Endomorphisms of Koszul complexes: formality and application to deformation theory
aa r X i v : . [ m a t h . AG ] D ec ENDOMORPHISMS OF KOSZUL COMPLEXES: FORMALITY ANDAPPLICATION TO DEFORMATION THEORY
FRANCESCA CAROCCI AND MARCO MANETTI
Abstract.
We study the differential graded Lie algebra of endomorphisms of theKoszul resolution of a regular sequence on a unitary commutative K -algebra R andwe prove that it is homotopy abelian over K but not over R (except trivial cases).We apply this result to prove an annihilation theorem for obstructions of (derived)deformations of locally complete intersection ideal sheaves on projective schemes. Introduction
Let R be a fixed unitary commutative ring, then the usual notion of differential graded(DG) Lie algebra over a field extends naturally to the notion of DG-Lie algebra over R (Definition 1.1). Recently DG-Lie algebras over commutative rings have received some at-tention in the study of degenerations of Batalin-Vilkoviski algebras [17] and also in thestudy of formality in families [16, 21, 27]. Also the notions of quasi-isomorphism, formalityand homotopy abelianity (i.e., formality plus the bracket trivial in cohomology) extendwithout difficulty to DG-Lie algebras over R , but the analysis of examples shows immedi-ately that these concepts become very restrictive when working over a ring, and should bereplaced with more convenient notions.In this paper we restrict our attention to homotopy abelianity and we introduce thenotion of numerically homotopy abelian (NHA) DG-Lie algebra (Definition 1.3) which seemsmore useful, at least for the application in deformation theory. Very briefly, the class of NHADG-Lie algebras is the smallest class satisfying the following conditions:(1) if L is abelian, then L is NHA;(2) if L → M is a morphism surjective in cohomology and L is NHA, then also M isNHA;(3) if L → M is a morphism injective in cohomology and M is NHA, then also L isNHA.It is plain that every homotopy abelian DG-Lie algebra is also numerically homotopyabelian, and every numerically homotopy abelian algebra has trivial bracket in cohomology.It is well known that over a field of characteristic 0 every numerically homotopy abelianDG-Lie algebra is also homotopy abelian and that there exist DG-Lie algebras with trivialbracket in cohomology that are not homotopy abelian.The first goal of this paper is to provide some examples, which occur in deformationtheory of locally complete intersections, of numerically homotopy abelian DG-Lie algebrasthat are not homotopy abelian: more precisely we shall prove that if f , . . . , f r ∈ R is aregular sequence contained in a proper ideal and K ∗ = K ( f , . . . , f r ) is its Koszul complex,then the DG-Lie algebras Hom ∗ R ( K ∗ , K ∗ ) of R -linear endomorphisms of K ∗ is not homotopyabelian over R , Date : December 18, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Koszul complex, Deformations of coherent sheaves, differential graded Liealgebras.
On the other side, we shall prove that the DG-Lie algebra L of graded endomorphismsfor the Koszul complex K ∗ (or its truncated K < ) for a regular sequence in R is numericallyhomotopy abelian (Theorems 2.2 and 2.4). This implies that if R is a K -algebra, with K field of characteristic 0, then L is homotopy abelian over K and therefore the associatedMaurer-Cartan/gauge functor is unobstructed.Clearly this fact has a number of consequences in deformation theory of locally completeintersections. For instance in Section 4 we shall prove the following result: Theorem 0.1.
