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DEFORMATIONS OF SHEAVES OF ALGEBRAS

VLADIMIR HINICH

arXiv:math/0310116v2 [math.AG] 12 Oct 2003

Abstract. A construction of the tangent dg Lie algebra of a sheaf of operad algebras on a site is presented. The requirements on the site are very mild; the requirements on the algebra are more substantial. A few applications including the description of deformations of a scheme and equivariant deformations are considered. The construction is based upon a model structure on the category of presheaves which should be of an independent interest.

0. Introduction 0.1. In this paper we study formal deformations of sheaves of algebras. The most obvious (and very important) example is that of deformations of a scheme X over a ?eld k of characteristic zero. In two di?erent cases, the ?rst when X is smooth, and the second when X is a?ne, the description is well-known. In both cases there is a di?erential graded (dg) Lie algebra TX over k such that formal deformations of X over the artinian local base (R, m) are described by the Maurer-Cartan elements of m ? TX , modulo a gauge equivalence. It is well-understood now that formal deformations over a ?eld of characteristic zero are governed by a di?erential graded Lie algebra. One of possible explanations of this phenomenon was suggested in [H3]: we expect deformation problems to have formal moduli (which is expected to be a “commutative” formal dg scheme). Then the representing dg Lie algebra corresponds to the formal moduli by Koszul (or bar-cobar) duality. Thus, the existence of dg Lie algebra governing deformations is equivalent to the representability (in “higher”, dg sense) of the deformation problem. However, in the two cases mentioned above (X smooth and X a?ne) the governing dg Lie algebra TX appears in seemingly di?erent ways. This can be shortly described as follows. 0.1.1. X is smooth. A?ne smooth scheme X has no formal deformations. Its trivial deformation UR with an artinian local base (R, m) admits the automorphism group exp(m?TX ) which is nothing but the value at R of the formal group corresponding to TX = Γ(X, TX ). Descent theorem of [H1] asserts in this situation that for a general smooth scheme X the dg Lie algebra TX governing the deformations of X can be calculated by the formula TX = RΓ(X, TX ).

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0.1.2. X is a?ne. Let X = Spec(A) for a commutative k-algebra A. It is convenient to consider A as a dg commutative k-algebra concentrated at degree zero. Then the deformation theory of dg algebras [H4] suggests the following recipe of calculation of TX . Let P → A be a co?brant (some call it free or semi-free) resolution of A in the model category of commutative dg k-algebras. Deformations of A and of P are equivalent; deformations of P appear as perturbations of the di?erential which are described by the Maurer-Cartan elements of the Lie algebra of derivations of P . Thus, one has TX = Der(P, P ). 0.1.3. We wish to describe in a similar way deformations of a sheaf of algebras. The ?rst problem seems to be the lack of co?brant resolutions for sheaves of algebras. This turns out to have a very pleasant solution: the category of complexes of presheaves admits a model category structure describing the homotopy theory of complexes of sheaves. A similar model category structure exists for sheaves of operad algebras in characteristic zero. This allows us to de?ne deformation functor and to construct the corresponding dg Lie algebra in a way similar to the one described in 0.1.2. The construction is local, so, as a result, we obtain a presheaf of dg Lie algebras. We use the construction mentioned in 0.1.1 to get a global dg Lie algebra. 0.2. Sheaves vs presheaves. Let X be a site and F : X op → A be a presheaf on X with values in a category A having a notion of weak equivalence (for instance, complexes, simplicial sets, categories or polycategories). The notion of sheaf is not very appropriate here: we know this well, for instance, in the case A = Cat. This was probably the reason Jardine [Ja] suggested a model category structure on the category of simplicial presheaves. The idea was to extend the notion of weak equivalence so that a presheaf will be weakly equivalent to its shea??cation. Then the localization of the category of simplicial sheaves with respect to the weak equivalences can be described as the homotopy category of the category of presheaves. We adopt a similar point of view. We need a model category structure on presheaves of algebras which would allow us to construct “semi-free resolutions”. This model category structure is based upon a model category structure on the category C(Xk ) of complexes of presheaves of k-modules which is described by the following result. 0.2.1. Theorem. Let X be a site, k a ring and let C(Xk ) denote the category of complexes of presheaves of k-modules on X. 1. The category C(Xk ) admits a model category structure so that ? weak equivalences are maps f : M → N inducing a quasi-isomorphisms f a : M a → N a of shea??cations. ? co?brations are generated by maps f : M → M x; dx = z ∈ M(U) coresponding to adding a section to kill a cycle z over an object U ∈ X.

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2. A map f : M → N in the above model category structure is a ?bration i? f (U) : M(U) → N(U) is surjective for any U ∈ X and for any hypercover ? : V? → U of U ∈ X the corresponding commutative diagram

E

M(U)

ˇ C(V? , M)

c

N(U) is homotopy cartesian.

E

c

ˇ C(V? , N)

ˇ We remind the notion of hypercover in 1.2. Cech complex C(V? , M) of M with respect to a hypercover V? is de?ned as the total complex corresponding to the cosimplicial complex n → M(Vn ), see 1.3.8. Notice that we do not require the existence of limits in the site X. This is important for us since we want to be able to apply this to the category of a?ne open subsets of a scheme which usually does not admit a ?nal object. 0.2.2. The proof of Theorem 0.2.1 is given in 1.3.2. It is based on an explicit description of generating acyclic co?brations. Recently (see ??) we learned that the model structure described above (at least part 1 of Theorem 0.2.1) is known to specialists, see, for instance, [T], Appendix C. We decided, however, to present our proof since it is direct, general, and gives an explicit description of ?brations which we need in any case. A similar model category structure was used in [HS] for n-stacks. We also present in Appendix A a version of Theorem 0.2.1 for simplicial presheaves. This model category structure on simplicial presheaves has the same weak equivalences as Jardine’s [Ja] but the co?brations are generated by gluing cells and ?brations have a similar description using hypercovers. Under some mild restrictions on the X (any hypercover can be re?ned by a split hypercover) our model structure coincides with a one recently de?ned in [DHI], Theorem 1.3. 0.3. Higher deformation functor. Classical formal deformation functors can be usually described as follows. Let art(k) be the category of artinian local k-algebras (R, m) with residue ?eld k. Let C(R) for each R ∈ art(k) denote a groupoid of “objects over R”, so that for each map f : R → S in art(k) a base change functor α? : C(R) → C(S) is de?ned.

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Then the groupoid of formal deformations of an object A ∈ C(k) over R ∈ art(k) is de?ned as the ?ber of π ? : C(R) → C(k) at A. In higher deformation theory one extends the category art(k) allowing artinian algebras which are not necessarily concentrated at degree zero. In this paper we work with the category dgart≤0 (k) of non-positively graded di?erential artinian algebras, see [H3] for the explanation. One cannot expect that the deformation functor extended in this way has values in the category of groupoids: one should expect a higher version of groupoid appearing here. We use simplicial groupoids (or simplicial weak groupoids which is the same from the homotopical point of view) as a higher version of groupoid. Our higher formal deformation functors can be described as follows. Let dgart≤0 (k) be the category of non-positively graded commutative artinian local dg algebras over k with residue ?eld k. Let C(R) for each R ∈ dgart≤0 (k) denote a category of “objects over R”. We suppose there is a subcategory W(R) of weak equivalences in C(R). Let W(R) be the full Dwyer-Kan (hammock) localization of W(R) (see A.2 for the details). This is a simplicial groupoid and we de?ne the simplicial groupoid of formal deformations of A ∈ C(k) as the homotopy ?ber of the map W(R) → W(k) at A. For the description of deformations of sheaves of algebras we take C(R) to be the category of sheaves of R-algebras ?at over R. W(R) is the subcategory of quasi-isomorphisms of sheaves of algebras. 0.4. Main result. Let X be a site and let k be a ?eld of characteristic zero. Let O be an operad in the category of complexes of sheaves of k-modules on X and let A be a sheaf of O-algebras. Our main result, Theorem 3.5.5, presents (under some restrictions on X, O and A) the dg Lie algebra governing formal deformations of A. The construction goes as follows. According to Theorem 2.2.1, the category of presheaves of O-algebras admits a model structure generalizing the one de?ned in 0.2.1. Let P be a ?brant co?brant O-algebra weakly equivalent to A. Then the presheaf of derivations of P , TA := Der ? (P, P ), is a ?brant presheaf of dg Lie algebras on X. The dg Lie algebra governing deformations of A can be expressed then as holim TA . Here are the assumptions for which the result is proven. 0.4.1. Assumptions on X. ? The topos X ? admits enough points. ? The ?nal presheaf in X admits a ?nite hypercover.

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The second condition is of course ful?lled for sites admitting a ?nal object. However, in our main application X is the site of a?ne open subschemes of a scheme. In this case the condition is ful?lled for quasi-compact separated or ?nite dimensional schemes. 0.4.2. Assumptions on O. Complexes O(n) are non-positively graded. This condition does not seem to be really restrictive. Operads one encounters are usually obtained by a tensor product of a sheaf of rings (e.g, the structure sheaf) with a constant operad. 0.4.3. Restrictions on A. For each U ∈ X the cohomology H i(U, A) is supposed to vanish for i > 0. This condition means the following. Choose a ?brant resolution A → A′ of A. Then for each U ∈ X the complex A′ (U) has no positive cohomology. This is the most serious assumption. It is not in general ful?lled even in the case A is a sheaf of algebras. In fact, in this case the cohomology H i (U, A) has its usual meaning (cohomology of the sheaf A|U ) and it does not vanish in general for any U ∈ X. The situation is, however, slightly better then one could think. The reason is that once we are given a sheaf of algebras A in a topos X ? , we have a freedom in the choice of X. If X ? admits a generating family of sheaves U satisfying H i (U, A) = 0, we can choose X to be the site generated by this family. For instance, if A is a quasi-coherent sheaf on a scheme, one chooses X to be the category of a?ne open subschemes of the scheme, with the Zariski topology. 0.5. Applications. Direct application of Theorem 3.5.5 gives the following result (see 4.1.1 and 4.1.2). Let X be a scheme over a ?eld of characteristic zero. Suppose X admits a ?nite dimensional hypercover by a?ne open subschemes. Then the functor of formal deformations of X (or, more generally, of a quasicoherent operad algebra on X) is represented by a dg Lie algebra. The dg Lie algebra representing deformations of a quasicoherent sheaf of algebras, is usually di?cult to determine. Its cohomology, however, can be easily identi?ed with the Hochschild cohomology (for associative algebras), see 4.2. In a very special case of associative deformations of the structure sheaf of a smooth scheme, the tangent Lie algebra identi?es with the (shifted and truncated) complex of Hochschild cochains given by polydi?erential operators. The last application we present in this paper is to the description of equivariant deformations. Let A be a sheaf of algebras on a site X satisfying the conditions of Theorem 3.5.5, and let T be the dg Lie algebra governing the deformations of A. Suppose now that a formal group G acts on X and on A in a compatible way. Then G acts in a natural way on T and the G-equivariant deformations of A are governed by a dg Lie algebra RΓG (T ) whose i-th cohomology is H i (G, T ). This is proven in 4.4.

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0.6. Structure of the sections. In Section 1 we prove Theorem 0.2.1 describing the model category structure on the category of complexes of presheaves. We describe the functors RHom ? and RΓ using this model category structure. In Section 2 we present a model structure for the category of presheaves of operad algebras. In Section 3 we describe the deformation functor for sheaves of operad algebras on a site. Here the main Theorem 3.5.5 is proven. In Section 4 we present two examples: deformations of a scheme (or of a quasi-coherent algebra on a scheme) and equivariant deformations of a sheaf of algebras with respect to a discrete group. In Appendix A we present a necessary information about simplicial categories, Dwyer-Kan localization and its di?erent presentations for a model category. In Appendix A which is not used in the main body of the paper, we present a model category structure on the category of simplicial presheaves and provide a description of ?brations similar to that of Theorem 0.2.1. 0.7. Notation. In this paper N denotes the set of non-negative integers, ? the category of sets [n] = {0, . . . , n}, n ∈ N and of non-decreasing maps, and Ens denotes the category of sets. The category of simplicial sets is denoted ?op Ens. As well, we denote Ab the category of abelian groups, Cat the category of small categories, Grp the subcategory of groupoids. If A is an abelian category, C(A) is the category of complexes over A; if A is a tensor category, Op(A) is the category of operads in A. The notation Mod and Alg for the categories of modules and algebras is obvious. Σn is the symmetric group. 0.8. Relation to other works. This work extends the approach of [H4] to the sheaves of algebras. Both [H4] and the present work are based on an idea (which goes back to Halperin-Stashe? [HS], Schlessinger-Stashef [SchSt], Felix [F]) that deformations of an algebra can be described by perturbation of the di?erential in its free resolution. Since [HS, SchSt, F] a better understanding of the notion of deformation has been achieved, due to Drinfeld and Deligne, so that the language of obstructions is being substituted with the dg Lie algebra formulation of deformation theory. One has to mention Illusie [I] and Laudal [La] who constructed the obstruction theory for deformation of schemes, and Gerstenhaber-Schack [GS] who studied obstruction theory for presheaves of algebras. Obstruction theory for deformations of sheaves of associative algebras was studied in [G] and [Lu]. 0.9. Acknowledgements. A part of this work was made during my visits at MPIM and at IHES. I am grateful to these institutions for stimulating atmosphere and excellent working conditions. During the conference on polycategories at Nice (November, 2001) I knew that a part of the results on model category structures described here is known to specialists. I am very grateful to the organizers of the conference A. Hirshowitz, C. Simpson, B. Toen for the invitation.

DEFORMATIONS OF SHEAVES

7

1. Models for sheaves Let X be a site, X and X ? be the categories of presheaves (resp., sheaves) on X. If k is ? a commutative ring, Xk (resp., Xk ) denotes the category or preseaves (resp., sheaves) of k-modules on X.

? The categories Xk and Xk are tensor (=symmetric monoidal) categories. The shea??cation functor M → M a is exact and preserves the tensor product, see [SGA4], IV.12.10.

In this section we provide a model (=closed model category) structure for the category C(X ) of complexes of presheaves on X. This structure “remembers” the topology of X in a way that that the model category C(X ) becomes a powerful tool in doing homological algebra of sheaves on X. In the next section we will describe a similar model structure on the category of presheaves of algebras over a dg operad on X. 1.1. Coarse topology. 1.1.1. Theorem. The category C(Xk ) of presheaves of k-modules admits a model structure with weak equivalences de?ned as pointwise quasi-isomorphisms and ?brations as pointwise surjections. Since the category C(k) of complexes of k-modules is co?brantly generated (see [DHK], 7.4, for the de?nition of co?brantly generated model categories and [H2] for the model structure on C(k)), the result follows from the general observation of [DHK], 9.6. The CMC structure described above is also co?brantly generated. Let us recall the description of a generating collection of co?brations. Let U ∈ X. We will identify U with the presheaf represented by U. The generating co?bration given by a pair (U ∈ X, n ∈ Z) is de?ned as i : kx · U → ky · U ⊕ kz · U where n = |x| = |y|, n ? 1 = |z|, i(x) = y, dx = dy = 0, dz = y. The generating acyclic co?bration is de?ned for each (U ∈ X, n ∈ Z) as j : 0 → kx · U ⊕ ky · U

(1)

where n = |x|, n ? 1 = |y|, dx = 0, dy = x. The model category structure de?ned in 1.1.1 knows nothing about the topology of X. It corresponds to the coarse topology on X. In the general case the notion of hypercover is of a great importance.

