ∆ n ։ B ∆ n , and the bottom right from A ∆ n ։ B ∆ n and from A ∂ ∆ n ։ B ∂ ∆ n ; these are allsquare-zero extensions and F is homotopy-homogeneous. By induction (using F ( A K ′ ) ≃ ( M hK ′ F ( A )) × h ( M hK ′ F ( B )) F ( B K ′ ) and F ( A ∂ ∆ n ) ≃ ( M hn F ( A )) × ( M hn F ( B )) F ( B ∂ ∆ n )), we can rewrite this as saying that the following square isa homotopy pullback F ( A K ) −−−−→ F ( B K ) × hM hK ′ F ( B ) M hK ′ F ( A ) y y F ( B K ) × hF ( B ∆ n ) Kn F ( A ∆ n ) K n −−−−→ F ( B K ) × [ M hn F ( B ) NnK × F ( B ∆ n ) LnK ] [ M hn F ( A ) N n K × F ( A ∆ n ) L n K ] . Now just observe that this pullback defines F ( B K ) × M hK F ( B ) M hK F ( A ), as required. (cid:3) Definition 3.8.
Say that a functor F : d N ♭ → S is homotopy-surjecting if for all tinyacyclic extensions A → B , the map π F ( A ) → π F ( B )is surjective. Definition 3.9.
Define ¯ W : s S → S to be the right adjoint to Illusie’s total Dec functorgiven by DEC ( X ) mn = X m + n +1 . Explicitly,¯ W p ( X ) = { ( x , x , . . . , x p ) ∈ p Y i =0 X i,p − i | ∂ v x i = ∂ hi +1 x i +1 , ∀ ≤ i < p } with operations ∂ i ( x , . . . , x p ) = ( ∂ vi x , ∂ vi − x , . . . , ∂ v x i − , ∂ hi x i +1 , ∂ hi x i +2 , . . . , ∂ hi x p ) ,σ i ( x , . . . , x p ) = ( σ vi x , σ vi − x , . . . , σ v x i , σ hi x i , σ hi x i +1 , . . . , σ hi x p ) . In [CR], it is established that the canonical natural transformationdiag X → ¯ W X from the diagonal is a weak equivalence for all X . Corollary 3.10.
If a homotopy-homogeneous functor F : d N → S is homotopy-surjecting,then the functor ¯ W F : d N → S is homotopy-preserving.Proof. Consider the homotopy matching maps (for the Reedy model structure on bisim-plicial sets) F ( A ) n → F ( B ) n × M h∂ ∆ n F ( B ) M h∂ ∆ n F ( A )of F ( A ) → F ( B ) , for an acyclic little extension A → B . By Lemma 2.11, we may replace tiny acyclicextensions with little acyclic extensions in the definition of homotopy-surjections.By Proposition 3.7, the map above is weakly equivalent to F ( A ′ ) → F ( B ′ ) , where A ′ = A ∆ n , B ′ = B ∆ n × B ∂ ∆ n A ∂ ∆ n . Now, A ′ → B ′ is a little acyclic extension, sothe homotopy matching maps of F ( A ) → F ( B ) are surjective on π (as α is homotopy-surjecting).For any Reedy fibrant replacement f : R → F ( B ) of F ( A ) → F ( B ), the homotopymatching maps must also be surjective on π . However, for Reedy fibrations, matchingobjects model homotopy matching objects, so f is a Reedy surjective fibration, and hence ahorizontal levelwise trivial fibration (the matching maps being surjective). It is therefore adiagonal weak equivalence by [GJ] Proposition IV.1.7, and [CR] then shows that ¯ W f is alsoa weak equivalence. Lemma 2.15 then implies that ¯
W F preserves all weak equivalences. (cid:3)
EPRESENTABILITY OF DERIVED STACKS 25
Proposition 3.11. If F : d N ♭ → S is homotopy-homogeneous, then for A → B a littleextension in d N ♭ and K a contractible finite simplicial set, the map F ( A K ) → ( M hK F ( A )) × h ( M hK F ( B )) F ( B K ) is a weak equivalence in S .Proof. We adapt the proof of Proposition 3.7, proceeding by induction on the dimensionof K . If K is of dimension 0, the statement is automatically true.For any contractible finite simplicial set K , any morphism ∆ → K can be expressed asan iterated pushout of anodyne extensions Λ m,k → ∆ m . In particular, if K has dimension n , there is a contractible simplicial set K ′ ⊂ K of dimension n −
1, with the map K ′ → K an iterated pushout of the maps Λ n,k → ∆ n for various k . The proposition holds byinduction for K ′ and Λ n,k , and is automatically satisfied by ∆ n .Since the map A ∆ n → B ∆ n × B Λ n,k A Λ n ,k is an acyclic little extension, the proof ofProposition 3.7 adapts to show that the proposition is satisfied by K , as required. (cid:3) Corollary 3.12.