Let I ⊂ O X be a locally complete intersection ideal sheaf on a projectivescheme X over a field K of characteristic 0, and denote by O Z = O X / I the structure sheafon the closed subscheme Z defined by I . Let F be either a line bundle over Z or F = I ⊗ L for a line bundle L on X . Then: (1) the obstructions to deforming F are contained in the kernel of the natural map α : Ext X ( F , F ) → H ( X, E xt X ( F , F )) . In particular, if α is injective then F has unobstructed deformations; (2) the obstructions to derived deformations of F are contained in the kernel of thenatural map of graded vector spaces α : M n Ext n +2 X ( F , F ) → M n H ( X, E xt n +2 X ( F , F )) . In particular, if α is injective then the DG-Lie algebra Tot( L ( U , E • )) controllingdeformations of F is homotopy abelian over K . The underlying idea is that, since the DG-Lie algebra of endomorphisms of a Koszulresolution is a homotopy abelian over K , the restriction to an affine open set of a sheaf F as above has unobstructed (derived) deformations; it is then sufficient to consider themorphism from global to local deformations of F . To ensure that the passage from global tolocal is a “genuine” morphism of deformation theories, we shall see that the maps α , α areinduced in cohomology by a morphism of DG-Lie algebras, namely from the DG-Lie algebracontrolling global deformations to the one controlling local deformation. This is done byfixing a locally free resolution E • of F (hence the assumption X projective) to then exploit arecent result by Fiorenza-Iacono-Martinengo [6] proving that local and global deformationsof F are controlled by the quasi-coherent sheaf of DG-Lie algebras H om ∗ X ( E • , E • ). Remark . After the first version of this paper was posted on arXiv, F. Meazzini proposeda new approach to deformations of coherent sheaves via local resolution [28] that does notrequire the existence of locally free resolutions and that allows to replace in the abovetheorem the assumption X projective with X separated and noetherian scheme of finitedimension: however this weakening of assumption is technically quite complicated and forsimplicity of exposition we maintain here the assumption X projective.1. DG-Lie algebras over commutative rings
In what follows R will be a unitary commutative ring. Definition 1.1. A differential graded Lie algebra (DG-Lie algebra) over R is the dataof cochain complex of R -modules ( L, d ) equipped with an R -bilinear bracket [ − , − ] : L × L → L satisfying the following conditions:(1) [ − , − ] is homogeneous graded skewsymmetric. This means that:(a) [ L i , L j ] ⊂ L i + j ,(b) [ a, b ] + ( − ab [ b, a ] = 0 for every a, b homogeneous,(c) [ a, a ] = 0 for every homogeneous a of even degree; NDOMORPHISMS OF KOSZUL COMPLEXES 3 (2) (Leibniz identity) d [ a, b ] = [ da, b ] + ( − a [ a, db ];(3) (Jacobi identity) every triple of homogeneous elements a, b, c satisfies the equality[ a, [ b, c ]] = [[ a, b ] , c ] + ( − a b [ b, [ a, c ]] ;(4) (Bianchi identity) [ b, [ b, b ]] = 0 for every homogeneous b of odd degree.Morphisms of DG-Lie algebras are morphisms of cochain complexes of R -modules commut-ing with brackets. The resulting category is denoted by DGLA ( R ).Notice that the Jacobi identity implies the Bianchi identity whenever 3 is not a zero-divisor in R . A quasi-isomorphism of DG-Lie algebras is a morphism of DG-Lie algebraswhich is also a quasi-isomorphism of complexes. A DG-Lie algebra with trivial bracket iscalled abelian. Definition 1.2.
A DG-Lie algebra L is said to be homotopy abelian if there exists a finitezigzag of morphisms of DG-Lie algebras L (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ ❆❆❆❆❆❆❆❆ L ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ (cid:30) (cid:30) ❂❂❂❂❂❂❂❂❂ L n (cid:0) (cid:0) ✁✁✁✁✁✁✁✁✁ ❆❆❆❆❆❆❆❆ L L ··· ······ M such that every arrow is a quasi-isomorphism and M is abelian.As we said in the introduction, over a general ring the notion of homotopy abelianityseems to be too restrictive and it is convenient to consider a suitable more general class. Definition 1.3.
A DG-Lie algebra L is said to be numerically homotopy abelian if thereexists a finite zigzag of morphisms of DG-Lie algebras L f (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ g ❆❆❆❆❆❆❆❆ L f ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ g (cid:30) (cid:30) ❂❂❂❂❂❂❂❂❂ L nf n (cid:0) (cid:0) ✁✁✁✁✁✁✁✁✁ g n ❆❆❆❆❆❆❆❆ L L ··· ······ M such that: • the DG-Lie algebra M is abelian, • every morphism f i is surjective in cohomology, • every morphism g i is injective in cohomology.It is plain that every homotopy abelian DG-Lie algebra is also numerically homotopyabelian, and that for every numerically homotopy abelian DG-Lie algebra the bracket istrivial in cohomology. Remark . It is well known (see e.g. [7, Lemma 6.1], [14, Lemma 2.11] , [17, Prop. 4.11])that if R is a field of characteristic 0, then every numerically homotopy abelian DG-Liealgebra is homotopy abelian. For instance if M is an abelian Lie algebra and f : M → L is a morphism of DG-Lie algebras that is surjective in cohomology, when R is a field itis sufficient to consider the restriction of f to a suitable subcomplex N ⊂ M such that f : N → L is a quasi-isomorphism. The proof of the general case follows the same ideas butrequires the factorisation lemma and characteristic 0.One of the goals of this paper is to show that over general commutative rings the classof numerically homotopy abelian DG-Lie algebras strictly contains the class of homotopyabelian DG-Lie algebra. More precisely, we shall prove that the algebra of endomorphismsof the Koszul complex of a (non trivial) regular sequence is numerically homotopy abelianbut not homotopy abelian. A simple obstruction to homotopy abelianity is given by the FRANCESCA CAROCCI AND MARCO MANETTI following result, in which the symbol ⊗ LR denotes the usual derived tensor product ofcomplexes of R -modules. Lemma 1.5.
