aa r X i v : . [ m a t h . AG ] J u l The deformation theory of sheaves ofcommutative rings
Jonathan Wise Department of Mathematics, The University of British Columbia, Room 121, 1984Mathematics Road, Vancouver, B.C., Canada V6T 1Z2
Abstract
We define a sheaf of abelian groups whose cohomology is represented by thecotangent complex, permitting a rapid introduction to the theory of the cotan-gent complex in the same generality as it was defined by Illusie, but avoidingsimplicial methods. We show how obstructions to some standard deformationproblems arise as the classes of torsors under and gerbes banded by this sheaf.This generalizes results of Quillen, Rim, and Gaitsgory.
1. Introduction If f : X → Y is a smooth morphism of schemes, and Y ′ is a square-zeroextension of Y such that the ideal of Y in Y ′ is O Y , then the square-zeroextensions X ′ of X over Y ′ are obstructed by a class ω ∈ H ( X, T
X/Y ), where T X/Y is the relative tangent bundle. Should this class vanish, deformations forma torsor under H ( X, T
X/Y ) and automorphisms of any fixed deformation arein bijection with H ( X, T
X/Y ).This may be explained succinctly by allowing X to vary in the definition ofthe deformation problem: the deformations of open subsets of X form a gerbeover X , banded by T X/Y . The obstruction ω is then the class of this gerbe.If ω vanishes, the gerbe is trivial, meaning it is isomorphic to the classifyingstack of T X/Y -torsors, and then the statements regarding isomorphism classesand automorphisms come from the cohomological classification of torsors.If we relax the hypothesis that f be smooth, the above argument fails, buta similar description of obstructions, deformations and automorphisms persists[Il71, Th´eor`eme 2.1.7]: letting L X/Y denote the cotangent complex of X over Y , there is an obstruction ω ∈ Ext ( L X/Y , O X ) whose vanishing is equivalentto the existence of a deformation; if a deformation exists, all deformations form Present address: Department of Mathematics, Stanford University, 450 Serra Mall, Build-ing 380, Stanford, CA, USA 94305E-mail address: [email protected]
Preprint submitted to Elsevier May 29, 2018 torsor under Ext ( L X/Y , O X ) and automorphisms of any given solution arein bijection with Ext ( L X/Y , O X ).The strong resemblance between this result and the one obtained in thesmooth case hints that there may still be a relationship between deformationsand banded gerbes. Moreover, it is only the “local triviality” aspect of a gerbethat fails to apply in the non-smooth case: the solutions to the deformation prob-lem still form a pseudo-gerbe banded by T X/Y , in the sense that isomorphismsbetween any two solutions form a pseudo-torsor under T X/Y , but solutions arenot guaranteed to exist locally, and pairs of solutions are not guaranteed to belocally isomorphic. In other words, the failure of the “gerbe argument” in thenon-smooth case may be attributed to the fact that the deformation problem isnot locally trivial in the Zariski topology on X .All of this suggests that Illusie’s result may be interpreted in terms of gerbesif we can find a topology finer than the Zariski topology in which the defor-mation problem becomes locally trivial. For affine schemes, such a topologywas defined, apparently simultaneously, by Quillen [Qu70] and Rim [SGA7-1,VI.3]. Rim speculated [SGA7-1, VI.3.16] that it might be possible to define ananalogous topology for arbitrary schemes, and Quillen apparently made such adefinition [Il10] but never published it. In [Ga97], Gaitsgory defined a topologyon the category of associative algebras on a scheme and showed that it is fineenough to find local trivializations of deformation problems associated to quasi-coherent algebras. As Gaitsgory notes [Ga97, Section 0.4], his methods may beadapted easily to the commutative case, where they can be used to treat therelative deformation theory of a scheme that is affine over the base. We note,however, that if f : X → Y is a morphism of schemes then O X is, in general, notquasi-coherent as a f − O Y -algebra, so Gaitsgory’s results do not apply directlyto the deformation theory of schemes.The introduction of banded gerbes to explain the obstruction to the exis-tence of algebra extensions is due to Gaitsgory [Ga95]. That deformations, whenthey exist, can be viewed as torsors was observed by Quillen [Qu70, Proposi-tion 2.4 (iv)].In this paper, we will define a new topology on the category of all commu-tative rings in a topos and show that it is fine enough to trivialize the standarddeformation problems about commutative rings, but is still coarse enough toglue their solutions. We obtain cohomological obstructions to the existenceof solutions to these problems and a cohomological description of the solutions,should they exist. As we explain in Section 2, this can be used to apply the ideasof Gaitsgory, Quillen and Rim to the deformation theory of schemes. We alsocompare our approach to cotangent cohomology with Illusie’s, showing that our Quillen attributed his definition to Grothendieck with a pre-publication reference to SGA4that I could not trace. It may be that Quillen was only crediting Grothendieck with the ideaof using the topology generated by universal effective epimorphisms, and not specifically forthe definition of the cotangent cohomology in this way. On the other hand, Quillen’s topologyis identical to Rim’s, so it may also be that Quillen intended to refer to Rim’s expos´e in SGA7. classes agree as well.The ideas in this paper may be applied easily to similar deformation problemsof other algebraic objects. We leave these applications to the reader for now.We hope to explain some of them (such as stable maps and sheaves of modules)in future work. In [Wi11], we will explain how the theory developed here can beused in place of sheaves of simplicial commutative rings to develop the standardproperties of the cotangent complex [Il71, II.2].