If a homotopy-homogeneous functor F : d N → S is homotopy-surjecting,then the functor ¯ W F : d N → S is homotopy-homogeneous.Proof. Take a square-zero little extension A → B ; by Proposition 3.11, the relative homo-topy partial matching object M h Λ n,k F ( A ) × hM h Λ n,k F ( B ) F ( B ) n is F ( A Λ n,k ) × hF ( B Λ n,k ) F ( B ∆ n ) . Since A ∆ n → A Λ n,k × B Λ n,k B ∆ n is an acyclic little extension, homotopy-surjectivity of F thus implies that the homotopy partial matching map F ( A ) n → M h Λ n,k F ( A ) × hM h Λ n,k F ( B ) F ( B ) n gives a surjection on π .If we take a Reedy fibrant replacement R for F ( A ) over F ( B ), this says that R n → M Λ n,k R × M Λ n,k F ( B ) F ( B ) n is surjective on π — since it is (automatically) a fibration, this implies that it is surjectivelevelwise.Thus f : R → F ( B ) is a Reedy fibration and a horizontal levelwise Kan fibration, so[GJ] Lemma IV.4.8 implies that diag f is a fibration, so for any map C → B ,(diag F ( A )) × h (diag F ( B )) (diag F ( C )) ≃ (diag R ) × (diag F ( B )) (diag F ( C ))= diag ( R × ( F ( B ) F ( C )) ≃ diag ( F ( A ) × hF ( B ) F ( C )) ≃ diag F ( A × B C ) , the penultimate equivalence following because R → F ( B ) is a Reedy fibrant replacementfor F ( A ), and the final one because F is homotopy-homogeneous and A → B is square-zero.Finally, [CR] shows that ¯ W X and diag X are weakly equivalent for all X , so( ¯ W F ( A )) × h ( ¯ W F ( B )) ( ¯ W F ( C )) ≃ ¯ W F ( A × B C ) . Any square-zero extension A → B in d N with kernel K can be expressed as thecomposition of the little extensions A/ ( I n +1 A K ) → A/ ( I nA K ), making ¯ W F homotopy-homogeneous. (cid:3)
Lemma 3.13.
For a homotopy-preserving functor F : d N → S , the natural transformation F → ¯ W F is a weak equivalence.Proof.
The transformation comes from applying ¯ W to the maps F ( A ) → F ( A ) of bisim-plicial sets coming from the canonical maps A → A ∆ n .Since A → A ∆ n is a weak equivalence, the maps F ( A ) → F ( A ) are also weak equiva-lences levelwise, so F = ¯ W F → ¯ W F is a weak equivalence (as ¯ W sends levelwise weakequivalences to weak equivalences). (cid:3) Representability.Definition 3.14.
Given a homotopy-surjecting homotopy-homogeneous functor F : d N ♭R → S , A ∈ Alg H R , x ∈ F ( A ), and an A -module M , define D ix ( F, M ) as follows.For i ≤
0, set D ix ( F, M ) := π − i ( F ( A ⊕ M ) × hF ( A ) { x } ) . For i >
0, setD ix ( F, M ) := π ( F ( A ⊕ M [ − i ]) × hF ( A ) { x } ) /π ( F ( A ⊕ cone( M )[1 − i ]) × hF ( A ) { x } ) . Note that homotopy-homogeneity of F ensures that these are abelian groups for all i ,and that the multiplicative action of A on M gives them the structure of A -modules. Lemma 3.15.