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1.2. Hypercovers. Let us recall a few standard notions connected to hypercovers. The context presented here is slightly more general than that of [SGA4], Expos? V. e Let X be a site. Notice that we do not require that ?ber products and ?nite products exist in X.

1.2.1. An object K ∈ X is semi-representable if it is isomorphic to a coproduct of representable presheaves. A simplicial presheaf K? is called hypercover of X if (HC0) For each i ≥ 0 Ki is semi-representable. (HC1) For each n ≥ 0 the canonical map Kn+1 → (coskn (K))n+1 is a cover of presheaves (i.e. its shea??cation is surjective). (HC2) The canonical map of presheaves K0 → ? is a cover. Let L ∈ X . A simplicial presheaf K? endowed with an augmentation ? : K0 → L is called a hypercover of L if it de?nes a hypercover of the site X/L.

1.2.2. Let K? be a simplicial presheaf on X. We denote by C? (K, k), or simply C? (K), the complex of normalized chains of K. This is an object of C(Xk ). If L is a presheaf of sets considered as a disrete simplicial presheaf, C? (L) = kL is the free presheaf of vector spaces generated by L. The following lemma is of crucial importance for us. Lemma. (cf. [SGA4], V.7.3.2(3)). Let ? : K? → L be a hypercover. Then the induced map C? (K? ) → kL induces a quasi-isomorphism of shea??cations.

1.3. General case. Now we will de?ne another CMC structure on C(Xk ) which “remembers” about the topology on X.

1.3.1. Theorem. 1. The category C(Xk ) of presheaves of k-modules admits a CMC structure with co?brations as in 1.1.1 and weak equivalences de?ned as maps f : M → N such that the shea??cation f a is a quasi-isomorphism of complexes of sheaves. 2. A map f : M → N is a ?bration i? f (U) : M(U) → N(U) is surjective for each U ∈ X and for any hypercover ? : V? → U of U ∈ X the corresponding commutative diagram

DEFORMATIONS OF SHEAVES

9

M(U) (2)

c

E

ˇ C(V? , M)

N(U) is homotopy cartesian.

E

c

ˇ C(V? , N)

1.3.2. The de?nition of Cech complex corresponding to a hypercover is given in 1.3.8. The ?rst part of Theorem 1.3.1 is proven in 1.3.3–1.3.7. It is based on an expicit description of the collection of generating acyclic co?bration. The second part of the theorem is proven in 1.3.9. 1.3.3. The generating set of acyclic co?brations is numbered by pairs (? : V? → U, n) where ? is a hypercover and n an integer. An acyclic co?bration j : K → L corresponding to a pair (?, n) as above is de?ned as follows. The presheaf K is shifted by ?n cone of the mapping C? (?) : C? (V? ) → U. This means that K n = k · U and K n?i?1 = k · Vi/ Vi , j = 0, . . . , i ? 1 are the degenerations.

j

k · sj Vi?1 for i ≥ 0, where sj : Vi?1 →

The presheaf L is de?ned as the cone of idK , with the obvious canonical embedding j : K → L. We will write sometimes K?,n and L?,n for the complexes K and L corresponding to a pair (? : V? → U, n). 1.3.4. Lemma. equivalence. The map j : K → L of presheaves constructed above is a weak

Proof. The map j is obviously injective. The shea??cation of its cokernel is contractible by [SGA4], V.7.3.2(3). 1.3.5. Note. The shea??cation of K is also contractible by [SGA4], V.7.3.2(3).

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1.3.6. Lemma. Let a map f : M → N of presheaves satisfy the right lifting property with respect to all generating acyclic co?brations and let f a be a (pointwise) quasiisomorphism. Then f is pointwise surjective quasi-isomorphism. Proof. The map f is pointwise surjective by 1.1.1. Therefore, we have to prove that for any U ∈ X the map f (U) is a quasi-isomorphism. We can put N = 0 without loss of generality. Let a ∈ M(U)n be a cycle. We have to prove a is a boundary. For this we will construct a hypercover ? : V? → U and a map f : K?,n → M whose restriction to the n-th component is the map a : U → M n . Then by the right lifting property f lifts to a map L?,n → M which proves a is a boundary. We proceed by induction: since M a has zero cohomology and da = 0, there exists a cover ? : V0 → U such that ?? (a) is a boundary. Fix a0 ∈ M(V0 ) such that da0 = ?? (a). Suppose, by the induction hypothesis, there exist sections ai ∈ M(Vi ), i = 0, . . . , n such that ai vanishes on the degeneracies M(sj Vi?1 ) and dai = (?1)j d? (ai?1 ). Then one can choose j a cover Vn+1 → coskn (V? )n+1 for which the cycle (?1)i d? (an ) becomes a boundary. i 1.3.7. Let J be the collection of generating acyclic co?brations. We de?ne ?brations as the maps of presheaves satisfying the RLP with respect to the elements of J. We denote by J the collection of maps which can be obtained as a countable direct composition of pushouts of coproducts of maps in J. We call the maps from J standard acyclic co?brations. They are co?brations and weak equivalences by Lemma 1.3.4. According to Lemma 1.3.6, ?brations which are weak equivalences are precisely acyclic ?brations in the sense of 1.1.1 (i.e., pointwise surjective quasi-isomorphisms). Let f : A → B be a map of presheaves. The existence of decomposition f = pi where p is an acyclic ?bration and i is a co?bration follows from Theorem 1.1.1. A small object argument, see [DHK], II.7.3, implies the existence of a decomposition f = qj where j ∈ J and q is a ?bration. Suppose now f is a co?bration and a weak equivalence and choose a decomposition f = qj as above. The map q is therefore a ?bration and a weak equivalence, therefore, by Lemma 1.3.6, q is an acyclic ?bration in the sense of 1.1.1. Therefore, q satis?es the RLP with respect to f . This implies that f is a retract of j. We proved that any acyclic co?bration is a retract of a standard acyclic co?bration which yields the ?rst part of the theorem. The second part of the theorem is explained in 1.3.9 below. 1.3.8. Let V? be a simplicial presheaf and let M ∈ C(Xk ). The collection (3) n → Hom(Vn , M)

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ˇ ˇ is a cosimplicial object in C(Xk ). Cech complex of M, C(V? , M), is de?ned as the normalization of (3), so that (4) ˇ C n (V? , M) = {f ∈

p+q=n

Hom(Vp , M q )|f vanishes on the degenerate simplices }.

In particular, any hypercover ? : V? → U gives rise to a map ˇ M(U) → C(V? , M). Note that one has an obvious isomorphism ˇ (5) C(V? , M) = Hom? (C? (V? ), M). 1.3.9. Proof of Theorem 1.3.1(2). Any ?bration is pointwise surjective since the maps (1) are acyclic co?brations. From now on we suppose that f is pointwise surjective. Put K = Ker(f ). The condition (2) is now equivalent to the claim that the map ˇ K(U) → C(V? , K) is a weak equivalence. This is equivalent to the requirement that the complex Hom? (K?,0, K) has trivial cohomology or, equivalently, that the map Hom? (K?,0 , M) → Hom? (K?,0 , N) is a quasi-isomorphism. This, in turn, can be interpreted as the right lifting property of f with respect to the generating acyclic co?brations K?,n → L?,n . 1.3.10. Note. The model category structure in C(Xk ) described in the theorem, depends essentially on the site X (i.e., on the generating family of the topos X ? ). For instance, a quasi-coherent sheaf is not ?brant in the Zariski site of a scheme. However, it is ?brant when considered as a presheaf on the site of a?ne open subschemes of the scheme. The following observation will be useful in the sequel. 1.3.11. Lemma. Let X be a site and let U ∈ X. Let j : X/U → X be the natural embedding. The restriction functor j ? : C(Xk ) → C((X/U)k ) preserves ?brations and weak equivalences. If X admits ?nite products, j ? preserves co?brations. Proof. Preservation of weak equivalences is immediate. Preservation of ?brations follows immediately from Theorem 1.3.1(2). To prove that j ? preserves co?brations we will check that the right adjoint functor j? : C((X/U)k ) → C(Xk ) preserves acyclic ?brations. One has j? (M)(W ) = M(U × W ) for M ∈ C((X/U)k ); acyclic ?brations are just pointwise surjective quasi-isomorphisms. This implies the lemma.

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1.4. Cohomology and RΓ. 1.4.1. Let M ∈ C(Xk ) and let M be a ?brant resolution of M. We de?ne the i-th cohomology presheaf of M, Hi (M), to be the presheaf Hi (M)(U) = H i (M (U)). The following lemma shows the result does not depend of the choice of the resolution. 1.4.2. Lemma. Let f : M → N be a weak equivalence of ?brant objects in C(Xk ). Then for each U ∈ X and i ∈ Z the map H i(M(U)) → H i(N(U)) is bijective. Proof. Any acyclic ?bration induces a pointwise quasi-isomorphism, so we can suppose that f is an acyclic co?bration. Since M is ?brant, f is split by a weak equivalence g : N → M. Therefore, the map H i(f (U)) is injective. This reasoning can be also applied to the map g instead of f , to conclude that H i (g(U)) is injective as well. 1.4.3. Global sections. Suppose ?rst that X admits a ?nal object ? ∈ X. Then the derived global sections RΓ(M) can be de?ned as M(?) where M is a ?brant resolution of M. If X does not admit a ?nal object, one de?nes the derived global sections functor as follows. Choose a hypercover V? of X. We de?ne ˇ RΓ(M) = C(V? , M), see 1.3.8. By formula (5) and 1.3.5, the result does not depend on the choice of hypercover V? and of the ?brant resolution M . 1.5. Presheaves of modules. Let O be a presheaf of dg associative k-algebras on X. We denote Mod(O, X ) the category of (presheaves of dg) O-modules. 1.5.1. Theorem. The category Mod(O, X ) admits a model structure for which a map f : M → N of presheaves of O-modules is a weak equivalence (resp., a ?bration) i? it is a weak equivalence (resp., a ?bration) of presheaves of k-modules. The proof of the theorem is easily deduced from the following lemma. 1.5.2. Lemma. Let j : K → L be a generating acyclic co?bration corresponding to a pair (?, n) as in 1.3.3, M be a O-module and f : K → M be a map of complexes of presheaves. Then the induced map M →M is a weak equivalence.

O?K

O?L

DEFORMATIONS OF SHEAVES

13

Proof. The map in question being injective, it is enough to study the cokernel which is isomorphic (up to a shift) to O ? Coker(j). Its shea??cation is isomorphic to Oa ? Coker(j)a . The complex Coker(j)a is acyclic by Lemma 1.2.2. Since its components are ?at, the tensor product is acyclic as well. The following lemma shows that weakly equivalent associative algebras give rise to equivalent derived categories of modules. 1.5.3. Let now an algebra homomorphism f : O → O′ be given. One de?nes in a standard way a pair of adjoint functors f ? : Mod(O, X ) → Mod(O′ , X ) : f? ← which induces a pair of derived functors Rf ? : Mod(O, X ) → Mod(O′ , X ) : f? = Lf? . ←

(6)

Lemma. Let f : O → O′ be a weak equivalence of presheaves of associative algebras. Then the adjoint pair (6) establishes an equivalence of the derived categories of modules. Proof. One has to check that of M is a co?brant O-module then the natural map M → f? (f ? (M)) is a weak equivalence. The claim immediately reduces to the case M = O ? k · U for U ∈ X. Then f? (f ? (M)) = O′ ? k · U and the claim is obvious. 1.6. Inner Hom ? . The model category Mod(O, X ) admits an extra structure similar to that of simplicial model category of Quillen [Q1]. 1.6.1. De?nition. Let M, N ∈ Mod(O, X ). The inner Hom Hom ? (M, N) ∈ C(Xk ) O assigns to each U ∈ X the complex of k-modules de?ned as Hom ? (M, N)(U) = O

← V →V ′ →U

lim

Hom? ′ ) (M(V ′ ), N(V )). O(V

Here Hom? ′ ) is the usual inner Hom in the category of complexes of O(V ′ )-modules. O(V We will write as well Hom? (M, N) for the complex of the global sections of Hom ? (M, N). O O

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1.6.2. Lemma. Let α : M → M ′ be a co?bration and β : N → N ′ be a ?bration in Mod(O). Then the natural map Hom ? (M ′ , N) → Hom ? (M, N) ×Hom ? (M,N ′ ) Hom ? (M ′ , N ′ ) is a ?bration in C(Xk ). It is a weak equivalence if α or β is a weak equivalence. Standard adjoint associativity isomorphism Hom? (X ?k Y, Z) ?→ Hom? (X, Hom ? (Y, Z)) O k O reduces the claim to the following. 1.6.3. Lemma. (Here ? = ?k ). Let α : A → A′ be a co?bration in C(Xk ) and let β : M → M ′ be a co?bration in Mod(O, X ). Then the induced map (7) A ? M′

A?M ?

A′ ? M → A′ ? M ′

is a co?bration. It is an acyclic co?bration if α or β is. Proof. For the ?rst claim it is enough to check the case α and β are generating co?brations. This is a very easy calculation. For the second claim note that the cokernel of (7) is isomorphic to Coker(α) ? Coker(β). Since the shea??cation commutes with the tensor product and Coker(α) is ?at, the result is immediate.

? 1.7. Comparing to sheaves. Let O be a dg algebra in Xk .