2. Summary of results
In this section we shall state our results in the context of schemes in order togive them a geometric appearance; statements in the generality of ringed topoifollow in the body of the text. In order to deduce the statements about schemesgiven in this section from the algebraic statements that follow, one need onlyobserve that infinitesimal extensions of a scheme are equivalent to infinitesimalextensions of their structure sheaves, in either the Zariski or ´etale topology.Suppose that f : X → Y is a morphism of Z -schemes, and consider theproblem of extending f to a fixed square-zero extension X ′ of X over Z withideal J (cf. [Il71, Probl`eme III.2.2.1.2]). Diagramattically, we are attempting tocomplete the commutative diagram of solid arrows X / / (cid:15) (cid:15) Y (cid:15) (cid:15) X ′ / / > > }}}} Z with a dashed arrow making both triangles commute. In Section 5, we shalldefine a site g − O Z − Alg /f − O Y , which we show in Section 6 is fine enough toensure that the deformation problem is locally trivial. In the statement of thetheorem, we abbreviate the name of the site to O Z − Alg / O Y . Theorem.
The extensions
Hom X ( X ′ , Y ) of f to f ′ : X ′ → Y form a torsoron O Z − Alg / O Y under the sheaf of abelian groups Der O Z ( O Y , J ) (defined inSection 6). The class of this torsor in H ( O Z − Alg / O Y , Der O Z ( O y , J )) obstructsthe existence of a lift. Provided that the obstruction vanishes, all lifts form atorsor under Der O Z ( O Y , J ) . Now consider a morphism of schemes f : X → Y and a fixed extension Y ′ of Y with ideal I . Also assume given a homomorphism ϕ : f ∗ I → J for somequasi-coherent sheaf J on X . We search for a completion of the diagram X / / ___ (cid:15) (cid:15) X ′ f ′ (cid:15) (cid:15) (cid:31)(cid:31)(cid:31) Y / / Y ′ (1)3n which X ′ is a square-zero extension of X by the ideal J and the inducedmorphism f ′∗ I → J coincides with ϕ (cf. [Il71, Probl`eme III.2.1.2.1]). Weshow in Section 8 that this deformation problem also becomes locally trivial in f − O Y − Alg / O X . Once again abbreviating the name of the site to O Y / − Alg / O X ,we obtain Theorem.
The completions of Diagram (1) form a gerbe on O Y − Alg / O X banded by Der O Y ( O X , J ) . The class in H (cid:0) O Y − Alg / O X , Der O Y ( O X , J ) (cid:1) of thisgerbe obstructs the existence of a solution to this problem. If the obstruction van-ishes, solutions form a torsor under H (cid:0) O Y − Alg / O X , Der O Y ( O X , J ) (cid:1) , and theautomorphisms group of any solution is H (cid:0) O Y − Alg / O X , Der O Y ( O X , J ) (cid:1) . In the case where I = 0 and Y = Y ′ , an extension always exists, so thisimplies Corollary.
There is an equivalence of categories between the category of com-pletions of the diagram X / / ___ (cid:15) (cid:15) X ′ ~ ~ } } } } Y, by a scheme X ′ that is a square-zero extension of X with ideal J and the cate-gory of torsors on O Y − Alg / O X under the sheaf of abelian groups Der O Y ( O X , J ) .Isomorphism classes are in bijection with H (cid:0) O Y − Alg / O X , Der O Y ( O X , J ) (cid:1) andautomorphisms of any object are in bijection with H (cid:0) O Y − Alg / O X , Der O Y ( O X , J ) (cid:1) . How do these obstructions compare to those defined by Illusie? In Section 9,we prove
Theorem.
The cotangent complex L X/Y [Il71, II.1.2.3] represents cohomologyof the sheaves
Der O Y ( O X , J ) on the site O Y − Alg / O X , in the sense that Ext p ( L X/Y , J ) = H p (cid:0) O Y − Alg / O X , Der O Y ( O X , J ) (cid:1) . This shows that our obstruction groups are the same as Illusie’s. In [Wi11],we show that the obstruction classes agree with Illusie’s by the identificationabove.