Let H be a homotopy abelian DG-Lie algebra over a ring R . Then the classof the bracket is trivial in the group Ext R ( H ⊗ LR H, H ) .Proof. The result is clearly true whenever the bracket of H is trivial. If H → K is aquasi-isomorphism of DG-Lie algebras, the commutative diagram H ⊗ R H [ − , − ] / / (cid:15) (cid:15) H (cid:15) (cid:15) K ⊗ R K [ − , − ] / / K gives a commutative diagram in the homotopy category of complexes of R -modules H ⊗ LR H [ − , − ] / / (cid:15) (cid:15) H (cid:15) (cid:15) K ⊗ LR K [ − , − ] / / K with the vertical arrows isomomorphisms. Thus the upper horizontal arrow is homotopicallytrivial if and only if the lower horizontal arrow is.For later use it is useful to give also a longer but more constructive proof of the lemma:every quasi-isomorphism of DG-Lie algebras H → K lifts to a morphism of complexesbetween cofibrant resolutions P / / η (cid:15) (cid:15) H (cid:15) (cid:15) Q / / K where cofibrant is intended with respect to the projective model structure [13, Thm. 2.3.11].This gives a commutative diagram of complexes P ⊗ R P [ − , − ] / / η ⊗ η (cid:15) (cid:15) H (cid:15) (cid:15) Q ⊗ R Q [ − , − ] / / K with the vertical arrows quasi-isomorphisms. Since P ⊗ R P and Q ⊗ R Q are cofibrantcomplexes the above diagram induces a pair of quasi-isomorphismsHom ∗ R ( P ⊗ R P, H ) −−→ Hom ∗ R ( P ⊗ R P, K ) ←−− Hom ∗ R ( Q ⊗ R Q, K )and then the class of the bracket of H is trivial in Ext R ( H ⊗ LR H, H ) = H (Hom ∗ R ( P ⊗ R P, H )) if and only if the same holds for K . (cid:3) The following example remarks that we really need to consider the class of the bracketin the derived category. Suppose L is homotopy abelian over R , but not cofibrant as acomplex , i.e. with respect to the projective module structure on complexes of R -modules[13]. In the latter case we can have that [ − , − ] ∈ H (Hom ∗ R ( L ⊗ R L, L )) is not homotopicto zero.
Example 1.6.
Given an integral domain R and a non invertible element t ∈ R, denote A = R/ ( t ) and consider the DG-Lie algebra over R : L : Rx d −→ Ry d −→ Az, deg( x ) = 0 , deg( y ) = 1 , deg( z ) = 2 , NDOMORPHISMS OF KOSZUL COMPLEXES 5 dx = ty, dy = z, [ x, y ] = y, [ x, z ] = z, [ y, y ] = 0 . The verification of Leibniz, Bianchi and Jacobi is completely straightforward. Although itis acyclic, and then homotopy abelian, the bracket[ − , − ] : L ⊗ R L → L is not homotopic to 0 and then its class is not trivial in H (Hom ∗ R ( L ⊗ R L, L )). In fact,if the bracket is the coboundary of a R -linear map q : L ⊗ R L → L of degree −
1, then q ( x, z ) = 0 since tq ( x, z ) = q ( x, tz ) = 0, and the relation y = [ x, y ] = dq ( x, y ) + q ( ty, y ) + q ( x, z ) ∈ tRy gives a contradiction. Corollary 1.7.
Let R be a commutative unitary ring and let F ∗ : F − n ∂ −→ · · · ∂ −→ F − ∂ −→ F be a complex of length n > of finitely generated free R -modules. Assume that there existsa proper ideal I ⊂ R such that ∂ ( F i ) ⊂ IF i +1 for every i . Then the DG-Lie algebra H = Hom ∗ R ( F ∗ , F ∗ ) is not homotopy abelian over R .Proof. By the assumption on the length of the complex we have F , F − n = 0 and wecan choose a morphism of R -modules β : F − n → F such that β ( F − n ) IF . Since H = Hom ∗ R ( F ∗ , F ∗ ) is a bounded complex of free R -modules, we haveExt R ( H ⊗ LR H, H ) = H (Hom ∗ R ( H ⊗ R H, H )) . Moreover, d ( H i ) ⊂ IH i +1 for every index i : in fact for every f ∈ H i and every generator e ∈ F j we have ( df ) e = ∂ ( f e ) − ( − i f ( ∂e ) ∈ IF i + j +1 . Assume that H is homotopy abelian over R , then by Lemma 1.5 there exists an R -bilinearmap h : H × H → H of degree − x, y ] = dh ( x, y ) + h ( dx, y ) + ( − deg( x ) h ( x, dy ) , x, y ∈ H .