Recall that one can de?ne the functor RHom ? on the category Mod(O, X ? ) using SpalO tenstein’s notion of K-injective complex of sheaves [Sp]. A complex I ∈ Mod(O, X ? ) is called K-injective if it satis?es the RLP with respect to injective quasi-isomorphisms of sheaves. Thus, one de?nes RHom ? (M, N) as Hom ? (M, I) where N → I is a K-injective resoluO O tion. 1.7.1. Lemma. Here O, M, N are as above. Let M ′ → M be a co?brant resolution of M in Mod(O, X ) and N → N ′ be a ?brant resolution of N in Mod(O, X ). Then Hom ? (M ′ , N ′ ) and RHom ? (M, N) are equivalent. O O Proof. Let N → I be a K-injective resolution of N. Any K-injective complex is ?brant as a complex of presheaves. Therefore, Hom ? (M ′ , N ′ ) and Hom ? (M ′ , I) are O O weakly equivalent ?brant complexes of presheaves. On the other hand, Hom ? (M ′ , I) = O ? Hom ? ((M ′ )a , I) ?→ Hom ? (M, I) since I is K-injective. O O

DEFORMATIONS OF SHEAVES

15

1.7.2. Remark. Subsection 1.4 and Lemma 1.7.1 show that the standard homological algebra of sheaves can be rephrased in the language of complexes of presheaves endowed with the model structure de?ned in Theorem 1.5.1. This has an advantage over the standard approach with sheaves since the analog of Theorem 1.5.1 takes place for presheaves of algebras as well. 1.7.3. We have to mention the following consequence of 1.7.1. Let M be a co?brant and N be a ?brant object in Mod(O, X ). Let U ∈ X and j : X/U → X be the natural embedding. According to Lemma 1.3.11 the functor j ? preserves ?brations and weak equivalences, but does not necessarily preserve co?brations. However, the following takes place. Proposition. The complex Hom ? U (j ? (M), j ? (N)) “calculates the RHom ? ”. More O| precisely, if M ′ → j ? (M) is a co?brant resolution, then the induced map Hom ? U (j ? (M), j ? (N)) → Hom ? U (M ′ , j ? (N)) O| O| is a weak equivalence. Proof. Let N → I be a K-injective resolution of N. Then j ? N → j ? I is a K-injective resolution of j ? N. This implies that all morphisms in the composition below are equivalences.

(8) Hom ? U (j ? (M), j ? (N)) = j ? Hom ? (M, N) → j ? Hom ? (M, I) = O O| O Hom ? U (j ? (M), j ? (I)) → Hom ? U (M ′ , j ? (I)) ← Hom ? U (M ′ , j ? (N)). O| O| O|

2. (Pre)sheaves of operad algebras In this section we describe a model structure on the category of presheaves of algebras over a Σ-split operad, in the case when the corresponding topos has enough points. The structure is based on the model structure on the category of complexes of presheaves described in Theorem 1.3.1. We also discuss the category of modules over an operad algebra, derivations and modules of di?erentials. 2.1. Homotopical amenability. Mimicing [H5], De?nition 2.2.1, we de?ne homotopically amenable presheaves of operads.

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2.1.1. Notation. Let X be a site and let k be a commutative ring. A presheaf of operads on X is, by de?nition, an operad in the tensor category C(Xk ) of complexes of presheaves of k-modules. If O ∈ Op(C(Xk )) is such a presheaf, we denote by Alg(O, X ) the category of (presheaves of) O-algebras. 2.1.2. De?nition. An operad O ∈ Op(C(Xk )) is called homotopically amenable if for each A ∈ Alg(O, X ) and for each generating acyclic co?bration j : K → L in C(Xk ) with K = K?,n and L = L?,n where ? is a hypercover of U ∈ X and n ∈ Z, see 1.3.3, the map α de?ned by the cocartesian diagram F (O, K) (9)

c

jE

F (O, L)

A is a weak equivalence.

α E c A′

Here F (O, K) denotes the free O-algebra generated by K. 2.1.3. The following result is standard. Theorem. Let O be a homotopically amenable operad in C(Xk ). Then the category Alg(O, X ) of O-algebras admits a model structure with weak equivalences and ?brations de?ned as for C(Xk ) in 1.3.1. 2.2. Σ-split operads. In the case X = Ens Σ-split operads de?ned in [H2], 4.2, are homotopically amenable. Recall that a Σ-splitting of an operad O is de?ned as a collection of Σn -equivariant splittings of the canonical maps O(n) ? kΣn → O(n), o ? σ → oσ satisfying some extra compatibility properties, see [H2], 4.2.4 for the precise de?nition. The de?nition of Σ-split operad makes sense in any tensor category. Thus, we can speak about Σ-split presheaves of operads on X. It is worthwhile to mention two big classes of Σ-split operads. ? If k ? Q then all operads in C(Xk ) are Σ-split. ? If A is an asymmetric operad then n → A(n)?Σn is a Σ-split operad. In particular, the operad for associative algebras is Σ-split over Z.

DEFORMATIONS OF SHEAVES

17

2.2.1. Theorem. Let X be a site having enough points. Then any Σ-split operad O ∈ Op(C(Xk )) is homotopically amenable. Proof. It is convenient to shea?fy all the picture. If O is a presheaf of Σ-split dg operads, Oa is a sheaf of Σ-split dg operads and Aa is an Oa -algebra. Shea??cation also commutes with the free algebra functor and with the coproducts. We have to check that any generating acyclic co?bration j : K → L gives rise to a weak equivalence α in the diagram (9). To check this it is su?cient to check that any ?ber functor φ : X ? → Ens transforms the shea??cation αa into a quasi-isomorphism. A ?ber functor transforms the diagram (9) into a cocartesian diagram over a ring k with an acyclic complex K and with L being the cone of idK . Note that K = φ(K?,n) is a non-positively graded acyclic complex of k-modules. Moreover, K admits an explicit presentation, see [SGA4], IV.6.8.3, as a ?ltered direct limit of non-positively graded acyclic complexes of free k-modules. Since the free algebra functor commutes with ?ltered colimits, it is enough for us to check that the map α in the diagram (9) is a weak equivalence provided K is a non-negatively graded complex of free k-modules, L = cone(idK ) and O is Σ-split operad in C(k). But this latter claim follows from the homotopical amenability of Σ-split operads over k. In what follows the following de?nition will be used. 2.2.2. De?nition. A map f : A → B is called a standard co?bration (resp., a standard acyclic co?bration) if it can be presented as a direct limit B = lim Ai with A0 = A and

→ i∈N

Ai+1 being a coproduct of generating co?brations (resp., of generating acyclic co?brations) over Ai . We have the following 2.2.3. Proposition. Any co?bration is a retract of a standard co?bration. Any acyclic co?bration is a retract of a standard acyclic co?bration. This, in fact, is true for any co?brantly generated closed model category. In the rest of this section we suppose that the operad O is homotopically amenable. 2.3. Modules. In this subsection we sketch a presheaf version of [H2], Section 5. 2.3.1. Enveloping algebra. Enveloping algebra U(A) of an operad algebra A ∈ Alg(O, X ) is de?ned in a usual way. This is an associative algebra in the category of complexes C(Xk ), such that U(A)-modules are just the modules over the operad algebra A.

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VLADIMIR HINICH

In particular, the category of modules Mod(O, A) admits a model structure as in 1.5.1. Moreover, a presheaf Hom ? (M, N) is de?ned for a pair (M, N) of A-modules. A 2.3.2. Weakly equivalent operad algebras have sometimes non-equivalent derived categories of modules even for X = Ens, see [H2], Section 5. To get a “correct” derived category of modules, one has to work with co?brant algebras (or with co?brant operads and ?at algebras, see [H2], 6.8). 2.3.3. Lemma. Suppose O is Σ-split. Let f : A → B be an acyclic co?bration of co?brant O-algebras. Then the induced map U(f ) : U(A) → U(B) is a weak equivalence. Proof. The proof is basically the same as that of Corollary 5.3.2, [H2]. Everything reduces to the case A is generated by a ?nite number of sections xi , i = 1, . . . , n, over Ui ∈ X and B is the colimit of a diagram A ← F (M) → F (idM ), where M is a coproduct of shifts of representable presheaves and has acyclic shea??cation. The enveloping algebra U(A) admits a ?ltration numbered by multi-indices d : {1, . . . , n} → N, with associated graded pieces given by the formula grd (U(A)) = O(|d| + 1) ?Σd where, as usual, |d| = di , Σd = Σdi ? Σ|d| . Ui?di ,

The enveloping algebra U(B) admits a ?ltration indexed by pairs (d, k) with d as above and k ∈ N. The associated graded piece corresponding to (d, k) takes form grd,k (U(B)) = O(|d| + k + 1) ?Σk ×Σd M[1]?k ? Ui?di .

Since O is Σ-split and M a is ?at and acyclic, the shea??cations of grd,k (U(B)) are acyclic for k > 0. This proves the lemma. 2.3.4. Corollary. Let O be Σ-split. Any weak equivalence f : A → B of co?brant O-algebras induces a weak equivalence U(f ) of the enveloping algebras. Proof. We already know the claim in the case f is an acyclic co?bration. Therefore, it su?ces to prove it for acyclic ?brations. Let f : A → B be an acyclic ?bration of co?brant algebras. There exist a map g : B → A splitting f : f g = idB . This implies that for any weak equivalence f : A → B the induced map H(U(A)a ) → H(U(B)a ) of the homologies of the shea??cations splits. Applying this to g : B → A we deduce that it is in fact invertible. 2.4. Di?erentials and derivations. In this subsection we present a presheaf version of parts of [H2], 7.2, 7.3.

DEFORMATIONS OF SHEAVES

19

2.4.1. De?nition. Let O ∈ Op(C(Xk )) be a presheaf of operads on X, α : B → A be a map in Alg(O, X ) and let M be an A-module. The presheaf Der ? O (A, M) of OB derivations over B from A to M is de?ned as the subpresheaf of Hom ? (A, M) consisting k of local sections which are O-derivations from A to M vanishing at B. By de?nition Der ? O (A, M) is a subcomplex of Hom ? (A, M). The functor B k M → Der ? O (A, M) is representable in the following sense. B 2.4.2. Lemma. There exists a (unique up to a unique isomorphism) A-module ?A/B Hom ? (?A/B , M) ?→ Der ?O (A, M). A B The proof of Lemma 2.4.2 is standard, see Proposition 7.2.2 of [H2]. The following lemma is the key to the calculation of ?A/B . 2.4.3. Lemma. Let α : B → A be a map of O-algebras, M ∈ C(Xk ). Let f : M → A be a map in C(Xk ) and let A′ = A M, f be de?ned by the cocartesian diagram F (O, M) E F (O, cone(idM )) .

E A′ A Put U = U(O, A) and U ′ = U(O, A′ ). The map ? ? f : M → ?A/B de?nes f ′ : U ′ ? M → U ′ ?U ?A/B . c c

?

together with a global derivation ? : A → ?A/B inducing a natural isomorphism in C(X )

Then the module of di?erentials ?A′ /B is naturally isomorphic to the cone of f ′ . For the proof see Lemma 7.3.2 of [H2]. 2.4.4. Proposition. Let O be homotopically amenable. Let α : B → A be a co?bration in Alg(O, X ). Then ?A/B is co?brant in Mod(O, X ). Proof. One can easily reduce the claim to the case when A is generated over B by a section a over U ∈ X subject to a condition da = b ∈ B(U). In this case Der ? O (A, M) = B Hom ? (U, M) so that ?A/B is isomorphic to U(A) ? U. k 2.4.5. Corollary. Let O be homotopically amenable. Let α : B → A is a co?bration in Alg(O, X ) and let M be a ?brant A-module. Then Der ? O (A, M) ∈ C(Xk ) is ?brant. B

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VLADIMIR HINICH

2.4.6. Let C → B → A be a pair of morphisms in Alg(O, Xk ). For each A-module M every derivation ? : A → M over C de?nes a derivation ? ? f : B → M over C. This de?nes a canonical map ?f : U(B) ?U (A) ?B/C → ?A/C . The following proposition shows that the module of di?erentials ?A/B has a correct homotopy meaning for co?brant morphisms B → A. It generalizes Proposition 7.3.6 of [H2] where the case X = Ens is considered. 2.4.7. Proposition. Let C → B → A be a pair of morphisms in Alg(O, Xk ). If f is a weak equivalence and α, f ? α are co?brations than the map ?f is a weak equivalence. Proof. First of all one proves the claim in the case f is an acyclic co?bration. This easily follows from Lemma 2.4.3. Then one proves the assertion in the case the algebras A and B are ?brant. Here the proof of Proposition 7.3.6 of [H2] can be repeated verbatim. Finally, the general case reduces to the case A and B are ?brant passing to ?brant resolutions of A and B. 2.4.8. Let O be homotopically amenable. Let A be a co?brant O-algebra on X and let M be a ?brant A-module. According to Corollary 2.4.5, Der ? (A, M) ∈ C(Xk ) is ?brant. We want to show that this object behaves well under localizations. Let U ∈ X and let j : X/U → X be the obvious embedding. We claim that the presheaf Der ? (j ? (A), j ? (M)) = j ? Der ? (A, M) ∈ C((X/U)k ) has “the correct homotopy meaning”. As in 1.7.3, the only problem is that the functor j ? does not always preserve co?brations. Therefore, we claim the following. Lemma. map Let f : A′ → j ? (A) be a co?brant resolution of j ? (A). Then the restriction Der ? (j ? (A), j ? (M)) → Der ? (A′ , j ? (M)) is a weak equivalence. Proof. It is enough by Lemma 1.7.3 to check that the map (10) is a weak equivalence. Recall that A is co?brant. One easily reduces the claim to the case A is standard co?brant. This means that A is presented as a colimit of An , n ∈ N with A0 = O(0) (the initial O-algebra) and An+1 de?ned as the colimit of a diagram An ← F (O, Mn ) → F (O, cone(idMn )), Mn being a direct sum of representable presheaves and their shifts. ?f : U(j ? (A)) ?U (A′ ) ?A′ → ?j ? (A)

α f

α

f

DEFORMATIONS OF SHEAVES

21

Then j ? (A) is the colimits of j ? (An ) with j ? (An+1 ) isomorphic to the colimit of the diagram j ? (An ) ← F (O, j ? (Mn )) → F (O, cone(idj ? (Mn ) )).

′ If we choose co?brant resolutions Mn → j ? (Mn ), one de?nes recursively the collection of co?brant algebras A′n+1 as colimits of the diagram ′ ′ A′n ← F (O, Mn ) → F (O, cone(idMn )).

Then by induction on n one checks using Lemma 2.4.3 that the map ?fn : U(j ? (An )) ?U (A′n ) ?A′n → ?j ? (An ) is a weak equivalence for each n. Passing to a limit, we get a weak equivalence (10) for a special choice of A′ = lim A′n .