3. Review of topologies, sites, and topoi If X is an object of a category C , a sieve of X is a subfunctor of the functorof points of X . It is frequently preferable to view a sieve as a fibered subcategoryover C of the category C/X of objects of C over X . We shall pass back andforth between these perspectives without comment. See [SGA4-1, I.4] for moreabove sieves (French: cribles ).The sieve of X generated from a collection S of maps Y → X is (as a fiberedcategory) the collection of all Z → X that factor through some Y → X in S .4 topology on a category C is a collection J ( X ) of sieves of X for each object X of C . These are generally called the covering sieves of the topology. Thecollections are required to comply with the conditions T1 (change of base), T2(local character), and T3 (inclusion of the sieves generated by identity maps) of[SGA4-1, D´efinition II.1.1]. We shall usually describe the sieves in each J ( X )by giving generators. These generators are called covering families .Every category possesses a canonical topology [SGA4-1, D´efinition II.2.5],the finest in which all representable functors are sheaves.If C is a category with a topology, a family S of objects of C is called acollection of topological generators of C if every object X of C is covered by asieve generated by maps from objects of S to X [SGA4-1, D´efinition II.3.0.1].We shall depart from the definitions [SGA4-1, I.1.1.5] and [MLM94, III.2.1] andcall C , together with its topology, a site if C possesses a set of topologicalgenerators.If C is a category with a topology, a functor F : C ◦ → Sets is called a sheafif the natural map F ( X ) → Hom(
R, F )is a bijection for every X in C and for every covering sieve R of X [SGA4-1,D´efinition II.2.1]. Here, Hom( R, F ) refers to maps of presheaves. If C is thefull subcategory of C spanned by a collection of generators of C with its inducedtopology, then the restriction map identifies the category of sheaves on C withthe category of sheaves on C . If C is a site then C is generated topologicallyby a small subcategory, which ensures that the sheaves on C form a category,which is frequently denoted e C .Any presheaf on a site C has an associated sheaf. If X is an object of C then the associated sheaf of the functor represented by X is denoted ǫ ( X ). Thisdetermines a functor ǫ : C → e C .A site C is called a topos if its topology is the canonical one and the functor ǫ : C → e C described above is an equivalence. Once again, our definition isslightly different from [SGA4-1, D´efinition IV.1.1]; it is equivalent to [MLM94,Definition III.4.3]. Every topos E has a final object, which we will denote bythe same letter E .A morphism of topoi f : C → C ′ is a pair of functors f ∗ : C ′ → C and f ∗ : C → C ′ such that f ∗ is right adjoint to f ∗ and f ∗ is exact. Every toposadmits an essentially unique morphism to the topos Sets .We shall depart again from [SGA4-1, IV.4.9.1] in the definition of a mor-phism of sites, and declare that a morphism between sites is a morphism betweentheir associated topoi. This extrinsic definition can be made intrinsic (see e.g.[MLM94, Theorem VII.10.1]), but we will be content to recall here that a mor-phism of sites can be induced from a cocontinuous functor C → C ′ or from aleft exact continuous functor C ′ → C .A functor f : C → C ′ between sites is cocontinuous if, for any X ∈ C andany covering sieve R of f ( X ), the collection of all Y → X such that f ( Y ) ∈ R is a covering sieve [SGA4-1, D´efinition III.2.1, Proposition III.2.3]. A left exactfunctor g : C ′ → C is continuous if and only if it takes covering families to cover-5ng families [SGA4-1, D´efinition III.1.1, Proposition III.1.3, Proposition III.1.6].If f is left adjoint to g then cocontinuity of f coincides with continuity of g [SGA4-1, Proposition III.2.5].By [Gr57, Th´eor`eme 1.10.1], every topos has enough injective sheaves ofabelian groups. This permits us to define the derived functors of f ∗ (applied tosheaves of abelian groups) for any morphism of topoi f : C → C ′ . In the casewhere C ′ = Sets , these derived functors are denoted F H p ( C, F ).Suppose that A u −→ B v −→ C is a sequence of morphisms of topoi. Since u ∗ hasan exact left adjoint it preserves injectives and we obtain a spectral sequencefor the composition of functors [Gr57, Th´eor`eme 2.4.1] R p v ∗ (cid:0) R q u ∗ F (cid:1) ⇒ R p + q ( vu ) ∗ F for any sheaf of abelian groups on A . In terms of derived categories, we have Rv ∗ ◦ Ru ∗ = R ( vu ) ∗ . In the case where C = Sets the spectral sequence abovespecializes to H p ( B, R q u ∗ F ) ⇒ H p + q ( A, F ) . See [SGA4-2, Expos´e V] for more about the Cartan–Leray spectral sequences.If X • is a hypercover of E [SGA4-2, V.7.3.1] then there is a spectral sequence[SGA4-2, V.7.4.0.3] H p ( X • , H q ( F )) ⇒ H p + q ( E, F ) . More specifically, if I • is a resolution of F by sheaves that are acyclic for eachof the X p , then the double complex Γ( X • , I • ) computes the cohomology of F .We refer the reader to [SGA4-2, V.7] for more details.