Consider now the following two elements α, β ∈ H :(1) α ∈ H = Hom R ( F ∗ , F ∗ ) is defined as the identity on F and α ( F i ) = 0 for i < β ∈ H n = Hom nR ( F ∗ , F ∗ ) extends the above β : F − n → F in the unique possibleway, i.e., β ( F i ) = 0 for i > − n .Then [ α, β ] = αβ − βα = β, dβ = 0 , dα ∈ IH . By the R -bilinearity of h we have β = [ α, β ] = dh ( α, β ) + h ( dα, β ) + h ( α, dβ ) ∈ IH n and therefore β ( F − n ) is contained in the submodule IF , which is a contradiction with thechoice of β . (cid:3) Corollary 1.8.
Let R be a commutative unitary ring and let K ∗ be the Koszul complex ofa sequence f , . . . , f n ∈ R , n > : K ∗ : 0 → K − n → · · · → K − → K , K − i = ^ i R n . If the ideal ( f , . . . , f n ) ⊂ R is proper, then the DG-Lie algebra H = Hom ∗ R ( K ∗ , K ∗ ) is nothomotopy abelian over R .Proof. Immediate consequence of Corollary 1.7. Notice however that Hom ∗ R ( K ∗ , K ∗ ) maybe homotopy abelian over some subring S ⊂ R even if ( f , . . . , f n ) is a proper ideal. (cid:3) FRANCESCA CAROCCI AND MARCO MANETTI The endomorphism algebra of Koszul complexes
We have already noticed that, except in trivial cases, the DG-Lie algebra L of endomor-phisms of the Koszul complex of a sequence f , . . . , f n ∈ R is not homotopy abelian. Inthis section we shall prove that if f , . . . , f n is a regular sequence, then L is numericallyhomotopy abelian.For every R -module E we shall denote by E ∨ = Hom R ( E, R ) its dual module. Recallthat the internal product: ^ ∗ E ∨ × ^ ∗ E y −−→ ^ ∗ E, is the R -bilinear map defined recursively by the formulas: f y ( v ∧ · · · ∧ v b ) = b X i =1 ( − i − f ( v i ) v ∧ · · · ∧ b v i ∧ · · · ∧ v b , ( f ∧ · · · ∧ f a ) y ( v ∧ · · · ∧ v b ) = ( f ∧ · · · ∧ f a − ) y ( f a y ( v ∧ · · · ∧ v b )) . Let us introduce the R -linear contraction operators: i ψ : ^ ∗ E → ^ ∗ E, i ψ ( ω ) = ψ y ω, ψ ∈ ^ ∗ E ∨ ;it is immediate to see from the definition above that i ψ ∧ η = i ψ i η for every ψ, η ∈ V ∗ E ∨ . Lemma 2.1.
In the above setup, consider V ∗ E as a graded R -algebra, where the elementsof V a E have degree − a . Then: (1) for every ψ ∈ E ∨ the operator i ψ is the unique R -linear derivation of V ∗ E ofdegree +1 such that i ψ ( v ) = ψ ( v ) for every v ∈ E ; (2) [ i ψ , i η ] = 0 for every ψ, η ∈ V ∗ E ∨ .Proof. The first item is clear. For the second, if ψ, η ∈ E ∨ then [ i ψ , i η ] is a derivation ofdegree +2 of V ∗ E , and therefore [ i ψ , i η ] annihilates a set of generators of the R -algebra V ∗ E . For the general case ψ, η ∈ V ∗ E ∨ the vanishing of [ i ψ , i η ] follows from the relations:(1) i α i β = − i β i α for every α, β ∈ E ∨ ;(2) i ψ ∧ η = i ψ i η for every ψ, η ∈ V ∗ E ∨ ;(3) [ i ψ , i η i γ ] = [ i ψ , i η ] i γ + ( − deg( ψ ) deg( η ) i η [ i ψ , i γ ] for every ψ, η, γ ∈ V ∗ E ∨ . (cid:3) From now on we assume that E is the free R -module generated by e , . . . , e n . We shallconsider V ∗ E ∨ as a differential graded commutative algebra, equipped with the trivialdifferential, and with the elements in V i E ∨ of degree i . For every f ∈ E ∨ we have ( i f ) = i f ∧ f = 0 and then, setting K − i = V i E , we get a complex of free R -modules: K ∗ : 0 → K − n i f −−→ K − n i f −−→ · · · i f −−→ K − i f −−→ K = R, usually called the Koszul complex of the sequence f = f ( e ),. . . , f n = f ( e n ). Theorem 2.2.