→

2.5. Simplicial structure. Similarly to the case X = ? described in [H2], sect. 4.8, we can de?ne a simplicial structure on Alg(O, Xk ) provided k ? Q. 2.5.1. Let S be a ?nite simplicial set and let A ∈ Alg(O, Xk ). We de?ne the presheaf AS by the formula AS (U) = ?(S) ? A(U) where ?(S) denotes the commutative dg k-algebra of polynomial di?erential forms on S. 2.5.2. For A, B ∈ Alg(O, Xk ) de?ne Hom? (A, B) ∈ ?op Ens by the formula Hom?op Ens (S, Hom? (A, B)) = Hom(A, B S ) where S ∈ ?op Ens is ?nite. The following theorem says that the simplicial structure de?ned satis?es Quillen’s axiom (SM7). 2.5.3. Theorem. Let α : A → B be a co?bration and β : C → D is a ?bration in Alg(O, Xk ). Then the natural map Hom? (B, C) → Hom? (A, C) ×Hom? (A,D) Hom? (B, D) is a Kan ?bration. It is a weak equivalence if α or β is a weak equivalence. Theorem 2.5.3 results from the following

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VLADIMIR HINICH

2.5.4. Lemma. Let α : K → L be an injective map of ?nite simplicial sets and let β : C → D be a ?bration in Alg(O, Xk ). Then the natural map (11) C L → C K ×DK D L is a ?bration. It is weak equivalence if α or β is a weak equivalence. Proof. The map (11) can be rewritten as ?(L) ? C → ?(K) ? C ×?(K)?D ?(L) ? D. it is pointwise surjective since β is pointwise surjective. This also implies that (11) is an acyclic ?bration provided α or β is a weak eqivalence. Since (11) is pointwise surjective, it is enough to check that its kernel is ?brant. The kernel easily identi?es with the tensor product Kα ? Kβ where Kα = Ker{α : ?(L) → ?(K)} and Kβ = Ker(β).

Q Presheaf Kβ is ?brant. Complex of k-modules Kα has form k ?Q Kα where the complex Q Kα has ?nite dimensional cohomology. One can therefore write Kα = H ⊕ C where H is the ?nite dimensional cohomology of Kα and C is contractible. Then H ? Kβ is ?brant as a ?nite direct sum of (shifts of) Kα ; C ? Kβ is pointwise contractible, and therefore ?brant.

2.5.5. Functor Tot. Let A? be a cosimplicial object in Alg(O, X ). The algebra Tot(A? ) is de?ned by the standard formula Tot(A? ) = lim p→q (Aq )? .

←

p

2.6. Descent.

? 2.6.1. Let V? be a hypercover of X and let O be an operad in Xk . Put On = O|Vn . Let Algf (resp., Algf ) denote the category of ?brant O-algebras on X (resp., ?brant n On -algebras on X/Vn ).

The assignment n ?→ Algf de?nes a cosimplicial object in Cat (more precisely, a category n co?brant over ?). We denote by Algf (V? ) the following category. The objects of Algf (V? ) are collections {An ∈ Algf } together with weak equivalences An → φ? (Am ) corresponding n to each φ : m → n in ?, satisfying the standard cocycle condition. The functor ?? : Algf → Algf (V? ) assigns to each algebra on X the collection of its restrictions. This functor preserves weak equivalences. We de?ne the functor α? : Algf (V? ) → Algf by the formula α? (An , φ) = Tot{n → ιn (An )}. ?

DEFORMATIONS OF SHEAVES

23

Here the functor Tot is de?ned as in 2.5 and ιn is the direct image of the localization ? functor ιn : X/Vn → X, [SGA4], III.5. 2.6.2. Proposition. The functors α? and α? induce an adjoint pair of equivalences on the corresponding Dwyer-Kan localizations of Algf (V? ) and Algf with respect to quasiisomorphisms. Proof. The functors involved do not depend on the operad O. Therefore, it is enough to prove the claim for the trivial operad so that Alg(O) is just the category of complexes. ? This is done in [SGA4], Expos? Vbis, for C + (Xk ) and in [HS], Sect. 21, for unbounded e complexes.

3. Deformations Let k be a ?eld. In this section we de?ne a functor of formal deformations of a sheaf of operad algebras over k. In the case when X has enough points and k ? Q we present a dg Lie algebra which governs (under some extra restrictions on the sheaf of algebras A) the described deformation problem. 3.1. Deformation functor.

? 3.1.1. Let X be a site, O ∈ Op(C(Xk )) be a sheaf of dg operads on X and let A ∈ Alg(O, X ? ) be a sheaf of O-algebras. The functor we de?ne below describes formal deformations of the sheaf of O-algebras A.

3.1.2. Bases of deformations. Fix a commutative algebra homomorphism π : K → k (it will usually be the identity id : k → k of a ?eld of characteristic zero). Let dgart≤0 (π) denote the category of commutative non-positively graded dg algebras R endowed with morphisms α : K → R, β : R → k, such that Ker(β) is a nilpotent ideal of ?nite length. The category dgart≤0 (π) is the category of allowable bases for formal deformations of A. We suppose that the operad O is obtained by the base change from a ?xed operad OK ∈ ? Op(C(XK )), so that O = OK ? k. For each R ∈ dgart≤0 (π) we put OR = OK ? R. 3.1.3. Fix R ∈ dgart≤0 (π). Let Algf? (OR , X ? ) denote the category of sheaves of OR f? algebras ?at as sheaves of R-modules. Let W? (R, X) be the subcategory of weak equivalences in Algf? (OK ? R, X ? ) (i.e. quasi-isomorphisms of complexes of sheaves). In what follows we will usually omit X from the notation. Recall that Dwyer-Kan construction [DK1, DK2, DK3] assigns to a pair (C, W) where W is a subcategory of C, a hammock localization LH (C, W) which is a simplicial category.

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VLADIMIR HINICH

We will apply this construction to pairs of coinciding categories (W, W). The Dwyer-Kan construction gives in this case a simplicial groupoid. We will denote it W and call a weak groupoid completion, see Appendix A for details. The base change functor M → M ?R k induces a functor between the weak groupoid completions (12) 3.1.4. De?nition.

f? β ? : W? (R) → W? (k).

Deformation functor of O-algebra A is the functor Def A : dgart≤0 (π) → sGrp

from the category dgart≤0 (π) to the category of simplicial groupoids sGrp de?ned as the homotopy ?ber of (12) at A ∈ W? (k). Since (12) is a functor between ?brant simplicial categories, the homotopy ?ber is represented by the ?ber product

f? ? ×W? (k) W? (k)? ×W? (k) W? (R),

1

see A.1.12. This allows one to consider Def A as a functor with values in sGrp and not in the corresponding homotopy category. 3.2. Properties of Def A . We list below some properties of the functor Def A which justify the above de?ntion. 3.2.1. The functor Def A does not depend on the speci?c choice of the site X; only the topos X ? counts. 3.2.2. Suppose that X is a one-point site, so that we are dealing with deformations of an operad algebra A. Suppose also that k is a ?eld of characteristic zero and π = idk : K = k → k. Recall that in [H4] a deformation functor is de?ned for this setup. In the de?nition the simplicial category of co?brant algebras and weak equivalences is used instead of the hammock localization. We prove in Proposition A.3.4 that these two functors are equivalent. 3.2.3. Descent. Let V? be a hypercover of X and let A be a sheaf of operad algebras on X. Put An = A|Vn . The assignment n ?→ Def An de?nes a cosimplicial object in sCat. One has a natural descent functor Def A → holim{n → Def An }, see A.1.10. We claim the functor described is an equivalence. In fact, since Def A is de?ned as a homotopy ?ber, the right-hand side of (3.2.3) is the homotopy ?ber of the map f? f? holim{n → W? (R, Vn )} → holim{n → W? (k, Vn )}.

DEFORMATIONS OF SHEAVES

25

It is enough therefore to check that the functor

f? f? W? (R, X) → holim{n → W? (R, Vn )}

is an equivalence. Since all simplicial categories involved are simplicial groupoids, it is enough by A.2.5 to check that this functor induces an equivalence of the nerves. The functor holim commutes with the nerve functor, see [H3], Prop. A.5.2, or A.1.11. Moreover, the nerve of a Bous?eld-Kan localization is equivalent to the nerve of an original category. Therefore, the claim follows from Proposition 2.6.2 and A.3.4. 3.2.4. Connected components. Here we assume that the topos X ? admits enough points. We assume as well that k is a ?eld of characteristic zero and π = idk . Suppose that R, A and O are concentrated at degree zero, so that we have a classical deformation problem. We claim that Def A (R) is equivalent to the groupoid Def cl (R) of A ?at R-deformations of A. Let B be a R-?at sheaf of (dg) R ? O-algebras such that the reduction β ? (B) is quasiisomorphic to A. We claim that B is concentrated in degree zero and H 0 (B) is a ?at deformation of A. In fact, since X ? has enough points, the claim can be veri?ed ?berwise. The case X ? = Ens is explained in [H4], 5.1. The assignment B → H 0 (B) de?nes, therefore, a simplicial functor Def A (R) → Def cl (R). A We claim this functor is an equivalence. In fact, it is enough by A.2.5 to check the functor f? f? induces an equivalence of the nerves. Let W? (R)0 denote the full subcategory of W? (R) consisting of algebras whose cohomology is ?at and concentrated in degree zero , and let Algiso (R) denote the groupoid of ?at R ? O-algebras concentrated in degree zero. It is enough to check that the functor

f? H 0 : W? (R)0 → Algiso (R)

induces an equivalence of the nerves. To prove this, consider a third category W c (R) consisting of co?brant presheaves P of R ? O-algebras. Shea??cation de?nes a functor

f? a : W c (R) → W? (R).

We denote by W c (R)0,f? the full subcategory of W c (R) consisting of complexes of presheaves f? whose shea??cation belongs to W? (R)0 . The restriction de?nes the ?nctors

f? a : W c (R)0,f? → W? (R).

and H 0 ? a : W c (R)0,f? → Algiso (R). Both functors a and H 0 ? a induce an equivalence of nerves by Quillen’s Theorem A [Q2] and Theorem A.3.2. This proves the assertion.

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VLADIMIR HINICH

3.3. Reformulation in terms of presheaves. From now on we assume that k is a ?eld of characteristic zero and π = idk . We assume as well that the topos X ? admits enough points. 3.3.1. Notation. The category Alg(O ? R, X ) admits a model structure de?ned in 2.1.3. We denote the subcategory of weak equivalences by W(R). The notation W c (R), W f (R), cf W cf (R) and W? (R) has the same meaning as in A.3.4.

f? Proposition 3.3.5 below claims that the weak groupoid W? (R) admits an equivalent description in terms of the weak groupoid of the category of presheaves. This description is functorial in R. This allows one to conveniently describe the homotopy ?bre of (12).

Note the following technical lemma. 3.3.2. Lemma. Let A be a co?brant O ? R-algebra and let A# be the corresponding presheaf of R-modules. There is an increasing ?ltration {F iA, i ∈ I} of A# indexed by a well-ordered set I such that the associated graded factors gri (A) = F i (A)/ j<i F j A are isomorphic to R ? Xi with X i ∈ C(Xk ). Proof. We can assume that A is a direct limit over a well ordered set of generating acyclic co?brations. In fact, a general co?brant algebra is a retract of an algebra of this type, and the property claimed in the lemma is closed under retractions. Let, therefore, the algebra A be freely generated by a collection of sections xj over Uj ∈ X of degree dj numbered by a well-ordered set J. We de?ne I to be the collection of functions m : J → N nonzero at a ?nite number of elements of J. The set I is endowed with the lexicographic order: m < m′ i? ?k ∈ J : m(k) < m′ (k) and m(j) = m′ (j) for j > k. The increasing ?ltration is numbered by elements of I and the associated graded factors have the form ?m ?m R ? O(n) ?Σm k[?d1 ] · U1 1 ? . . . ? k[?dk ] · Uk k where n = mi , Σm = Σmi ? Σn , Σn being the symmetric group.

3.3.3. Lemma. tions.

1. The functor β ? : Alg(OR , X ) → Alg(O, X ) preserves co?bra-

2. Let A ∈ Algc (OR , X ). Then A is ?brant if and only if β ? (A)is ?brant.

DEFORMATIONS OF SHEAVES

27

Proof. 1. The ?rst claim is obvious for generating co?brations. The general case follows from the fact that β ? commutes with colimits and retracts. 2. According to Lemma 3.3.2, A# admits an increasing ?ltration {F i A, i ∈ I} with associated graded factors having a form gri A = R ? X i . This implies that A(U) is a co?brant R-module for each U ∈ X. A is ?brant i? for any hypercover ? : V? → U in X the natural map ˇ (13) A(U) → C(V? , A) ˇ is a weak equivalence. In Lemma 3.3.4 below we prove that C(V? , A) is a co?brant Rmodule. The Cech complex of β ? (A) is just the reduction of the Cech complex of A modulo the maximal ideal of R. Therefore, if (13) is a weak equivalence, the reduction modulo the maximal ideal of R is also a weak equivalence. In the other direction, there exists a ?nite ?ltration of R by k-subcomplexes such that the associated graded factors are isomorphic to k up to shift. Therefore, the cone of (13) admits a ?nite ?ltration with acyclic associated graded factors. 3.3.4. Lemma. Let A ∈ Mod(R, Xk ) admit an increasing ?ltration {F i A, i ∈ I} with associated graded factors of form gri (A) = R ? X i , ˇ for some X i ∈ C(Xk ). Then for each hypercover ? : V? → U the Cech complex C(V? , A) is a co?brant R-module. Proof. Cech complex of A is the total complex corresponding to the bicomplex A0 → A1 → . . . where An is the collection of sections of A(Vn ) vanishing on the degenerate part of Vn (see (4) for the precise de?nition). Each An is ?ltered and δ n preserves the ?ltration. De?ne Bn = ker(δ n ). Bn are ?ltered R-modules with associated graded factor having the form

i R ? Yn , i i i Yn = ker(δ n : Xn → Xn+1 ). δ0 δ1

The restriction of the di?erential δn on Bn vanishing, the corresponding total subcomplex is isomorphic to n Bn [?n]. Let us show it is a co?brant R-module. In fact, each Bn admits an increasing ?ltration with R-free associated graded factors. Since R is artinian, a product of free R-modules is free. Therefore, n Bn [?n] admits as well an increasing ?ltration with R-free associated graded factors. Therefore, it is R-co?brant. The quotient is the total complex of the bicomplex A0 /B0 → A1 /B1 → . . .

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VLADIMIR HINICH

which has also a vanishing horisontal di?erential. Each quotient An /Bn admits a ?ltration i i with the associated graded factors having form R ? Xn /Yn which is also R-co?brant. Lemma is proven.

cf f? The following proposition claims that the simplicial categories W? (R) and of W? (R) are canonically equivalent.

3.3.5. Proposition. groupoids

There is a canonical in R collection of equivalences of weak

cf cf f? W? (R) → W? (R) ← W cf (R) → W? (R). c b a

Proof. The map c is a canonical map from a simplicial category to its hammock localcf ization. It is an equivalence since W? (R) is a weak groupoid, see A.2.3. The map b is equivalence by part (ii) of Proposition A.3.4. The map a is induced by the shea??cation. First of all, by Lemma 3.3.2 the shea??cation of A ∈ W c (R) is R-?at. Therefore, the functor a is de?ned. To prove a is an equivalence, it su?ces, by Corollary A.2.5, to check that the functor f? a : W cf (R) → W? (R) induces an equivalence of the nerves. Proposition A.3.4 (2) asserts that the map W cf (R) → W c (R) induces an equivalence of the nerves.

f? The shea??cation functor a : W c (R) → W? (R) has its image in W? (R). The resulting functor f? a : W c (R) → W? (R) induces an equivalence of the nerves by Theorem A.3.2 and Quillen’s Theorem A, see [Q2].