For all
F, A, M as above, there are canonical isomorphisms D ix ( F, M ) ∼ = D ix ( ¯ W F , M ) , where the group on the left-hand side is defined as in Definition 3.14, and that on the rightas in Definition 1.13.In particular, if F is homotopy-preserving, then Definitions 3.14 and 1.13 are consistent.Proof. We begin by noting that ¯
W F is indeed homotopy-preserving and homotopy-homogeneous, by Corollaries 3.10 and 3.12. Since F ( A ) = ¯ W F ( A ) for all A ∈ Alg H R , itfollows immediately that D ix ( F, M ) ∼ = D ix ( ¯ W F , M ) for all i ≤
0. Now for i > ix ( ¯ W F , M ) = π ( ¯ W ( T x ( F /R )( M [ − i ]))= π ( T x ( F/R )( M [ − i ])) /π ( F (( A ⊕ M [ − i ]) ∆ × A ∆1 A ) × hF ( A ) { x } )= π ( T x ( F/R )( M [ − i ])) /π ( T x ( F/R )( M [ − i ] ∆ )) , where the quotient is taken by the map ∂ − ∂ coming from the projections M ∆ → M . If d N ♭R = s N ♭R , then M [ − i ] ∆ = M [ − i ] ⊕ cone( M )[1 − i ], so D ix ( ¯ W F , M ) ∼ = D ix ( F, M ). When d N ♭R = dg + N ♭R , we have more work to do. In this case, M [ − i ] ∆ = τ ≥ ( M [ − i ] ⊗ Ω n ). Thekey observation to make is that M [ − i ] ⊕ cone( M )[1 − i ] can be expressed as a retract of τ ≥ ( M [ − i ] ⊗ Ω n ) over M ⊕ M given by m ⊗ ( m, m ⊗ x n (0 , m ) for n >
0, and m ⊗ x n dx (0 , dm/ ( n + 1)). Thus( ∂ − ∂ ) : π ( T x ( F/R )( M [ − i ] ∆ ) → π ( T x ( F/R )( M [ − i ]))has the same image as π ( T x ( F/R )(cone( M )[1 − i ]) → π ( T x ( F/R )( M [ − i ])), soD ix ( ¯ W F , M ) ∼ = D ix ( F, M ).Finally, if F is homotopy-preserving, then Lemma 3.13 shows that the map F → ¯ W F is a weak equivalence, making the definitions consistent. (cid:3)
Theorem 3.16.
Let R be a derived G-ring admitting a dualising module (in the sense of [Lur] Definition 3.6.1) and take a functor F : d N ♭R → S satisfying the following conditions. (1) F is homotopy-surjecting, i.e. it maps tiny acyclic extensions to surjections (on π ). (2) For all discrete rings A , F ( A ) is n -truncated, i.e. π i F ( A ) = 0 for all i > n . EPRESENTABILITY OF DERIVED STACKS 27 (3) F is homotopy-homogeneous, i.e. for all square-zero extensions A ։ C and allmaps B → C , the map F ( A × C B ) → F ( A ) × hF ( C ) F ( B ) is an equivalence. (4) π F : Alg H R → S is a hypersheaf for the ´etale topology. (5) π π F : Alg H R → Set preserves filtered colimits. (6)
For all A ∈ Alg H R and all x ∈ F ( A ) , the functors π i ( π F, x ) : Alg A → Set preserve filtered colimits for all i > . (7) for all finitely generated integral domains A ∈ Alg H R , all x ∈ F ( A ) and all ´etalemorphisms f : A → A ′ , the maps D ∗ x ( F, A ) ⊗ A A ′ → D ∗ fx ( F, A ′ ) are isomorphisms. (8) for all finitely generated A ∈ Alg H R and all x ∈ F ( A ) , the functors D ix ( F, − ) :Mod A → Ab preserve filtered colimits for all i > . (9) for all finitely generated integral domains A ∈ Alg H R and all x ∈ F ( A ) , thegroups D ix ( F, A ) are all finitely generated A -modules. (10) for all complete discrete local Noetherian H R -algebras A , with maximal ideal m ,the map π F ( A ) → lim ←− h F ( A/ m r ) is a weak equivalence.Then ¯ W F is the restriction to d N ♭R of a geometric derived n -stack F ′ : s Alg R → S (resp. F ′ : dg + Alg R → S ), which is almost of finite presentation. Moreover, F ′ is uniquelydetermined by F (up to weak equivalence).Proof. By Corollaries 3.10 and 3.12, ¯
W F is homotopy-preserving and homotopy-homogeneous. Since π F = π F , the map π F → π ¯ W F is a weak equivalence. Lemma3.15 then shows that D ix ( F, M ) ∼ = D ix ( ¯ W F , M ), so ¯
W F satisfies all the conditions ofTheorem 2.17. (cid:3)
Example . If X is a dg manifold (in the sense of [CFK1]), then the functor X : dg + N ♭R → Set given by X ( A ) = Hom(Spec A, X ) satisfies the conditions of Theorem 3.16,so X : dg + N ♭R → S is a geometric derived 0-stack.In fact, X is just the hypersheafification of X . This follows because X is a geometricderived 0-stack, so X ♯ = X , and there is thus a map f : X ♯ → X . Since X ♯ is ageometric derived 0-stack (as can be shown for instance by observing that it is equivalentto the derived stack Gpd( X ) ♯ of [Pri3] § f must be anequivalence.This example will be adapted further in [Pri2], constructing geometric derived n -stacksfrom DG Lie algebras similar to those used in [CFK2] and [CFK1]. References [Art] M. Artin. Versal deformations and algebraic stacks.
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