In the above notation, the contraction map i : ^ ∗ E ∨ → Hom ∗ R ( K ∗ , K ∗ ) is a morphism of differential graded algebras. If K ∗ is exact then i is surjective in coho-mology and therefore the DG-Lie algebra Hom ∗ R ( K ∗ , K ∗ ) is numerically homotopy abelian.Proof. The fact that i is a morphism of differential graded algebras follows immediatelyfrom the formulas i α ∧ β = i α i β and [ i f , i α ] = 0. If K ∗ is exact and I = i f ( K − ) ⊂ R is theideal generated by f , . . . , f n , then the composition of the projections p : K ∗ → K = R and q : R → R/I induces a quasi-isomorphism of complexes qp : K ∗ → R/I and therefore,
NDOMORPHISMS OF KOSZUL COMPLEXES 7 since K ∗ is cofibrant, the map qp : Hom ∗ R ( K ∗ , K ∗ ) → Hom ∗ R ( K ∗ , R/I ) is a surjective quasi-isomorphism. Since Hom ∗ R ( K ∗ , R/I ) has trivial differential, to conclude the proof it is nowsufficient to show that the morphism: pq i : ^ ∗ E ∨ → Hom ∗ R ( K ∗ , R/I )is surjective. This is clear since q i : V ∗ E ∨ → Hom ∗ R ( K ∗ , R ) is an isomorphism of graded R -modules and p : Hom ∗ R ( K ∗ , R ) → Hom ∗ R ( K ∗ , R/I ) is surjective. (cid:3) Let now p ≤ K
For every ψ, η ∈ V ∗ E ∨ we have: (1) π i ψ = π i ψ jπ ∈ Hom ∗ R ( K ∗ , K
If the Koszul complex K ∗ is exact, then the morphism of DG-Lie algebras i < : ∗ ^ E ∨ → Hom ∗ R ( K < , K < ) is surjective in cohomology, and then Hom ∗ R ( K < , K < ) is numerically homotopy abelian.Proof. Denote as above by I ⊂ R the ideal generated by ( f , . . . , f n ), then the map i f : K − → I gives a quasi-isomorphism of complexes p : K < → I [1] and then, since K < is cofibrant the induced map(2.1) Hom ∗ R ( K < , K < ) → Hom ∗ R ( K < , I [1])is a quasi-isomorphism. In particular the cohomology of Hom ∗ R ( K < , K < ) is concentratedin degrees 0 , . . . , n −
1. Since i < : ^ n − E ∨ → Hom R ( ^ n E, E ) = Hom n − R ( K < , K < ) , is an isomorphism and Hom ∗ R ( K < , K < )) is concentrated in degrees ≤ n −
1, the map V n − E ∨ → H n − (Hom ∗ R ( K < , K < )) is surjective.Let now 0 ≤ m < n − α ∈ Hom mR ( K < , K < ) be a cocycle,and consider its components α − m − − i : K − m − − i → K − − i , i = 0 , . . . , n − − m, together with the morphism β = i f α − m − : K − m − → R .
FRANCESCA CAROCCI AND MARCO MANETTI
Since α is a cocycle, α − m − i f = ± i f α − m − and then β i f = ± ( i f ) α − m − = 0. Equiv-alently β is a cocycle of degree m + 1 in the dual Koszul complex Hom ∗ R ( K ∗ , R ). Since m + 1 < n , by the self-duality of the Koszul complex [4, Prop. 17.15], the cocycle β is alsoa coboundary and there exists γ : V m E → R such that β = ( − m γ i f . In view of theisomorphism V m E ∨ ≃ ( V m E ) ∨ there exists η ∈ V m E ∨ such that i η ( x ) = γ ( x ) for every x ∈ K − m .For every x ∈ K − m − we have β ( x ) = ( − m i η i f ( x ) and then i f ( α − i η )( x ) = β ( x ) − i f i η ( x ) = β ( x ) − ( − m i η i f ( x ) = 0 . We have therefore proved that α and i γ have the same image in Hom ∗ R ( K < , I [1]), andthe conclusion follows by (2.1). (cid:3) Deformations of coherent sheaves via locally free resolutions
In this section we briefly recall the construction of the DG-Lie algebra controlling thedeformation theory of a coherent sheaf given in [6, 28]. Here and in the following we denoteby K a fixed field of characteristic 0, by Set , Grpd and
Art K the categories of Sets,Groupoids and local Artin K -algebras with residue field K , respectively. For every A ∈ Art K we shall denote by m A its maximal ideal.3.1. Deligne groupoids.