3.4. Fibration lemma. 3.4.1. In this subsection we assume the following properties. ? For each n ∈ N the complex of sheaves O(n) is non-positively graded, O(n) ∈ C ≤0 (X ? ). ? The site X admits a ?nal object.

cf cf Let W? (R)≤0 denote the full simplicial subcategory of W? (R) consisting of algebras A satisfying the extra condition Hi (A) = 0 for i > 0.

DEFORMATIONS OF SHEAVES

29

3.4.2. Lemma. Suppose that the condition of 3.4.1 on O and X are ful?lled. Then the functor (14) is a ?bration in sCat. We recall the model category structure on sCat in A.1. The proof of Lemma 3.4.2 is similar to that of Lemma 4.2.1 of [H4]. It is presented in 3.4.3 – 3.4.6 below. The properties of derivations studied in 2.4 play an important role in the proof. 3.4.3. Let B be a co?brant presheaf of O-algebras. We claim that B and α? β ? (B) are isomorphic as presheaves of graded (without di?erential) algebras. In fact, any co?brant presheaf is a retract of a standard co?brant (see 2.2.2) presheaf of algebras, so we can assume that B is standard. This means, in particular, that B is freely generated, as a graded presheaf of algebras, by a collection of generators bi which are sections of B over some objects Ui ∈ X. In this case the claim is obvious. Therefore, any co?brant presheaf B of O-algebras is isomorphic to a presheaf of form (α? (A), d + z) where (A, d) = β ? (B) and z ∈ m ? Der(A, A) satis?es the Maurer-Cartan equation. Here m = Ker(β : R → k) is the maximal ideal of R and Der denotes the global sections of the presheaf Der ? . 3.4.4. A morphism of simplicial categories is a ?bration if it satis?es the conditions (1), (2) of De?nition A.1.8. Let us check the condition (1). Let f : A → B be a weak equivalence of ?brant co?brant algebras in Alg(O, X ). Let TA = Der ? (A, A) and similarly for TB . The global sections of the derivation algebras are denoted TA and TB correspondingly. Let one of two elements a ∈ MC(m ? TA ), b ∈ MC(m ? TB ) be given. We have to check that there exists a choice of the second element and a map (15) g : (α? (A), d + a) → (α? (B), d + b) in Alg(R ? O, X ) lifting f . Note that under the restrictions of 3.4.1 any algebra (α? (A), d + a) belongs to W cf (R). We can consider separately the cases when f is an acyclic ?bration or an acyclic co?bration. In both cases we will be looking for the map (15) in the form

?1 g = γB ? α? (f ) ? γA cf cf β ? : W? (R)≤0 → W? (k)≤0

where γA ∈ exp(m ? TA )0 and similarly for γB . A map (15) should commute with the di?erentials d + a and d + b. This amounts to the condition f? (γA (a)) = f ? (γB (b)), where the natural maps (16) TA ?→ Derf (A, B) ←? TB

f? f?

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VLADIMIR HINICH

are de?ned as the global sections of the standard maps TA = Hom ? (?A , A) ?→ Hom ? (?A , B) ←? Hom ? (?B , B) = TB . The maps f? and f ? in (16) are weak equivalences as global sections of weak equivalences between the ?brant presheaves TA , TB , Der ? f (A, B). Recall that we assume that f is either acyclic co?bration or acyclic ?bration. 3.4.5. Lemma. Let f : A → B be a weak equivalence of ?brant co?brant algebras. Suppose that X admits a ?nal object. Suppose that either (af) f is an acyclic ?bration or (ac) f is a standard acyclic co?bration. Then there exists a commutative square Tf

? d ? d h g? d ? d ? ? ? ? d f? f E Derf (A, B) '

f? f?

TB TA where Tf is a dg Lie algebra and g,h are Lie algebra quasi-isomorphisms. Proof. Note ?rst of all that the maps f ? , f? are weak equivalences. This follows, as for the “absolute” case of [H2], 8.1, from the presentation TA = Hom? (?A , A); TB = Hom? (?B , B); Derf (A, B) = Hom? (?A , B).

We construct the dg Lie algebra Tf as follows. Case 1. f is acyclic ?bration. Let I = Ker(f ). De?ne Tf to be the subalgebra of TA consisting of (global) derivations preserving I. Since A is co?brant, any global derivation of B can be lifted to A. Therefore, the map h : Tf → TB is surjective. The kernel of h consists of global derivations on A with values in I. The presheaf Der ? (A, I) = Hom? (?A , I) is acyclic ?brant. Therefore, its global sections are acyclic since X admits a ?nal object. Case 2. f is a standard acyclic co?bration. We de?ne Tf as the collection of (global) derivations of B preserving f (A). Let us check that the map g : Tf → TA is a surjective quasi-isomorphism. The algebra B is obtained from A by a sequence of generating acyclic co?brations corresponding to hypercovers of X.

DEFORMATIONS OF SHEAVES

31

Let B = lim Bi with B0 = A and Bi obtained from the cocartesian diagram

→

F (Ki) F (φi)

c

F (jiE )

F (Li )

Bi?1

E

c

Bi ,

where ji : Ki → Li are acyclic co?brations of presheaves and F ( ) is the free algebra functor. Put Ti = {δ ∈ Der(Bi , B)|δ(f (A)) ? f (A)}. Then T0 = TA and Tf = lim Ti . We claim ← that the maps gi : Ti → Ti?1 are acyclic ?brations. Then g : Tf → TA is also an acyclic ?bration. To prove surjectivity of gi ?x a derivation δ : Bi?1 → B. Derivations of Bi extending δ correspond to commutative diagrams of presheaves

Ki φi

c

ji

E

Li

Bi?1

δ

E

c

B

Since the map ji is an acyclic co?bration and B is ?brant, there is no obstruction to extending δ. Surjectivity of gi is proven. Now the kernel of gi identi?es with Hom? (Li /Ki , B). The presheaf Hom ? (Li /Ki , B) being acyclic and ?brant, its global sections Hom? (N/M, B) are acyclic. Lemma 3.4.5 is proven. Recall now (see A.4) that for a dg Lie algebra g the formal groupoid Delg : dgart≤0 (k) → Grp is de?ned as the transformation groupoid of the group exp(m ? g)0 acting on the set of Maurer-Cartan elements of (m ? g)1 . The set of connected components π0 (Delg(R)) is a weak homotopy invariant of g. Thus, the maps g and h induce bijections π0 (DelTA (R)) ←? π0 (DelTf (R)) ?→ π0 (DelTB (R)). of the sets of components. This proves the condition (1) of A.1.8.

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3.4.6. Let us check the condition (2) of A.1.8. Let A, B ∈ W cf (R) and let A = β ? (A), B = β ? (B). We have to check that the map (17) Hom? (A, B) → Hom? (A, B) is a Kan ?bration. The algebra B can be considered as OR -algebra with R acting on B through β : R → k. The canonical map B → B is pointwise surjective and both B and B are ?brant. Therefore, it is a ?bration by Theorem 1.3.1 (2). Since A is co?brant, the map (18) Hom? (A, B) → Hom? (A, B) is a Kan ?bration. But the maps (17) and (18) coincide. This proves the condition (2) of A.1.8. Fibration Lemma 3.4.2 is proven. 3.5. The main theorem. 3.5.1. Assumptions. In this subsection we assume that the following conditions. ? For each n ∈ N the complex O(n) of sheaves is non-positively graded, O(n) ∈ C ≤0 (X ? ). ? The O-algebra A satis?es the property Hi (A) = 0 for i > 0. 3.5.2. Lemma. Under the assumptions of 3.5.1, there exists an algebra A′ weakly equivalent to A such that ? A′ is ?brant and co?brant; ? A′ ∈ C ≤0 (Xk ). Proof. Let A → B be a ?brant resolution of A. One has H i(B) = 0 for i > 0. Let C = τ ≤0 (B). The canonical map C → B is therefore pointwise quasi-isomorphism. Then Theorem 1.3.1 (2) asserts that C is ?brant as well. One can choose a co?brant resolution A′ of C having generators only in non-positive degrees. The algebra A′ satis?es all necessary properties. 3.5.3. Let A be as above. Let A′ be an algebra whose existence is guaranteed by Lemma 3.5.2. We de?ne the local tangent Lie algebra of A by the formula TA = Der ? (A′ , A′ ). This is a ?brant presheaf of dg Lie algebras. De?ne, ?nally, global tangent Lie algebra of A by the formula TA = RΓTA

DEFORMATIONS OF SHEAVES

33

where the functor RΓ assigning a dg Lie algebra to a presheaf of dg Lie algebras, is understood in the following sense (compare to 1.4.3). Choose a hypercover V? of X. The collection n → Hom(Vn , TA ) is a cosimplicial dg Lie algebra. The corresponding Tot functor produces a dg Lie algebra which is denoted RΓTA . The result does not depend on the choice of V? . Note that if the site X admits a ?nal object ?, TA is equivalent to TA (?). 3.5.4. A simplicial presheaf K? is called ?nite dimensional if it coincides with its n-th skeleton for some n. In Theorem 3.5.5 below we require the site X admit a ?nite dimensional hypercover. This condition is void if X admits a ?nal object. In the case X is the site of a?ne open subschemes of a scheme S, the requirement is ful?lled if S is quasi-compact separated or ?nite dimensional scheme. 3.5.5. Theorem. Suppose that the conditions of 3.5.1 on O and A are ful?lled. Suppose also that the site X admits a ?nite dimensional hypercover. Then the deformation functor Def A is equivalent to Del TA where TA is the (global) tangent Lie algebra of A. 3.5.6. Remark. Fibration Lemma 3.4.2 implies Theorem 3.5.5 if X admits a ?nal object. In fact, let A be a ?brant co?brant O-algebra with Ai = 0 for i > 0. By Lemma 3.4.2 the homotopy ?ber of (14) is equivalent to its usual ?ber. Fix (R, m) ∈ dgart≤0 (k). The ?ber of (14) at A is the simplicial groupoid de?ned by the perturbations of the di?erential in R?A. This is precisely the simplicial Deligne groupoid Del T (R) de?ned in [H4] (see A.4), with T = Der(A, A). Proof of the theorem. In a few words, the proof is the following. By Remark 3.5.6 the result is proven in the case X admits a ?nal object. To prove the result in general, one uses the descent properties of the objects involved: that of the deformation functor according to 3.2.3, and that of the Deligne groupoid by [H1]. The requirement on the existence of a ?nite dimensional hypercover is due to a similar requirement in the proof of [H1], Theorem 4.1. Here are the details. Choose A to be a ?brant co?brant algebra such that Ai = 0 for i > 0. Choose a ?nite dimensional hypercover ? : V? → X. Let An = A|Vn . This is a ?brant algebra on X/Vn ; it is not necessarily co?brant but it “behaves as if it were co?brant”. We have TA = RΓ(TA ). Let Tn = TA (Vn ) = Der(An , An ).

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VLADIMIR HINICH

According to [H1] (more precisely, according to its simplicial version Proposition A.4.5), there is an equivalence Del TA → holim{n → Del Tn }. For each n we de?ne canonically a functor ρn : Del Tn → Def An which will turn out to be an equivalence. It is convenient to interpret here Def An as the homotopy ?ber of the map W →W where W denotes the simplicial groupoid W? (R) obtained by Dwyer-Kan localization from the simplical category W? (R) of R ? O|Vn -algebras, and W = W? (k). This means, according to A.1.12, that Def An (R) = ? ×W W

?1

×W W.

The functor ρn perturbs the di?erential. Here is its description. Let (R, m) ∈ dgart≤0 (k). An object z of Del Tn (R) is a Maurer-Cartan element of m ? Tn . Then ρn (z) = (R ? An , d + z) where d denotes the di?erential of the algebra A. This de?nes ρn on objects. Let now γ : z → z ′ be an m-morphism in Del Tn . This means that γ ∈ exp(?m ? m ? Tn ) and γ(z) = z ′ . Thus, γ induces a map γ : ρn (z) → ?m ? ρn (z ′ ) which gives an m-morphism in W? (R) and, therefore, in W = W? (R). Let us check now that ρn is an equivalence. Choose a co?brant resolution π : A′ → An . Let T ′ = Der(A′ , A′ ). According to Lemma 2.4.8, the maps T ′ → Der(A′ , An ) ← Tn induced by π are weak equivalences and, moreover, φ is surjective. De?ne a dg Lie algebra g by the cartesian diagram

φ ψ

g

ψ′

φ′

E

Tn

ψ

.

c

c

T′

φ

E

Der(A′ , An )

DEFORMATIONS OF SHEAVES

35

The maps φ′ and ψ ′ are, therefore, quasi-isomorphisms of dg Lie algebras. We have the following functors: ? Ψ : Del g(R) → Del T ′ (R) induced by ψ ′ ; ? Φ : Del g(R) → Del Tn (R) induced by φ′ ; ? ρn : Del Tn (R) → Def An (R) de?ned in (3.5.6). The functors Ψ and Φ are equivalences. De?ne an arrow r : Del T ′ (R) → Def An (R) as follows. Recall that Def An (R) is the ?ber of the map W

?1

×W W ?→ W.

Each object z ∈ Del T ′ (R) gives rise to an algebra (R ? A′ , d + z) whose reduction is A′ . This gives an object in Def An (R) presented by the morphism A′ → An . The action of r on morphisms is obvious. Look at the diagram

Del g(R)

Φ

Ψ

E

Del T ′ (R) = Def A′ (R)

r

ρn Del Tn (R) E Def An (R)

c

c

The diagram is not commutative. However, there is a homotopy connecting rΨ with ρn Φ assigning to each object z ∈ Del g(R) a morphism de?ned by the quasi-isomorphism (R ? A′ , d + π(z)) ?→ (R ? An , d + χ(z)) induced by π : A′ → An . The map r is an equivalence (of homotopy ?bers at An and at A′ ). Therefore, ρn is also an equivalence. The collection of functors ρn sums up to an equivalence holim(ρ) : holim{n → Del TA (Vn ) } → holim{n → Def An }. We have already mentioned that the left hand side is equivalent to Del TA . The right-hand side is equivalent to Def A by the descent property 3.2.3.