4. Review of torsors and gerbes
We restrict attention to torsors and gerbes under abelian groups. Supposethat E is a topos and G is a sheaf of abelian groups on E . A G -torsor is a sheaf F on E with an action a : G × F → F of G such that1. (pseudo-torsor) the map ( a, p ) : G × F → F × F is an isomorphism ofsheaves, and2. (local triviality) F covers E .If F satisfies only the first condition then F is called a pseudo-torsor under G .The second condition says that, locally in E , the sheaf F admits a section. Sincea section of a torsor trivializes it, the second condition says that F is locallyisomorphic to G as a sheaf with G -action. Theorem (cf. [Gi71, Remarque III.3.5.4]) . Isomorphism classes of torsors un-der an abelian group G are in bijection with H ( E, G ) . Isomorphisms betweenany two torsors form a pseudo-torsor under H ( E, G ) . In particular, automor-phisms of a torsor are in canonical bijection with H ( E, G ) . gerbe on E banded by G is a stack F [Gi71, D´efinition 1.2.1] on E with anaction of G on the morphism sheaves of G that is compatible with compositionand satisfies1. (pseudo-gerbe) for any two sections ξ, η of F over U , the sheaf Isom( ξ, η )is a G -pseudo-torsor on U ,2. (local triviality for morphisms) any two sections of F over U are localllyisomorphic,3. (local triviality for objects) F covers E .If F satisfies only the first condition, we call F a pseudo-gerbe banded by G .The second condition ensures that Isom( ξ, η ) forms a G U -torsor for each pair ofsections ξ, η ∈ Γ( E/U, F ). As in the case of torsors, the final condition meansthat sections of F exist locally in E . Since a section of F induces an equivalenceof banded gerbes between F with BG , the classifying stack of G -torsors, we saythat a banded gerbe is locally isomorphic to BG . Theorem (cf. [Gi71, Th´eor`eme IV.3.4.2]) . Equivalence classes of gerbes on E banded by an abelian group G are in bijection with H ( E, G ) . If F is agerbe banded by G then sections of a F form a pseudo-torsor under H ( E, G ) .Isomorphisms between any two sections of F form a torsor under H ( E, G ) . Inparticular, automorphisms of any section of F are in canonical bijection with H ( E, G ) .
5. The topos of commutative rings
All rings and algebras are commutative and unital.Now let (
E, A ) be a ringed topos [SGA4-1, IV.11.1.1], and let B an A -algebra.Let A − Alg ( E ) /B (or A − Alg /B for short) be the category of pairs ( U, C ) where U ∈ E and C is an A U -algebra with a map to B U . A morphism of A − Alg /B from ( U , C ) to ( U , C ) is a map f : U → U of E and a map C → f ∗ C of A U -algebras commuting with the projections to B U . Definition.
A family of maps ( U i , C i ) → ( U, C ) in A − Alg /B is covering if, forany V → U and any finite set of sections Λ ⊂ Γ( V, C ), there exists, locally in V , a map V → U i for some i and a lift of Λ to Γ( V, C i ).To understand this topology, it may be helpful to consider the case where E is a point. In that case, A → B is a ring homomorphism, and a family of A -algebra maps C i → C over B is considered to be covering if every finite setof elements of C can be lifted to some C i .Now let E be an arbitrary topos and suppose that R is a sieve of ( U, C )in A − Alg ( E ) /B . If V → U is a map of E and Λ ⊂ Γ( V, C ) is finite, let(
V, A V [Λ]) → ( U, C ) be the induced map of A − Alg /B . Let Q (Λ) the collection7f all W → V such that it is possible to complete the diagram( W, A W [Λ]) / / (cid:15) (cid:15) (cid:31)(cid:31)(cid:31) ( V, A V [Λ]) (cid:15) (cid:15) ( U ′ , C ′ ) / / ( U, C )with ( U ′ , C ′ ) in R . Then for R to be a covering sieve means that Q (Λ) is acovering sieve of V in E for every Q (Λ) arising as above.It is immediate from this description of the topology that any ( U, C ) in A − Alg /B is covered by the collection of all ( V, A V [ S ]) → ( U, C ) where S isan arbitrary finite set. Therefore the pairs ( V, A V [ S ]) generate the topology of A − Alg /B and we are free to say that in this topology, any A -algebra is locally afinitely generated polynomial ring. Furthermore, we obtain a set of topologicalgenerators for A − Alg /B by taking the collection of all ( V, A V [ S ]) such that V lies in a set of topological generators for E . Remark.
This topology is slightly more complicated than the ones used byGaitsgory, Quillen, and Rim. Analogues of those topologies would work here,but this topology has a technical advantage in its possession of a set of topolog-ical generators. This permits us to make use of topoi without making recourseto universes.
6. Deformation of homomorphisms
Let A be a sheaf of rings on E and B → C a homomorphism of A -algebras.Suppose that C ′ is a square-zero extension of C , as an A -algebra, with ideal J .Consider the problem of lifting the homomorphism B → C to a map B → C ′ (cf. [Il71, Probl`eme III.2.2.1.1]).Putting B ′ = C ′ × C B , this problem immediately reduces to that of findinga section of B ′ over B . We denote the set of such sections Hom AB ( B, B ′ ).The difference between any two sections of B ′ over B is an A -derivation from B into J . Denoting the set of all such derivations by Der A ( B, J ) we may saythat Hom AB ( B, B ′ ) is a Der A ( B, J )-pseudo-torsor. If B is allowed to vary weobtain a sheaf Der A ( B, J ) on A − Alg /B , represented by the object ( E, B + J ),where B + J is the trivial square-zero extension of B by J . Then B ′ representsa Der A ( B, J )-pseudo-torsor on A − Alg /B .In fact, B ′ is a Der A ( B, J )-torsor on A − Alg /B , since the map ( E, B ′ ) → ( E, B ) is covering in A − Alg /B . The cohomological classification of torsors nowimplies Theorem 1.