For every DG-Lie algebra L = ⊕ i ∈ Z L i over K , we denote by C L : Art K → Grpd the action groupoid of the gauge actionexp( L ⊗ m A ) × MC L ( A ) ∗ −→ MC L ( A ) , where A ∈ Art K , MC L ( A ) = { x ∈ L ⊗ m A | dx + 12 [ x, x ] = 0 } is the set of Maurer-Cartan element and the gauge action may be defined by the formula,see e.g. [9, 25, 26]: e a ∗ x := x + X n ≥ [ a, − ] n ( n + 1)! ([ a, x ] − da ) . We also denote by Def L = π ( C L ) : Art K → Set the deformation functor associated to L ,namely: Def L ( A ) = MC( L ⊗ m A )exp( L ⊗ m A ) , A ∈ Art K . Given two objects x, y of C L ( A ), the morphisms between them areMor C L ( x, y ) = { e a ∈ exp( L ⊗ m A ) | e a ∗ x = y } . The irrelevant stabilizer of an element x ∈ MC L ( A ) is the subgroup of I ( x ) ⊆ Mor C L ( x, x )defined in the following way: I ( x ) = n e du +[ x,u ] (cid:12)(cid:12)(cid:12) u ∈ L − ⊗ m A o . Then the the Deligne groupoid Del L : Art K → Grpd is defined by taking the same objectsas C L and by morphismsMor Del L ( x, y ) = Mor C L ( x, y ) I ( x ) ∼ = Mor C L ( x, y ) I ( y ) . We refer to [6, 15] for full details and for the proof that Del L is properly defined. The abovefunctors are homotopy invariant in the sense described by the following theorem: Theorem 3.1.
Let f : L → M be a quasi-isomorphism of differential graded Lie algebras.Then NDOMORPHISMS OF KOSZUL COMPLEXES 9 (1)
The induced morphism on deformation functors f : Def L → Def M is an isomor-phism. (2) The induced morphism
Del L → Del L is an equivalence of groupoids. If furthermore L and M are positively graded, the equivalence holds at the level of action groupoids C L → C M .Proof. This is nowadays well known: different proofs can be found for instance in [9, 18,23, 25]. (cid:3)
Totalization of semicosimplicial DG-Lie algebras.
Let L : L / / / / L / / / / / / L / / / / / / / / . . . be a semicosimplicial DG-Lie algebra: this means that every L i is a DG-Lie algebra andthe arrows in the diagram are the morphisms of DG-Lie algebras δ k : L n − → L n , k = 0 , . . . , n , subjected to the semicosimplicial identities δ l δ k = δ k +1 δ l , for any l ≤ k .The cochain complex C ( L ) associated to L is defined as the differential graded vectorspace C ( L ) = Y n ≥ L n [ − n ] , d + δ . More precisely, for every degree p ∈ Z we have C ( L ) p = Q n ≥ L p − nn , and the differential d + δ : C ( L ) p → C ( L ) p +1 is the sum of: • d = P n ≥ ( − n d n , where d n is the differential of L n ; • δ = P k ≥ ( − k δ k .For every integer n ≥
0, denote byΩ n = K [ t , . . . , t n , dt , . . . dt n ](1 − P t i , P dt i )the differential graded algebra of polynomial differential forms on the standard simplexof dimension n . Then the collection Ω • = { Ω n } n ≥ has a natural structure of simplicialdifferential graded algebra, and the face operators δ ∗ k : Ω n → Ω n − , k = 0 , . . . , n , are the morphisms of differential graded algebras such that δ ∗ k ( t i ) = t i if i < k i = kt i − if i > k The (Thom-Whitney-Sullivan) totalization of the semicosimplicial DG-Lie algebra L isthe DG-Lie subalgebra of Q n ≥ Ω n ⊗ L n defined byTot( L ) = ( x n ) ∈ Y n ≥ Ω n ⊗ L n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( δ ∗ k ⊗ Id ) x n = ( Id ⊗ δ k ) x n − for every 0 ≤ k ≤ n By a well known theorem by Whitney, see e.g. [8, 20, 29] and references therein, thereexists a natural pair of quasi-isomorphisms of complexes C ( L ) E −→ Tot( L ) I −→ C ( L ) suchthat IE = Id . In particular, the cohomology of the totalization Tot( L ) is the same of thecohomology of the cochain complex C ( L ). Remark . For later use we point out that the composition of I : Tot( L ) → C ( L ) with theprojection C ( L ) → L is a morphism of DG-Lie algebras, since it is the natural projectionTot( L ) → L . The complex C ( L ) admits the complete and exhaustive filtration F p = Q n ≥ p L n [ − n ] whose the first page in the associated spectral sequence is E p,q = H p ( L p ).Moreover the natural map H n (Tot( L )) = H n ( C ( L )) ։ E ,n ∞ ֒ → E ,n = H n ( L )coincides with the map induced in cohomology by the projection Tot( L ) → L .3.3. Descent of Deligne groupoids.