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VLADIMIR HINICH

4. Examples 4.1. Deformations of schemes. Let X be a scheme over a ?eld k of characteristic a? zero. Denote by XZar the site of a?ne open subschemes of X with the Zariski topology. a? a? The topoi corresponding to the sites XZar and XZar being equivalent, we can use XZar to a? describe deformations of X. The structure sheaf O considered as an object in C((XZar )k ), is ?brant. Therefore, our general result Theorem 3.5.5 is applicable under some ?niteness conditions. We get the following. 4.1.1. Corollary. Let X be a scheme over a ?eld k of characteristic zero. Suppose that X admits a ?nite dimensional hypercover by a?ne open subschemes. Then the functor of formal deformations of X, Def X , is equivalent to the simplicial Deligne groupoid Del T where T is the dg Lie algebra of global derivations of a co?brant resolution of O.

E O be a co?brant resolution of O. Since O is ?brant and p is an Proof. Let p : A acyclic ?bration, A is ?brant as well. The conditions 3.5.1 are ful?lled, so Theorem 3.5.5 gives the claim in question.

The same reasoning provides a similar description of formal deformations of a quasicoherent sheaf of algebras. 4.1.2. Corollary. Let X be a scheme over a ?eld k of characteristic zero. Suppose that X admits a ?nite dimensional hypercover by a?ne open subschemes. Let A be a quasicoherent sheaf of algebras over a linear operad P . Then the functor of formal deformations of P -algebra A is equivalent to the simplicial Deligne groupoid Del T where T is the dg Lie algebra of global derivations of a co?brant resolution of A. Note that here there is no connection between the P -algebra structure and the OX -module structure on A: we use the fact that A is quasi-coherent in order to deduce that Hi (A) = 0 for positive i. 4.2. Obstruction theory. The tangent Lie algebra T in Corollary 4.1.2 is not easy to determine. The classical obstruction theory task, the determination of the cohomology of T is a much easier problem. Assume we are dealing with deformations of associative algebras. Let A be a quasicoherent sheaf of associative OX -algebras. We wish to describe deformations of A as a k-algebra.

a? Consider A as a presheaf on the site XZar of a?ne open subschemes of X and let φ : E A be a co?brant resolution. According to our de?nition, TA = Der ? (P, P ) governs P local deformations of A. Let C ? (P, P ) be the Hochschild cochain complex for P . Let π : C ? (P, P ) E P be the obvious projection to P = C 0 (P, P ). The natural map

iP : Der ? (P, P ) → cone(π : C ? (P, P ) → P )

DEFORMATIONS OF SHEAVES

37

from the presheaf of derivations of P to the (shifted and truncated) Hochshild cochains presheaf, is a weak equivalence. According to Lemma 1.7.1, C ? (P, P ) represents RHom ? op (A, A), the (local) Hochschild A?A cohomology of A. This gives the exact sequence ...

E

HH i (A, A)

E

H i (X, A)

E

H i (T )

E

HH i+1(A, A)

E

...

where the global Hochschild cohomology HH i(A, A) is de?ned as H i (R HomA?Aop (A, A)) (compare to Lunts’ [Lu], Cor. 5.4). 4.3. Standard complex. It seems too naive to expect that the standard complex cone(C ? (A, A)

E

A))

represent the tangent Lie algebra TA . However, in a very special case of associative deformations of the structure sheaf of a smooth variety, the version of the standard complex based on cochains which are di?erential operators in each argument, gives a correct result. In a more detail, let A be the structure sheaf of a smooth algebraic variety X over k, ? and let Cdo (A, A) denote the subcomplex of C ? (A, A) consisting of cochains given by di?erential operators in each argument.

? Let us compare the Hochschild cochains C ? (P, P ) and Cdo (A, A). One has the following commutative diagram

T

? ? ? ? ? ? d d φ′′ d d ? d

? Cdo (A, A)

C ? (P, P )

d d φ′ d d ? d ? ? ? ?

ψ ?

?

C ? (P, A) where T is de?ned to make the diagram cartesian. The map φ′ is an acyclic ?bration since P ?n are co?brant complexes of presheaves and φ is an acyclic ?bration. Therefore, φ′′ is an acyclic ?bration of dg Lie algebras. By a version of Hochschild-Kostant-Rosenberg theorem proven in [Y], ψ is a weak equivalence of complexes of presheaves. This gives a ? weak equivalence of Lie dg algebras C ? (P, P )[1] and Cdo (A, A)[1].

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4.4. Equivariant deformations. Let G be a group and let g be a dg Lie algebra governing formal deformations of some object A (we are informal at the moment). If G acts on A, one should expect a G-action to be induced on g. One can expect that the equivariant deformations of A are governed by the dg Lie algebra RΓG (g) whose i-th cohomology is H i (G, g). We are able to prove this when G is a formal group, “an object” means “a sheaf of operad algebras”, under the restrictions of Theorem 3.5.5. In 4.4.1–4.4.5 we discuss the action of formal groups on sites. In 4.4.6 we describe the equivariant deformation functor. The formula for the equivariant tangent Lie algebra is deduced in 4.4.9. 4.4.1. In this subsection a formal group is a functor G : art(k)

E

Groups

from the category of artinian local k-algebras to the category of groups, commuting with the ?ber products. According to the formal Lie theory, the ?ber at 1 functor G(R) = Ker(G(R)

E

G(k))

is uniquely de?ned by the corresponding Lie algebra g (possibly, in?nite-dimensional). The group G(k) of k-points of G acts on g (adjoint action). Thus, a formal group G in our sense is described by a pair (G(k), g) consisting of a discrete group G(k), a Lie algebra g and an action of G(k) on g. A representation of a formal group G is a k-vector space V together with a collection of compatible operations G(R) E GLR (R ? V ). This can be rephrased in terms of the corresponding pair (G(k), g) as follows. To de?ne a representation of the formal group corresponding to (G(k), g) on a k-vector space V , one has to de?ne the action of G(k) and of g on V satisfying the compatibility condition γ(x)(v) = γ(x(γ ?1 v)), γ ∈ G(k), x ∈ g, v ∈ V.

4.4.2. Let X be a site and O be a sheaf of commutative k-algebras on X. An action of a formal group G on the ringed site (X, O) is a collection of compatible actions of groups G(R) on (X, R ? O) for R ∈ art(k). If a formal group G is described by a pair (G(k), g) as in 4.4.1, its action on (X, O) is given by an action of the discrete group G(k) on (X, O), action of g by vector ?elds on O(U) for each U ∈ X, subject to the compatibility γ(x)(γ(f )) = γ(x(f )).

DEFORMATIONS OF SHEAVES

39

In the case O = k (our assumption below) we will assume that the action of g on k is trivial, so that the action of G on X is reduced to the action of the discrete group G(k). 4.4.3. Now we are able to de?ne G-equivariant (pre)sheaves on X. Suppose a formal group G acts on X as above. A G-(pre)sheaf M is given by a (pre)sheaf M of k-vector spaces on X together with a compatible collection of structures of G(R)-module on Rmodule R ? M. If G is presented by a pair (G(k), g) as above, a G-module structure on M amounts to a compatible collection of maps γ : M(U) E M(γ U) (γ ∈ G(k)), a collection of actions g E End(M(U)) satisfying the condition γ(xm) = γ(x)γ(m). This implies the following Proposition. There is an explicitly de?ned ringed site (X/G, U) such that the category of G-(pre)sheaves of k-modules on X is equivalent to that of (X/G, U)-modules. Proof. The category X/G has the same objects as X; for U, V ∈ X one has HomX/G (U, V ) = {(γ, f )|γ ∈ G(k), f ∈ HomX (U,γ V )}. The composition of morphisms in X/G is de?ned in a standard way, so that the morphism de?ned by a pair (γ, f ) is denoted as γf . One has the identity f γ = γ γ f. The topology on X/G is generated by that on X. We de?ne the sheaf of rings U as the shea??cation of a presheaf U ′ de?ned below. Let U ∈ X. We set U ′ (U) to be the enveloping algebra of g. If f : U

E

V is a map in X, the corresponding map U ′ (V )

E

U ′ (U)

is identity. For γ :γ U

E

U the corresponding map U ′ (U)

E

U ′ (γ U)

is induced by the automorphism of the Lie algebra g sending x ∈ g to γ(x). A straightforward check shows the pair (X/G, U) satis?es the required property. 4.4.4. A special case X = ? of the above construction gives the ringed site BG = (?/G, U). As a category, this is the classifying groupoid of G(k); the sheaf of rings is de?ned by the enveloping algebra Ug endowed with the adjoint G(k)-action. Sheaves on this site are G-modules.

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4.4.5. If M is a presheaf of complexes on X/G, we denote by M # the presheaf on X obtained by forgetting the G-structure. If M is ?brant, M # is ?brant as well; the global sections RΓ(X, M # ) admit a natural G-structure, that is de?ne a sheaf on BG (this is E BG induced by the morphism just the higher direct image of the morphism X/G E ?). We denote this sheaf RΓ(X, M). One has X RΓ(X/G, M) = RΓG (RΓ(X, M)). 4.4.6. Equivariant deformation functor. Let X be a site and let a formal group G act on f? X (through G(k)). Let O be a G-equivariant operad. Let W?,G (R), R ∈ dgart≤0 (k), be the weak groupoid of R-?at sheaves of equivariant R ?O-algebras. Let A be an O-algebra in C(X ? , k) endowed with a G-action. Similarly to 3.1.4, we de?ne the equivariant deformation functor Def A,G : dgart≤0 (k) as the homotopy ?ber at A of the functor

f? W?,G (R)

E

sGrp

E

f? W?,G (k)

induced by the projection R → k. 4.4.7. ... and its tangent Lie algebra. We assume that the conditions of Theorem 3.5.5 are satis?ed for O-algebra A. In particular, deformations of A are governed by the global tangent Lie algebra TA which can be calculated using a co?brant resolution P of A in the category Alg(O, Xk ) by the formula TA = RΓ(X, Der ? (P, P )). We wish to express the functor of equivariant deformations through TA . This can be done as follows. Consider the ringed site (X/G, U) described in 4.4.3. Sheaves (resp., presheaves) on (X/G, U) are precisely equivariant sheaves (resp., presheaves) on (X, k). We de?ne a new operad O#G in C((X/G)k ) as the one governing equivariant O-algebras on X. It is explicitly given by the formula O#G(n) = O(n) ? U ?n , with the operations uniquely de?ned by the G-action on O U ? O(n)

?n ?id

E

U n+1 ? O(n)

E

U ? O(n) ? U ?n

E

O(n) ? U ?n ,

where the second arrow swaps the arguments and the third one is de?ned by the G-action on O(n). This is a generalization of the twisted group ring construction.

DEFORMATIONS OF SHEAVES

41

4.4.8. Lemma. 1. The forgetful functor # : (X/G)U → Xk preserves weak equivalences, ?brations and co?brations. 2. The same is true for the forgetful functor # : Alg(O#G, (X/G) )

E

Alg(O, X ).

Proof. The proofs of both claims are identical. Since shea??cation commutes with #, weak equivalences are preserved. Co?brations are preserved since joining a section over U ∈ X corresponds, after application of #, to joining sections corresponding to a chosen basis of Ug over all γ(U), for γ ∈ G. Finally, since any hypercover in X/G is isomorphic to a hypercover in X, ?brations are also preserved. 4.4.9. Thus, A can be considered as a sheaf of algebras on X/G. By Lemma 4.4.8 the condition Hi (A) = 0 for i > 0 is valid also when A is considered as a sheaf on X/G. Therefore, Theorem 3.5.5 is applicable to A in the equivariant setting. Let us calculate the equivariant global tangent Lie algebra. Choose an equivariant ?brant co?brant representative P of A. We can calculate both equivariant and non-equivariant tangent Lie algebras using P . Thus, the equivariant local tangent Lie algebra is TA,G = Der ? X/G (P, P ) and the one in X is TA = Der ? X (P # , P # ).

# Note that TA = TA,G . Since TA,G is ?brant by 2.4.5, one has

RΓ(X/G, TA,G) = RΓG ? RΓ(X, TA,G ). Appendix A. Simplicial categories and all that A.1. Simplicial categories. Throughout the paper simplicial category means a simplicial object in the category Cat of small categories having a discrete simplicial set of objects. The category of simplicial categories is denoted by sCat. A.1.1. The functor π0 : sCat → Cat is de?ned by the formulas Ob π0 (X) = Ob X Homπ0 (X) (x, y) = π0 (HomX (x, y)).

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A.1.2. Any simplicial category can be considered as a simplicial object in Cat. Applying the nerve functor N : Cat → ?op Ens layer by layer and taking the diagonal, we get a simplicial set called simplicial nerve (or just nerve) N (C) of a simplicial category C. A.1.3. Model structure. In this paper we use a model category structure on sCat de?ned in [H3]. A.1.4. De?nition. A map f : C → D in sCat is called a weak equivalence if the map π0 (f ) : π0 (C) → π0 (D) induces a weak homotopy equivalence of the nerves and for each x, y ∈ Ob(C) the map Hom? (x, y) → Hom? (f (x), f (y)) of the simplicial Hom-sets is a weak equivalence. Sometimes the following notion of strong equivalence is useful. A.1.5. De?nition. A weak equivalence f : C → D is called strong equivalence if the functor π0 (f ) : π0 (C) → π0 (D) is an equivalence of categories. A.1.6. Co?brations in sCat are generated by the following maps: (cof-1) ? → ?, the functor from the empty simplicial category to a one-point category. (cof-2) For each co?bration K → L in ?op Ens the induced map from K01 to L01 . Here K01 denotes the simplicial category having two objects 0 and 1 with the only nontrivial maps K = Hom? (x, y). A.1.7. Theorem. ([H3]) The collections of co?brations and of weak equivalences de?ne a CMC structure on sCat. The maps in sCat satisfying RLP with respect to acyclic co?brations (=co?brations + weak equivalences) will be called ?brations. Recall for the sake of completeness the explicit de?nition of ?bration. A.1.8. De?nition. following properties A map f : C → D in sCat is called a ?bration if it satis?es the

(1) the right lifting property (RLP) with respect to the maps ? 0,1 : ?0 → ?1 from the terminal category ?0 = ? to the one-arrow category ?1 . (2) For all x, x′ ∈ Ob C the map f : Hom? (x, x′ ) → Hom? (f x, f x′ ) is a Kan ?bration.

DEFORMATIONS OF SHEAVES

43

A.1.9. Simplicial structure and Tot. The category sCat admits a structure of a simplicial model category. Let S be a simplicial set, C be a category and let N C be the nerve of C. The simplicial set Hom? (S, N C) is the nerve of a category which will be denoted by C S . Let X = {Xn } ∈ sCat, S ∈ ?op Ens. The collection

S n → Xn

forms a simplicial object in Cat. We de?ne X S to be the simplicial category given by the formulas (19) (20)

S Ob X S = Ob X0 S HomX S (x, y)n = HomXn (xn , yn )

S S where x, y are objects of X0 and xn , yn are their degeneracies in Xn .