Let ω ∈ H (cid:0) A − Alg /B,
Der A ( B, J ) (cid:1) be the class of B ′ as a torsorunder Der A ( B, J ) . Then ω = 0 if and only if B ′ admits a section as an A -algebraover B . In that case, the sections form a torsor under H (cid:0) A − Alg /B,
Der A ( B, J ) (cid:1) = Der A ( B, J ) . . Extensions of algebras Suppose that B is an A -algebra in the topos E and J is a B -module. Let Exal A ( B, J ) be the category of square-zero A -algebra extensions of B withideal J . These categories fit together into a fibered category Exal A ( B, J ) over A − Alg /B . We saw in Theorem 1 that any B ′ ∈ Exal A ( B, J ) represents a
Der A ( B, J )-torsor on A − Alg /B , so we obtain a fully faithful functor Exal A ( B, J ) → B Der A ( B, J ) (2)from
Exal A ( B, J ) to the classifying stack of
Der A ( B, J )-torsors on A − Alg /B . Lemma.
Every
Der A ( B, J ) -torsor is representable by a square-zero A -algebraextension of B by J . Suppose that P is a Der A ( B, J )-torsor. Let B ′ be the sheaf on E whosesections over U are pairs ( b, b ′ ) where b is a section of B over U , correspondingto a map b : A U [ x ] → B (denoted by the same letter), and b ′ is a section of b ∗ P over ( U, A U [ x ]).We can give B ′ a ring structure as follows. Suppose ( b, b ′ ) and ( c, c ′ ) are twosections of B ′ . Choose a cover R of ( E, B ) over which P is trivial. Then E canbe covered by objects U such that there is a ( U, C ) ∈ R and both b U and c U liftto Γ( U, C ). There is therefore a map( b U , c U ) : A U [ x, y ] → C and ( b U , c U ) ∗ P C is trivial because P C is trivial. A trivial Der A ( B, J )-torsoris certainly representable (by B + J ), so let B ′ C be an extension of C by J representing ( b U , c U ) ∗ P C . We are given maps b ′ U : A U [ x ] → B ′ C and c ′ U : A U [ y ] → B ′ C over C . Since B ′ U is a ring, these extend uniquely to a map( b ′ U , c ′ U ) : A U [ x, y ] → B ′ C over C . Restricting this, via the maps A U [ x ] ∼ = A U [ x + y ] → A U [ x, y ] and A U [ x ] ∼ = A U [ xy ] → A U [ x, y ], yields sections that wewill denote b ′ U + c ′ U and b ′ U c ′ U of ( b + c ) ∗ P C and ( bc ) ∗ P C , respectively. Thesegive us sections ( b U + c U , b ′ U + c ′ U ) and ( b U c U , b ′ U c ′ U ) of B ′ over U .The uniqueness of the constructions above implies that they patch togetherto give a ring structure on B ′ over E , which makes ( E, B ′ ) an object of A − Alg /B .The verifications of commutativity, associativity, etc. are essentially the same,using a trio of sections of C instead of a pair. To check that B ′ represents P , oneonly needs to produce an isomorphism between P and the object represented by B ′ under the assumption that P admits a section and B is a finitely generatedpolynomial algebra over A , since the pairs ( U, A U [ S ]) such that P U is trivial and S is finite generate the topology of A − Alg /B . Under these assumptions, theconstruction clearly provides the isomorphism, and this varies in a functorialway with free A -algebras B . Remark.
Let F be the fibered category of pairs ( U, B → C ) where B → C isan A -algebra morphism over U and morphisms are commutative squares. Theprojection F → A − Alg sending the object above to (
U, C ) makes F into afibered category over A − Alg and the proof of the lemma demonstrates that F is a stack over A − Alg . 9he cohomological classification of torsors now gives us
Theorem 2.
The functor (2) is an equivalence. Isomorphism classes in thecategory
Exal A ( B, J ) are therefore in bijection with H ( A − Alg /B,
Der A ( B, J )) .
8. Deformation of algebras
Let A be a sheaf of rings on E and B an A -algebra. Suppose that A ′ is anextension of A with ideal I (not necessarily square-zero). Fix a B -module J andan A → B homomorphism ϕ : I → J . Define Def A ( A ′ , B, ϕ ) to be the categoryof completions of the diagram0 / / I / / (cid:15) (cid:15) A ′ / / (cid:15) (cid:15) (cid:31)(cid:31)(cid:31) A / / (cid:15) (cid:15) / / J / / ___ B ′ / / ___ B / / B ′ of B by J (cf. [Il71, Probl`eme 2.1.2.1]). Allowing B tovary, these categories fit together into a fibered category Def A ( A ′ , B, ϕ ) over A − Alg /B . (Note that the special case I = 0 recovers Exal A ( B, J ) =
Def A ( A, B, B ′ and B ′′ are any objects in Def A ( A ′ , B, ϕ ) then the difference betweenany two isomorphisms between B ′ and B ′′ is a derivation B → J , i.e., an elementof Der A ( A ′ , B, ϕ ). It follows from the lemma of Section 7 that Der A ( A ′ , B, ϕ )is a stack, so this tells us that Def A ( A ′ , B, ϕ ) is a pseudo-gerbe banded by Der A ( B, J ), and
Def A ( A ′ , B, ϕ ) is a pseudo-gerbe over A − Alg /B , banded by Der A ( B, J ). In fact, we have
Proposition.