Let G : G / / / / G / / / / / / G / / / / / / / / . . . be a semicosimplicial groupoid, i.e., a semicosimplicial object in the category Grpd . De-noting as above by δ k the face operators in G , by semicosimplicial identities we get inparticular that δ δ = δ δ , δ δ = δ δ , δ δ = δ δ . Mimicking the construction of nonabelian 1-cocycles we define Z ( G ) as the set of pairs( l, m ) ∈ Obj( G ) × Mor( G )such that m : δ l → δ l and the cocycle diagram (3.1) δ δ l ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ δ m $ $ ❍❍❍❍❍❍❍❍❍ δ δ l δ m (cid:15) (cid:15) δ δ lδ δ l ❍❍❍❍❍❍❍❍❍ ❍❍❍❍❍❍❍❍❍ δ δ l δ m z z ✈✈✈✈✈✈✈✈✈ δ δ l is commutative in G . The total descent groupoid Tot( G ) is defined in the following way: • the set of objects of Tot( G ) is Z ( G ); • the morphisms between ( l , m ) , and ( l , m ) are morphisms a ∈ Mor( G ) between l and l making the following diagram: δ l m / / δ a (cid:15) (cid:15) δ l δ a (cid:15) (cid:15) δ l m / / δ l commutative in G . Theorem 3.3. [6, Theorem 7.6]
Let L : L / / / / L / / / / / / L / / / / / / / / . . . be a semicosimplicial DG-Lie algebra such that H j ( L i ) = 0 for every i and every j < .Consider the associated totalization Tot( L ) and the corresponding semicosimplicial Delignegroupoid: Del L : Del L / / / / Del L / / / / / / Del L / / / / / / / / . . . . Then there exists a natural isomorphism of functors
Art K → Set
Def
Tot( L ) ∼ = π Tot(Del( L )) . NDOMORPHISMS OF KOSZUL COMPLEXES 11
A result of this type appears already in [10, Theorem 4.1]; here under the strongerhypothesis N i = 0 for i < Deformations of coherent sheaves via locally free resolutions.
Let X be aseparated noetherian scheme over a field K of characteristic 0 and F a coherent sheaf on X . Every locally free resolution E • → F gives a sheaf H om ∗ O X ( E • , E • ) of DG-Lie algebrasover the sheaf of commutative rings O X . Let now U = { U i } i ∈ I be an affine open cover of X , then the ˇCech cochains of H om ∗ O X ( E • , E • ) in the cover U , i.e., Q i End ∗ O Ui ( E • | U i ) / / / / Q i,j End ∗ O Uij ( E • | U ij ) / / / / / / Q i,j,k End ∗ O Uijk ( E • | U ijk ) / / / / / / / / . . . has a natural structure of semicosimplicial differential graded Lie algebra whose cochaincomplex is the one computing ˇCech hypercohomology. For simplicity of notation denotethis semicosimplicial DG-Lie algebra by L ( U , E • ). Notice that for every multiindex α inthe nerve of the covering, the DG-Lie algebra End ∗ O Uα ( E • | U α ) has trivial cohomology innegative degree. In fact, since U α is affine, we haveEnd ∗ O Uα ( E • | U α ) = Hom ∗ O X ( U α ) ( E • ( U α ) , E • ( U α ))and E • ( U α ) → F ( U α ) is a projective resolution. Therefore for every i ∈ Z we have H i (Hom ∗ O X ( U α ) ( E • ( U α ) , E • ( U α )) ≃ Ext i O X ( U α ) ( F ( U α ) , F ( U α )) . In particular Theorem 3.3 applies to the semicosimplicial DG-Lie algebra L ( U , E • ): as aconsequence, Fiorenza, Iacono and Martinengo proved the following result: Theorem 3.4 ([6, Section 5]) . In the above notation the quasi-isomorphism class of theDG-Lie algebra
Tot( L ( U , E • )) depends only on F , i.e., it is independent of the choice of theresolution and of the affine cover. The functor Def
Tot( L ( U , E • )) is isomorphic to the functor Def F of (infinitesimal) deformation of the coherent sheaf F . Although technically complicated, the underlying idea of proof of Theorem 3.4 is quiteeasy and dates back to [3, 31], cf. also [11, 12]. First, by standard argument of deformationtheory, essentially contained mutatis mutandis in [1, 32], one prove that for every α inthe nerve of the covering, the Deligne groupoid of the DG-Lie algebra End ∗ O Uα ( E • | U α ) isisomorphic to the deformation groupoid of infinitesimal deformations of the coherent sheaf F over the affine scheme U α , see also [2] for a fully detailed proof. Then the total groupoidof the semicosimplicial Deligne groupoid associated to L ( U , E • ) compute the deformationsof F over X , since the commutativity of (3.1) can be interpreted as the usual cocycleconditions to glueing the local deformations of F into a global deformation.Assume for simplicity that U is a finite cover and consider its disjoint union Y = ` U i together the open covering map u : Y → X . Then the DG-Lie algebra Q i End ∗ O Ui ( E • | U i )controls the deformations of u ∗ F and the natural transformation of deformation functorsDef F → Def u ∗ F is induced by the projectionTot( L ( U , E • )) → Y i End ∗ O Ui ( E • | U i ) . Deformations of locally unobstructed coherent sheaves