Given a cosimplicial object X ? in sCat, we can now de?ne Tot(X ? ) by the usual formula (21) Tot(X ? ) = lim p→q (X q )? .

←

p

Note that our simiplicial structure (and functor Tot) on sCat is not standard. In [DHK] another de?ntion is given. The de?nitions coincide for simplicial groupoids. However, the de?nition of [DHK] does not provide sCat with a simplicial model structure, and this is the main reason we use the model structure described above. A.1.10. Homotopy limits. The functor Tot de?ned in (21) prescribes a de?nition of the homotopy limit as in, say, [DHK], Chapter XIV. If F : I → sCat is a functor, holim F is de?ned as Tot(F ) where the cosimplicial object F in sCat is de?ned by the standard formula Fn =

i0 →...→in ∈Nn (I)

F (in ).

In the case all F (i) are ?brant in sCat, the homotopy limit holim F represents the right derived functor R lim : Ho(sCatI ) → Ho(sCat). In this case the following holds. A.1.11. Proposition. (see [H3]) Let F : I → sCat be a functor such that F (i) are ?brant for all i ∈ I. Then the natural functor N (holim F (i)) is a weak equivalence.

E

holim(N (F (i)))

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VLADIMIR HINICH

A.1.12. Homotopy ?bers. In this paper we are particularly interested in homotopy ?bers. Let f : C → D be functor in sCat and let d ∈ D. We suppose also that the simplicial categories C and D are ?brant (for instance, simplicial groupoids), so that the homotopy ?ber represents the right derived functor of the usual ?ber product. In this case homotopy ?ber of f at d can be represented by the ?ber product ? ×D D ? ×D C 1 where the map ? → D is given by d ∈ D and the maps from D ? to D are given by the ends of the segment ?1 . This immediately follows from [H3], Prop. A.4.3 claiming, in particular, that the map 1 D ? → D × D is a ?bration for ?brant D. A.2. Weak groupoids. A.2.1. De?nition. groupoid. A simplicial category G is called a weak groupoid if π0 (G) is a

1

The following fact justi?es the above de?nition. A.2.2. Proposition. A simplicial category C is a weak groupoid if and only if it is strongly equivalent to a simplicial groupoid. Proof. The “if” part is obvious. According to [DK2], if C is a weak groupoid, the map C → LH (C, C) from C to the hammock localization of C is a strong equivalence. A.2.3. Corollary. A map f : V → W of weak groupoids is a weak equivalence i? its nerve N (f ) : N (V) → N (W) is a weak homotopy equivalence. Proof. Proposition A.2.2 reduces the claim to simplicial groupoids. In this case the result is proven, for instance, in [H4], 6.2.2, 6.2.3. A.2.4. De?nition. Let W be a simplicial category. Its weak groupoid completion W is the hammock localizaton LH (W, W), see [DK2]. One has a canonical map W → W from a simplicial category to its weak groupoid completion. Since simplicial localizations preserve the homotopy type of the nerve, see [DK1], 4.3, we obtain immediately the following A.2.5. Corollary. A map f : C → D of simplicial categories induces a weak equivalence of the weak groupoid completions C → D i? the nerve N (f ) : N (C) → N (D) is a weak homotopy equivalence.

DEFORMATIONS OF SHEAVES

45

A.3. Weak groupoid of a model category. Let C be a closed model category and let W be the subcategory of weak equivalences in C. One can assign to C a few di?erent weak groupoids. These are weak groupoid completions of the categories W, W c (the objects are co?brant objects of C, the morphisms are weak equivalences), and also W f and W cf . We denote them W, W c , W f and W cf . In the case C admits a simplicial structure so that the axiom (SM7) of [Q1] is satis?ed, cf one de?nes another weak groupoid W? whose objects are co?brant ?brant objects of C and n-morphisms are the ones lying in the components of weak equivalences. Similarly the weak groupoid W? is de?ned. In Proposition A.3.4 below we show that all these weak groupoids are strongly equivalent. A.3.1. Contractibility of resolutions. Homological algebra starts with an observation that resolutions are usually unique up to a homotopy which is itself unique up to homotopy. In this section we prove a generalization of this fact: the category of resolutions has a contractible nerve. This result will be used in the proof of equivalences A.3.4 below.

c Let C be a closed model category, M ∈ C. Let CM denote the full subcategory of the category C/M whose objects are the weak equivalences f : P → M with co?brant P . cf Similarly, CM consists of quasi-isomorphisms P → M with P co?brant and ?brant.

A.3.2.

c Theorem. 1. The nerve of the category CM is contractible.

cf 2. If M is ?brant, the nerve of CM is contractible. c Proof. Step A. First of all, we check that the nerve of CM is simply connected. Suppose c f : P → M and g : Q → M are two objects of CM . Present the map f g : P Q → M as a composition of a co?bration ι and an acyclic ?bration π

P

Q → R → M.

ι

π

c c The map π : R → M presents an object of CM . This proves that the nerve of CM is cf connected. Note that the same construction proves that the nerve of CM is connected if M is ?brant.

To prove that the nerves in question are simply connected, one can pass to groupoid completions and calculate the automorphism group of any object of the obtained groupoid. A standard reasoning shows that an acyclic ?bration f : P → M with co?brant P has no nontrivial automorphisms in the groupoid completion. Step B. Choose an acyclic ?bration f : P → M with co?brant P . This de?nes a functor

c c F : CP → CM

which carries a weak equivalence g : Q → P to f g : Q → M. Fix an object g : R → M c of C c (M). The ?bre category F/g can be easily identi?ed with the category CP ×M R .

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VLADIMIR HINICH

According to Step A, all ?bres of F are simply connected. Note that the nerve of C c (P ) is contractible since the category admits a ?nal object.

c Step C. We have to check that the reduced homology of the nerve of CM vanishes. The c construction of Step B allows to prove this by induction. In fact, let Hi (CM ) = 0 for all c i < n and for all objects M. Then Proposition A.3.3 below shows that Hn (CM ) = 0. This proves the theorem modulo Proposition A.3.3 below.

The following result is very much in the spirit of [Q2], Theorems A and B. A.3.3. Proposition. Let F : C → D be a functor. Suppose that

(a) the nerve of C is contractible. (b) Hi (F/d) = 0 for all i < n, d ∈ D. Then Hn (D) = 0. Proof. Consider the bisimplicial set T?? (used by Quillen in the proof of Theorem A, cf. [Q2], p. 95) de?ned by the formula Tpq = {cq → . . . c0 ; F (c0 ) → d0 → . . . → dp }, with the obvious faces and degeneracy maps. The diagonal of this bisimplicial set, diag T , is homotopy equivalent to the nerve of C (see [Q2], p. 95). Therefore, it has trivial homology. Recall that the homology of a simplicial set X can be calculated as follows. First, one considers ZX ∈ ?op Ab. Then one applies Dold-Puppe equivalence of categories Norm : ?op Ab → C ≤0 (Z). Finally, one has Hi (X) = H ?i(Norm(ZX)). If Y ∈ (?op )2 Ab, we denote Norm2 (Y ) the bicomplex obtained from Y by normalization in both directions. Lemma. Norm(diag(Y )) = Tot(Norm2 (Y )). The lemma is similar to Quillen’s lemma at p. 94, [Q2]. One checks it for representable Y = Zhpq where hpq = ?p × ?q , and then checks that both sides of the equality commute rs r s with the direct limits. Now consider the map Tpq =

d0 →...→dp

N (F/d0 ) →

d0 →...→dp

pt = Np (D).

DEFORMATIONS OF SHEAVES

47

Let Z be the bicomplex corresponding to the bisimplicial abelian group ZT?? . Denote by H vert and H hor the homology with respect to the vertical (of degree (0, 1)) and the horizontal (of degree (1, 0)) di?erential. By the assumptions of the proposition, one has vert vert Hq (Z) = 0 for q = 1, . . . , n ? 1 and H0 (Z) = Normp (N (D)). Look at the spectral sequence 2 hor vert Epq = Hp Hq (Z) ? Hn (Tot(Z)). According to the above lemma, the spectral sequence converges to zero. Our calculation 2 2 shows that Epq = 0 for q = 1, . . . , n ? 1 and Ep0 = Hp (N (D)). This implies Proposi2 ∞ tion A.3.3 since the map Hn (N (D)) = En0 → En should be injective. A.3.4. Proposition. (i) The weak groupoids W, W c , W f and W cf are equivalent.

(ii) Suppose C admits a simplicial structure and suppose that (SM7) and half the axiom (SM0) of [Q1], (existence of simplicial cylinder or path spaces), is ful?lled. Then the weak cf groupoids W? and W? are also equivalent to the above. Proof. (i) According to A.2.3 it is enough to prove the nerves of the four categories are weakly homotopically equivalent. This follows from Theorem A.3.2 by Quillen’s Theorem A, see [Q2].

cf (ii) Since W? is a weak groupoid, it is equivalent to its weak groupoid completion. In order to prove that the latter is equivalent to the weak groupoid completion of W cf , one has to cf cf compare the (simplicial) nerves of W? and of W cf = W0 . This will immediately follow cf cf once we prove that the multiple degeneracy s : W0 → Wn induces a weak equivalence cf cf N (s) : N (W0 ) → N (Wn ).

Let us suppose that the simplicial path functor exists. Then Homn (x, y) = Hom(x, y ? ). Therefore, the ?bre s/y is equivalent to the category of ?brant co?brant resolutions of n the object y ? . This object being ?brant, Theorem A.3.2 asserts that s/y is contractible. Once more Quillen’s Theorem A accomplishes the proof. The case when the simplicial cylinder functor exists, as well as the weak groupoid W? are treated similarly. A.4. Simplicial Deligne groupoid. In this paper weak groupoids appear as values of a formal deformation functor on artinian algebras. One looks for a presentation of such a functor with a dg Lie algebra. We recall below three functors assigned to a dg Lie algebra, of which the last one is used in this paper. A.4.1. Deligne groupoid (see [GM]). Let g be a Lie dg algebra and (R, m) ∈ dgart≤0 (k). Deligne groupoid Delg(R) has as objects the Maurer-Cartan elements of m ? g, 1 Ob Delg(R) = MC(m ? g) := {z ∈ (m ? g)1 |dz + [z, z] = 0}. 2 0 The group exp(m ? g) acts in a natural way on the set (A.4.1) and one de?nes HomDelg (R) (z, z ′ ) = {γ ∈ exp(m ? g)0 |z ′ = γ(z)}.

n

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VLADIMIR HINICH

This de?nition is homotopy invariant if one requires (m ? g)i = 0 for i < 0. A.4.2. Nerve of a dg Lie algebra (see [H1, H3]). Let g and R be as above. De?ne for n≥0 Σg(R)n = MC(?n ? m ? g) where ?n is the algebra of polynomial di?erential forms on the standard n-simplex. Then Σg(R) is a Kan simplicial set, its fundamental groupoid is canonically identi?ed with Delg(R) and the two are homotopically equivalent if m ? g is non-negatively graded. A.4.3. Simplicial Deligne groupoid (see [H4]). Here g, R are as before. De?ne simplicial groupoid Del g(R) as follows. Its objects are the objects of Delg(R). The collection of n-morphisms from z to z ′ coincides with the collection of such morphisms in the Deligne groupoid corresponding to the Lie dg algebra ?n ? g. The following lemma connects between the di?erent constructions. A.4.4. Lemma. 1. There is a natural weak equivalence of simplicial sets N (Del g(R)) ?→ Σg(R). 2. Deligne groupoid Delg naturally identi?es with the fundamental groupoid Π(Σg) of the nerve Σg. The composition Σg(R) → Π(Σg(R)) = Delg(R) is a weak equivalence if m ? g is non-negatively graded. A.4.5. Descent. All three functors mentioned above are de?ned by a nilpotent dg Lie algebra gR := m ? g: one has Delg(R) = Del(gR ), Σg(R) = Σ(gR ) and Del g(R) = Del (gR ) in an obvious notation. In what follows we will ?x once an forever the commutative dg algebra R and we will erase the subscript R from the notation. Let g? be a nilpotent cosimplicial dg Lie algebra. A natural morphism of simplicial sets Σ(Tot(g? )) → Tot(Σ(g? )) can be easily constructed. The main result of [H1] claims that this map is a homotopy equivalence provided g? is ?nitely dimensional in the cosimplicial direction. In the main body of the paper we need a similar result in the context of simplicial Deligne groupoids. Let us show it easily follows from the result of [H1]. Let us construct a map of simplicial groupoids (22) Del(Tot(g? )) → Tot(Del(g? )).

?

On the level of objects the map is constructed as follows. An object of the left-hand side is an element of lim p→q MC(?p ? gq ).

←

DEFORMATIONS OF SHEAVES

49

An element of MC(?p ? gq ) = Σp (gq ) gives rise to a sequence of p 1-simplices in Σ(gq ); passing to the fundamental groupoid we get a p-simplex in the groupoid Del(gq ). Since an object of the right-hand side of (22) is an element of lim p→q Ob Del(gq )? ,

←

p

the morphism (22) is de?ned on the level of objects. Fix z, z ′ ∈ MC(Tot(g? )). A n-map from z to z ′ on the left-hand side of (22) is given by an element γ ∈ exp(?n ? Tot(g? ))0 satisfying the equation z ′ = γ(z). The composition exp(?n ? Tot(g? ))0 → exp(Tot(?n ? g? )) = lim p→q exp(?p ? ?n ? gq )0 →

←

→ lim p→q Np (Del(?n ? gq ))

←

de?nes the map (22) for the n-morphisms. Proposition. Suppose g? is ?nite dimensional in the cosimplicial direction, i.e. there exists n such that the intersection of kernels of all codegeneracies vanishes in degrees > n. Then the morphism (22) de?ned above is an equivalence. To prove a map of simplicial groupoids is an equivalence, it is enough to check it induces a homotopy equivalence of the nerves. Applying the nerve functor to the both sides of (22), we get the morphism (A.4.5) which is an equivalence by [H1]. This proves the proposition.

Appendix A. Simplicial presheaves A model category structure similar to the one described in Section 1 exists also on the category of simplicial presheaves. This model category structure di?ers from the one de?ned in [Ja]. More precisely, weak equivalences are the same; we have much less co?brations and, consequently, much more ?brations. Let X be a site. Since the category ?op Ens of simplicial sets is co?brantly generated, the category of simplicial presheaves ?op (X ) admits a CMC structure in which co?brations are generated by co?bration in ?op Ens, see [DHK], 9.6 and 6.10. In this model structure a map f : A → B is a weak equivalence (resp., a ?bration) i? for each U ∈ X f (U) : A(U) → B(U) is a weak equivalence (resp., a ?bration). The collection of co?bration is generated by gluing simplices along a boundary over some U ∈ X. This is the model structure we have in mind in the case X is endowed with the coarse topology. In order to describe the model structure on ?op (X ) which remembers the topology of X, we rede?ne the notion of weak equivalence as in [Ja], leaving the notion of co?bration unchanged. Recall the following de?nition due to Joyal [Jo] and Jardine, [Ja].