Def A ( A ′ , B, ϕ ) is a gerbe over A − Alg /B , banded by Der A ( B, J ) . We must check that
Def A ( A ′ , B, ϕ ) admits a section locally in A − Alg /B , andthat any two sections are locally isomorphic. Since B is locally a polynomialalgebra over A , it’s sufficient for the local existence to show that Def A ( A ′ , B, ϕ )in that case. But then we could take B ′ = A ′ [ S ] ∐ I [ S ] J , the extension obtainedby pushing out the extension A ′ [ S ] of A [ S ] by I [ S ] by the canonical map I [ S ] → J . To prove that any two sections B ′ and B ′′ of Def A ( A ′ , B, ϕ ) are locallyisomorphic, we shall construct their difference B ′′′ = B ′′ − B ′ and show that itis a trivial extension of B as an algebra over A ′ − A ′ = A + I . Before makingthis precise, note that we may replace A ′ by A ′ /I , and therefore assume that I = 0, without changing the deformation problem. The ring B ′′ × B B ′ is anextension of B by the ideal J × J and there is a morphism of exact sequences,0 / / I × I / / (cid:15) (cid:15) A ′ × A A ′ / / (cid:15) (cid:15) A / / (cid:15) (cid:15) / / J × J / / B ′ × B B ′′ / / B / / .
10e push out these sequences via the difference maps I × I → I and J × J → J sending ( x, y ) to x − y . This yields a map of exact sequences0 / / I / / (cid:15) (cid:15) A + I / / (cid:15) (cid:15) A / / (cid:15) (cid:15) / / J / / B ′′′ / / B / / . Note that B ′′ can be recovered functorially from B ′ and B ′′′ by an additionprocedure inverse to the difference procedure just executed. Thus to show that B ′ and B ′′ are locally isomorphic, as extensions of B compatible with A ′ , it isequivalent to show that B ′′′ is locally isomorphic to the trivial extension B + J of B , with its trivial ( A + I )-algebra structure.The ( A + I )-algebra structure of B ′′′ is determined by the A -algebra structureinduced from the section A → A + I . It is therefore equivalent to show that anyextension of B by J as an A -algebra is locally isomorphic to B + J . This wasthe content of Theorem 1.The cohomological classification of banded gerbes now provides Theorem 3.
Let ω be the class in H (cid:0) A − Alg /B,
Der A ( B, J ) (cid:1) correspondingto the banded gerbe Def A ( A ′ , B, ϕ ) . Then ω = 0 if and only if Def A ( A ′ , B, ϕ ) isnon-empty.In that case, Def A ( A ′ , B, ϕ ) is isomorphic as a banded gerbe to the classifyingstack of Der A ( B, J ) -torsors. Hence, isomorphism classes in Def A ( A ′ , B, ϕ ) forma torsor under the group H (cid:0) A − Alg /B,
Der A ( B, J ) (cid:1) , and the automorphismgroup of any fixed object of Def A ( B, J ) is canonically isomorphic to the group Der A ( B, J ) = H (cid:0) A − Alg /B,
Der A ( B, J ) (cid:1) .
9. The cohomology of free algebras
Suppose that E is a topos, A a sheaf of algebras on E , and S a sheaf of setson E . Let J be a sheaf of A [ S ]-modules. We wish to compare the cohomologygroups of Der A ( A [ S ] , J ) on A − Alg /A [ S ] and J S on E/S . We construct severalsites to mediate between the A − Alg /A [ S ] and E/S .Let
Sets ( E ) and Sets ∗ ( E ), or Sets and
Sets ∗ for short, be the sites whosecommon underlying category is the category of pairs ( U, T ) where U is an objectof E and T is a sheaf of sets on U . A map ( U , T ) → ( U , T ) is a map f : U → U and a map T → f ∗ T . (From another point of view, this categoryis the category of arrows in E .)We shall say that a family of morphisms ( U i , T i ) → ( U, T ) is covering in
Sets ∗ if, for any f : V → U and any finite subset Λ ⊂ Γ( V, f ∗ T ), there is,locally in V , a factorization of f through g : V → U i , for some i , and a lift ofΛ to Γ( V, g ∗ T i ). The topology on Sets is defined in the same way, except Λ isrestricted to be a 1-element set.
Remark.
The topologies on
Sets and
Sets ∗ are genuinely different. In the casewhere E is the punctual topos (i.e., the category of sets), the category of sheaves11n Sets ( E ) may be identified with the category of sets; the category of sheaveson Sets ∗ ( E ) may be identified with the category of presheaves on the categoryof finite sets.The topology on Sets is finer than that on
Sets ∗ , so there is a morphism ofsites Sets → Sets ∗ (the identity functor is cocontinuous). This induces a mapΦ : Sets / ( E, S ) → Sets ∗ / ( E, S ), for any sheaf of sets S on E .There is also a functor Sets ∗ / ( E, S ) → A − Alg /A [ S ] which sends ( U, T ) to(
U, A U [ T ]). This functor is left exact, and by definition, it takes covers to coversso it is continuous by [SGA4-1, Proposition III.1.6] and we get a morphism ofsites Ψ : A − Alg /A [ S ] → Sets ∗ / ( E, S ).Finally, we have a cocontinuous functor
Sets ( E ) → E sending ( U, T ) to i ! T ,where i is the canonical morphism of topoi from E/U to E [SGA4-1, IV.5.1–2].This induces a morphism of sites Ξ : Sets / ( E, S ) → E/S . Putting all of thesemorphisms together, we obtain the following diagram of morphisms of sites.