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For A ∈ ?op X one de?nes π0 (A) as the shea??cation of the presheaf U → π0 (A(U)). Similarly, for n > 0 and a ∈ A(U)0 one de?nes πn (A, a) as the shea??cation of the presheaf V → πn (A(V ); a) de?ned on X/U. A.1. De?nition. A map of simplicial presheaves f : A → B is a weak equivalence if

? π0 (f ) : π0 (A) → π0 (B) is an isomorphism; ? for each a ∈ A(U)0 the map πn (f ) : πn (A; a) → πn (B; f (a)) is an isomorphism. A.2. Theorem. The category ?op (X ) endowed with the classes of weak equivalences described above and co?brations as for the coarse topology, is a closed model category. A nice feature of this model structure is the following description of ?brations (see Proposition A.2.12 below). Proposition. A map f : M → N ∈ ?op (X ) is a ?bration if and only if the following conditions are satis?ed. ? f (U) : M(U) → N(U) is a Kan ?bration for each U ∈ X. ? For each hypercover ? : V? → U the comutative diagram M(U)

E

ˇ C(V? , M) := Tot M(V? )

c

N(U) is homotopy cartesian.

E

c

ˇ C(V? , N)

A.2.1. Remark. The Cech complex appearing in the description of ?brations, is not homotopy invariant, even for pointwise ?brant presheaves. This means that that ?brantness is not necessarily preserved under pointwise weak equivalence of pointwise ?brant presheaves. A pointwise ?brant presheaf F is called to satisfy descent property with respect to a hypercover V? → U if F (U) is homotopy equivalent to holim F (V? ). In general, our ?brant presheaves do not satisfy the descent property. However, suppose ? V? is split, see [SGA4], Exp. Vbis, 5.1.1. This means that for each n the map Dn → Vn from the subpresheaf of degenerate n-simplices to the presheaf of all n-simplices, can be presented as a composition Dn → Dn ? Nn ?→ Vn .

?

DEFORMATIONS OF SHEAVES

51

Then, for each acyclic co?bration S → T of simplicial sets the induced map of simplicial presheaves

S×Dn

(S × Vn )

(T × Dn ) → T × Vn

is an acyclic co?bration. This implies that if F is a ?brant presheaf and V? is a split hypercover, then F (V? ) is a ?brant cosimplicial simplicial set in the sense of [BK], X.4. This implies that Tot F (V? ) is homotopy equivalent to holim F (V? ) and, therefore, F satis?es the descent property with respect to V? . This implies that if any hypercover in X is re?ned by a split hypercover, our model structure coincides with the one named UC/S in [DHI], Theorem 1.3. A.2.2. To prove the theorem, we describe a collection of morphisms which are simultaneously weak equivalences and co?brations. These morphisms are called generating acyclic co?brations. Theorem A.2 then follows from Lemma A.2.10 below claiming that weak equivalences satisfying the RLP with respect to all generating acyclic co?brations, are pointwise acyclic Kan ?brations. A.2.3. The following version of [SGA4], V.7.3.2, plays a very important role here. Let E be a topos, M, N ∈ ?op E. We will use the notion of weak equivalence in ?op E de?ned in A.1. Proposition. Let f : M → N be a morphism of simplicial objects in E. Suppose that

? fp is an isomorphism for p < n. ? fn is an epimorphism. ? morphisms M → coskn (M), N → coskn (N), are isomorphisms. Then f is a weak equivalence. Proof. The proof is essentially the same as in [SGA4], V.7.3.2. Embed E into a topos E ′ having enough points. If E = X ? , E ′ can be taken to be X . Let a? : E ′ → E : a? ← be the corresponding inverse and direct image functors. We construct f ′ : M ′ → N ′ as follows. M ′ = a? (M). ′ ′ Ni′ = a? (Ni ) for i < n; Nn = a? (f )(Mn ); N ′ = coskn (N ′ ). f ′ = a? (f ) : M ′ → N ′ . Then f = a? (f ′ ). The functor a? preserves weak equivalences. In fact, weak equivalences can be described using ?nite limits and arbitrary colimits which are preserved by the inverse image functor.

52

VLADIMIR HINICH

This reduces the claim to the case E has enough points. In this case f is a weak equivalence i? for each point x fx is a weak equivalence. Inverse image functor preserves the properties listed in the proposition. Thus everything is reduced to the case E = Ens where the claim is well-known.

A.2.4. We denote by ?+ the category obtained from ? by attaching an initial object ? = [?1]. In particular, any hypercover ? : V? → U de?nes an object V? ∈ ?op X with + V?1 = U. In what follows we will use the subscript (resp., the superscript) ? to denote the augmented simplicial (resp., cosimplicial) object. Let A? ∈ ?+ (?op Ens) and let V? ∈ ?op X . + ? We de?ne a simplicial presheaf A? ? V? by the formula ? (A ? V )n = lim p→q∈?+ Ap × Vq . n

→

A.2.5. Let us make a few calculations. Let, for instance, Ap = Kn for some K ∈ ?op Ens. n Then A ? V = K × V?1 . A.2.6. Another important example is Ap = Kn × ?p . n n One can easily see that (? ? V? ) = V? . Also, if K ∈ ?op Ens then for any A? one has ? (K × A? ) ? V? = K × (A? ? V? ). ? ? Thus we ?nally get (K × ?) ? V? = K × V? . An easy calculation shows that for a simplicial presheaf M one has ˇ Hom(A ? V? , M) = Hom(K, C(V? , M)). A.2.7. Generating acyclic co?brations for our model structure consist of two collections. The ?rst collection is numbered by U ∈ X and a pair of integers (i, n) such that 0 ≤ i ≤ n. It consists of maps Λn × U → ?n × U. i This collection de?nes the model category structure on the simplicial presheaves on X corresponding to the coarse topology. The second collection is numbered by hypercovers V? of X and integers n ≥ 0. It is de?ned as follows.

DEFORMATIONS OF SHEAVES

53

Let ?+ (don’t confuse with ?+ !) denote the cosimplicial simplicial space with ?+n = ?n+1 δ +i = δ i σ +i = σ i . The map i : ? → ?+ is de?ned by the “last face”: in = δ n+1 : ?n → ?n+1 . De?ne A(n) by the cocartesian diagram i ??n × ? E ??n × ?+ (23) ? ×?

n

c

E

c

A(n)

and put B(n) = ?n × ?+ . Then the natural map A(n) ? V → B(n) ? V is the basic acyclic co?bration corresponding to the pair (V, n). A.2.8. The object ?+ ∈ ?+ ?op Ens identi?es with the pushout of the diagram ? δ 0 × idE ? ?1 ? ?

E ?+ ? Under this identi?cation the map i : ? → ?+ is induced by

c

c

δ 1 × id? : ? → ?1 × ?. Therefore, ?+ ? V? can be calculated from the cocartesian diagram V? δ 0 × idV? E ?1 × V?

c

V?1 Lemma.

v E + c ? ? V?

The morphism v : V ?1 → ?+ ? V? is a weak equivalence.

54

VLADIMIR HINICH

Proof. The simplicial set ?1 being contractible, the map δ 0 × idV? is a weak equivalence. It is also pointwise injective. This means that it is a trivial co?bration in the sense of Jardine [Ja]. Then Proposition 2.2 of [Ja] asserts that v is a weak equivalence.

A.2.9. Lemma. Let V? be a hypercover and n ∈ N. The map A(n) ? V? → B(n) ? V? of simplicial presheaves is a co?bration and a weak equivalence. Proof. The map is co?bration since Vi are coproducts of representable presheaves. Let us check the acyclicity. Tensoring diagram (23) by V? and using Proposition 2.2 of [Ja] we get that A(n) ? V? is equivalent to V? . The tensor product B(n) ? V? is obviously equivalent to V?1 . By Proposition A.2.3 the map ? : V? → V?1 is a weak equivalence. This implies the lemma. The following lemma is the analog of Lemma 1.3.6.

A.2.10. Lemma. Let f : M → N be a weak equivalence. Suppose f satis?es the RLP with respect to all generating acyclic co?brations. Then f (U) : M(U) → N(U) is an acyclic Kan ?bration for all U ∈ X. Proof. The map f (U) is a ?bration for any U ∈ X. Fix U and ?x a section of N over U. Let F be the ?ber of f at the chosen section. We wish to prove that F (U) is acyclic. To prove this we have to check that any commutative diagram

??n .. .. .

E

. ..

F (U) . .

.. .. c ?n

can be completed with a dotted arrow. Below we denote A(n) by A. Since f is a weak equivalence, there exists a covering ? : V0 → U and a dotted arrow A0 → F (V0) making the diagram

DEFORMATIONS OF SHEAVES

55

A?1 = ??n

E

F (U) .. . . . .. . . .. . .. . .

E F (V0 ) . B . . ..

c

A commutative.

0

.. . .

..

.. . . .. .

Suppose, by induction, a k-hypercover ? : V≤k → U and a collection of compatible maps Ai → F (Vi), i ≤ k, has been constructed. This induces a map skk (A? )k+1 → F (Vk ). Since the shea??cation of the homotopy groups πi (F ) vanishes, there exists a covering Vk+1 → coskn (V )n+1 and a map Ak+1 → F (Vk+1) compatible with he above. Therefore, a hypercover ? : V? → U and a map A(n) ? V → F is constructed. According to hypothesis of the lemma, this latter can be extended to a map B(n) ? V → F which includes a map B ?1 = ?n → F (U). Lemma is proven. A.2.11. Let J be the collection of generating acyclic co?brations. Let, furthermore, J denote the collection of maps which can be obtained as a countable direct composition of pushouts of coproducts of maps in J. Fibrations are de?ned as the maps satisfying RLP with respect to J. Repeating the reasoning of 1.3.7, we prove this de?nes a model structure on ?op X . As in 1.3.7, acyclic co?brations in this model structure are retracts of elements of J. Theorem A.2 is proven. Note the following description of ?brations. A.2.12. Proposition. A map f : M → N ∈ ?op (X ) is a ?bration if and only if the following conditions are satis?ed. ? f (U) : M(U) → N(U) is a Kan ?bration for each U ∈ X. ? For each hypercover ? : V? → U the comutative diagram M(U)

E

ˇ C(V? , M) := Tot M(V? )

c

N(U) is homotopy cartesian.

E

c

ˇ C(V? , N)

56

VLADIMIR HINICH

Proof. A map f : X → Y of Kan simplicial sets is a weak equivalence i? the space X ×Y P (Y ) ×Y ? is acyclic Kan for each point ? → Y . Here P (Y ) is the path space of Y . The rest of the proof is a direct calculation. References

[BK] Bous?eld, Kan, Homotopy limits, completions and localizations, Lecture notes in mathematics, 304. [DHI] D. Dugger, S. Hollander, D. Isaksen, Hypercovers and simplicial presheaves, preprint math.AT/0205027 [DHK] W. Dwyer, P. Hirschhorn, D. Kan, Model categories and more general abstract homotopy theory, preprint available at www-math.mit.edu/~psh/#Mom. [DK1] W. Dwyer, D. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17(1980), 267– 284. [DK2] W. Dwyer, D. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18(1980), 17–35. [DK3] W. Dwyer, D. Kan, Function complexes in homotopical algebra, Topology 19(1980), 427–440. [F] Y. Felix, Classi?cation homotopique des espaces rationnels de cohomologie donn?e, Bull. Soc. e Math. Belg. S?r. B 31 (1979), no. 1, 75–86. e [G] D. Gaitsgory, Grothendieck topologies and deformation theory, II, Compositio Math., 106(1997), 321–348. [GS] M. Gerstenhaber, Schack, Algebraic cohomology and deformatioin theory, Deformation theory of algebras and structures and applications, 11–264, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 247, Kluwer, 1988. [GM] W. Goldman, J. Millson, The deformation theory of representations of fundamental groups of compact K¨hler manifolds, Publ. Math. IHES, 67 (1988), 43–96. a [HS] S. Halperin, J. Stashe?, Obstructions to homotopy equivalences, Adv. in Math. 32 (1979), no. 3, 233–279. [H1] V. Hinich, Descent of Deligne groupoids, Intern. Math. Res. Notices, 1997, 223-239. [H2] V. Hinich, Homological algebra of homotopy algebras, Comm. in Algebra, 25(10)(1997), 3291– 3323. [H3] V. Hinich, DG coalgebras as formal stacks, J. Pure Appl. Algebra, 162(2001), no. 2-3, 209–250. [H4] V. Hinich, Deformations of homotopy algebras, preprint math.AG/9904145, to appear at Communications in Algebra. [H5] V. Hinich, Virtual operad algebras and realization of homotopy types, J. Pure Appl. Algebra, 159(2001), 173–185. [HS] A. Hirschowitz, C. Simpson, Descente de n-champs, preprint math.AG/9807049 [I] L. Illusie, Complexe cotangent et d?formations, Lecture notes in mathematics, 239(1971) and e 283(1972). [Ja] J. F. Jardine, Simplicial presheaves, J. Pure Appl. Algebra, 47(1987), 35–87. [Jo] A. Joyal, lettre ` Grothendieck, le 11/4/84. a [La] Laudal, Formal moduli of algebraic structures, Lecture notes in mathematics, 754, 1979. [Lu] V. Lunts, Deformations of quasi-coherent sheaves of algebras, J. Algebra 259(2003), 59–86. [Q1] D. Quillen, Homotopy algebra, Lecture notes in mathematics., 43(1967). [Q2] D. Quillen, Higher algebraic K-theory, I, Lecture notes in mathematics., 341(1973), 85–147. [Sch] M. Schlessinger, Functors on Artin rings, Trans. A.M.S., 130(1968), 208–222. [SchSt] M. Schlessinger, J. Stashe?, Deformation theory and rational homotopy type, preprint. [SGA4] Th?orie des topos et cohomologie ?tale des sch?mas, S?minaire de la g?om?trie alg?brique du e e e e e e e Bois Marie, 63/64, SGA4, dirig? par M. Artin, A. Grothendieck, J.L. Verdier. Tome 1: Lecture e notes in mathematics, 269; Tome 2: Lecture notes in mathematics, 270(1972).

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[Sp] [T] [Y]

M. Spaltenstein, Resolutions of unbounded complexes. Compositio Math. 65 (1988), no. 2, 121– 154. B. Toen, Dualit? de Tannaka sup?rieure I: Structures mono¨ e e idales, Preprint MPIM 2000 (57). A. Yekutieli, The continuous Hochschild complex of a scheme, Canad. J. Math., 54(6), 2002, 1319–1337.

Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel E-mail address: hinich@math.haifa.ac.il

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