Sets / ( E, S ) Φ ' ' OOOOOOOOOOOO Ξ y y ssssssssss A − Alg /A [ S ] Ψ w w oooooooooooo E/S
Sets ∗ / ( E, S ) Proposition.
The morphisms Φ and Ψ are acyclic. First consider Ψ : A − Alg /A [ S ] → Sets ∗ / ( E, S ). To see that R p Ψ ∗ F = 0 for p > Sets ∗ / ( E, S ). We can therefore reduce to the casewhere S is the constant sheaf associated to some finite set S , since pairs ( V, T )where T is constant and finite generate the topology of Sets ∗ /E . We must showthat if α ∈ H p ( A − Alg /A [ S ] , Ψ ∗ ( U, T )) then α can be trivialized on some coverof ( U, T ). But Ψ ∗ ( U, T ) = (
U, A U [ T ]) and all covering sieves of ( U, A U [ T ]) in A − Alg /A [ S ] are pulled back from covering sieves of the final object of E , henceare pulled back from covering sieves of ( E, S ) in
Sets ∗ / ( E, S ). Since α cancertainly be trivialized on some covering sieve of ( U, A U [ T ]), any covering sieveof Sets ∗ / ( E, S ) that pulls back via Ψ to this one will trivialize α in Sets ∗ / ( E, S ).Now consider Φ :
Sets / ( E, S ) → Sets ∗ / ( E, S ). If R is a covering sieve of someobject ( U, T ) of
Sets / ( E, S ), then let R ′ be the collection of all finite disjointunions of objects of R . If R is a covering sieve of Sets / ( E, S ) which trivializesa cohomology class, then R ′ is a covering sieve of Sets ∗ / ( E, S ) which trivializesthe same cohomology class. Therefore this map is acyclic as well.
Lemma.
There is a canonical isomorphism Φ ∗ ( E, J ) ≃ Ψ ∗ Der A ( A [ S ] , J ) forany sheaf of A [ S ] -modules J . This is a matter of unwinding the definitions. For (
U, T ) ∈ Sets ∗ / ( E, S ), wehave Γ (cid:0) ( U, T ) , Φ ∗ ( E, J ) (cid:1) = Γ (cid:0) ( U, T ) , ( E, J ) (cid:1) = Hom U ( T, J U ) .
12n the other hand,Γ (cid:0) ( U, T ) , Ψ ∗ Der A ( A [ S ] , J ) (cid:1) = Γ (cid:0) ( U, A [ T ]) , Der A ( A [ S ] , J ) (cid:1) = Der A U ( A U [ T ] , J U ) = Hom U ( T, J U )using the universal property of A [ T ].Since Φ and Ψ are acyclic, this proves that R Φ ∗ ( E, J ) = R Ψ ∗ Der A ( A [ S ] , J ).We can therefore compute the cohomology of Der A ( A [ S ] , J ) by computing thecohomology of ( E, J ) on
Sets / ( E, S ). Lemma.
The natural map J → R Ξ ∗ Ξ ∗ J is an isomorphism. The projection
Sets / ( E, S ) → E/S sending (
U, T ) to T is left exact andinduces an exact left adjoint Ξ ! to Ξ ∗ [SGA4-1, Proposition I.5.4 4]. Therefore,Ξ ∗ preserves injectives. Since Ξ ∗ is also exact, this implies R Ξ ∗ Ξ ∗ J = Ξ ∗ Ξ ∗ J .The natural map J → Ξ ∗ Ξ ∗ J is certainly an isomorphism, since Ξ ∗ J = ( E, J )and Hom ( E,S ) (cid:0) ( E, T ) , ( E, J ) (cid:1) = Hom S ( T, J ).Let u : E/S → E and v : A − Alg /A [ S ] → E denote the projections. Thenthe lemma implies that Rv ∗ Der A ( A [ S ] , J ) = Ru ∗ J . Lemma.
There is a canonical isomorphism R Hom( Z S , J ) ≃ Ru ∗ u ∗ J for anysheaf of abelian groups J on E . Since u ∗ has an exact left adjoint on sheaves of abelian groups [SGA4-1,Proposition IV.11.3.1], both R Hom( Z S , J ) and Ru ∗ u ∗ J can be computed bytaking an injective resolution of J in E . It’s therefore sufficient to remark thatHom( Z S , J ) = Hom( S, J ), by definition.Putting all of the lemmas together, we find that Rv ∗ Der A ( A [ S ] , J ) = R Hom Z ( Z S , J ) = R Hom A [ S ] ( A [ S ] S , J ) = R Hom(Ω A [ S ] /A , J ) . Corollary. If J is injective and B is a free A -algebra then for every p > , thegroup H p ( A − Alg /B,
Der A ( B, J )) vanishes. Since Ω
B/A is functorial in B , this permits us to compute the cohomologyof Der A ( B, J ) for any sheaf B of A -algebras and any B -module J using hyper-ˇCech cohomology. Let B • be the standard simplicial resolution of B by free A -algebras [Il71, I.1.5.5 b, II.1.2.1.1]. By [Il71, I.1.5.3], the resolution B • → B is a homotopy equivalence on the underlying sheaves of sets. This implies inparticular that it is a hypercover, so corollary above implies that Rv ∗ Der A ( B, J )is computed by the complex C with C p,q = v ∗ Der A ( B p , J q )for any injective resolution J • of J . Taking B • = P A ( B ) we obtain the cotangentcomplex as L B/A = Ω B • /A ⊗ B • B by [Il71, II.1.2.3], and then Rv ∗ Der A ( B, J ) ≃ v ∗ Der A ( B • , J • ) = Hom B • (Ω B • /A , J • )= Hom B (Ω B • /A ⊗ B • B, J • ) ≃ R Hom( L B/A , J ) . J tothe cohomology of Der A ( B, J ): Theorem 4.
Let v denote the projection A − Alg ( E ) /B → E . If J is any B -module, then there is an isomorphism in the derived category of sheaves of B -modules Rv ∗ Der A ( B, J ) ≃ R Hom( L B/A , J ) . In particular,
Ext p ( L B/A , J ) = H p (cid:0) A − Alg ( E ) /B, Der A ( B, J ) (cid:1) for all p .
10. Acknowledgements
I am happy to thank Dan Abramovich, Patrick Brosnan, Barbara Fantechi,Martin Olsson, Ravi Vakil, and Angelo Vistoli for helpful conversations aboutthese topics. Although a goal of this work has been to circumvent some ofthe technical aspects of [Il71], Illusie’s treatise has been a continual inspiration.Vistoli’s remark that “with uncanny regularity [deformations are] a cohomologygroup of a certain algebraic object, and [obstructions are] the cohomology groupof the same object in one degree higher” [Vi97, p. 2] was a particular impetusfor this project. Dan Abramovich provided numerous suggestions that improvedthe exposition of this paper considerably (and helped me to avoid one seriouserror). I am also grateful to Bhargav Bhatt for informing me of Gaitsgory’swork.This research was partly supported by nsf-msprf 0802951.
References [Ga95] Dennis Gaitsgory. Operads, Grothendieck topologies and Deforma-tion theory. arXiv:alg-geom/9502010, February 1995.[Ga97] Dennis Gaitsgory. Grothendieck topologies and deformation the-ory II.
Compositio Math. , 106(3):321–348, 1997.[Gi71] Jean Giraud.
Cohomologie non ab´elienne . Springer-Verlag, Berlin,1971. Die Grundlehren der mathematischen Wissenschaften, Band179.[Gr57] Alexander Grothendieck. Sur quelques points d’alg`ebre homologique.
Tˆohoku Math. J. (2) , 9:119–221, 1957.[Il71] Luc Illusie.
Complexe cotangent et d´eformations. I . Lecture Notes inMathematics, Vol. 239. Springer-Verlag, Berlin, 1971.14Il10] Luc Illusie. Reminiscences of Grothendieck and his school.
Notices ofthe AMS , 57(9), October 2010. With Alexander Beilinson, SpencerBloch, and Vladimir Drinfeld.[MLM94] Saunders Mac Lane and Ieke Moerdijk.
Sheaves in geometry andlogic . Universitext. Springer-Verlag, New York, 1994. A first intro-duction to topos theory, Corrected reprint of the 1992 edition.[Qu70] Daniel Quillen. On the (co-) homology of commutative rings. In
Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol.XVII, New York, 1968) , pages 65–87. Amer. Math. Soc., Providence,R.I., 1970.[SGA4-1]
Th´eorie des topos et cohomologie ´etale des sch´emas. Tome 1: Th´eoriedes topos . Lecture Notes in Mathematics, Vol. 269. Springer-Verlag,Berlin, 1972. S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie1963–1964 (SGA 4), Dirig´e par M. Artin, A. Grothendieck, et J.L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B.Saint-Donat.[SGA4-2]
Th´eorie des topos et cohomologie ´etale des sch´emas. Tome 2 . Lec-ture Notes in Mathematics, Vol. 270. Springer-Verlag, Berlin, 1972.S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1963–1964 (SGA4), Dirig´e par M. Artin, A. Grothendieck et J. L. Verdier. Avec lacollaboration de N. Bourbaki, P. Deligne et B. Saint-Donat.[SGA7-1]
Groupes de monodromie en g´eom´etrie alg´ebrique.
Lecture Notes inMathematics, Vol. 288. Springer-Verlag, Berlin, 1972. S´eminaire deG´eom´etrie Alg´ebrique du Bois-Marie 1967–1969 (SGA 7 I), Dirig´epar A. Grothendieck. Avec la collaboration de M. Raynaud et D. S.Rim.[Vi97] Angelo Vistoli. The deformation theory of local complete intersec-tions. arXiv:alg-geom/9703008 , 1997.[Wi11] Jonathan Wise. The deformation theory of sheaves of commutativerings II. arXiv:1102.2924arXiv:1102.2924