A coherent sheaf F on a scheme X is said to be locally unobstructed if there exists anopen affine cover U = { U i } such that every restriction F | U i has unobstructed deformations. The notion of obstruction to deformations of F | U i makes sense even if the space of first order defor-mations is not finite dimensional, and therefore the corresponding functor of Artin rings does not satisfySchlessinger’s conditions, see [5, 26]. The construction of the previous section allows to give a proof in the framework of dif-ferential graded Lie algebras of the following result about the vanishing of the obstructionsof a locally unobstructed coherent sheaf under the local-to-global Ext spectral sequence.
Theorem 4.1.
Let F be a locally unobstructed coherent sheaf on a projective scheme X over a field of characteristic 0. Then the obstructions to deforming F are contained in thekernel of the natural map α : Ext X ( F , F ) → E , = H ( X, E xt X ( F , F )) . In particular, if α is injective then F has unobstructed deformations.Proof. Let U = { U i } be a open affine cover such that every restriction F | U i has unobstructeddeformations. Fix a locally free resolution E • → F and let L ( U , E • ) be the semicosimplicialDG-Lie algebra as in Theorem 3.4. According to Remark 3.2 the pages E p,qr , r ≥
2, of thespectral sequence may be computed by considering the cochain complex C ( L ( U , E • )). Since E p,q = 0 for every p < H ( X, E xt nX ( F , F )) = E ,n ⊆ E ,n = Y i H ( U i , E xt nU i ( F , F )) , and then it is sufficient to prove that every obstruction is annihilated by the mapExt X ( F , F ) = H (Tot( L ( U , E • ))) → Y i H ( U i , E xt U i ( F , F )) . induced by the morphism of DG-Lie algebrasTot( L ( U , E • )) → L ( U , E • ) = Y i End O Ui ( E •| U i ) . Since the DG-Lie algebra End O Ui ( E •| U i ) controls the infinitesimal deformations of F | U i andmorphisms of DG-Lie algebras commute with obstruction maps, it follows that obstructionsto deformations of F are annihilated by every map Tot( L ( U , E • )) → End O Ui ( E •| U i ). (cid:3) The expert reader immediately recognize that the use of DG-Lie algebras in the proofof Theorem 4.1 is not strictly necessary and the same result can be proved by classicalmethods. In fact, for every index i the composite mapExt X ( F , F ) → H ( X, E xt X ( F , F )) → H ( U i , E xt U i ( F , F ))is the obstruction map associated to the natural transformation of deformation functorsDef F → Def F | U i and the functor Def F | U i is unobstructed by assumption. However, theapproach via DG-Lie algebras not only it is interesting on its on, but also provide an imme-diate generalization of the theorem to derived deformations, i.e., to deformations of F overdifferential graded local Artin rings. Without entering into details of derived deformationtheory, we only recall that, over a field of characteristic 0, there exists an equivalence ofcategories, between the homotopy category of DG-Lie algebras and the category of deriveddeformation problems [22, 30]: when the derived deformation problem is represented by aset-valued functor in the category of DG-Artin rings, the staightforward generalization ofMaurer-Cartan equation and gauge equivalence gives the required equivalence, cf. [24, 30].Moreover unobstructed derived deformation problems corresponds precisely to homotopyabelian DG-Lie algebras (this is implicitly proved in [24, Sec. 7] since the minimal L ∞ model is reconstructed from derived obstruction maps; for an explicit statement we referto [19, Thm. 6.3]). NDOMORPHISMS OF KOSZUL COMPLEXES 13
A remark about obstructions to derived deformations. If L is a DG-Lie algebra the notionsof MC L ( A ) and Def L ( A ) extend naturally to the case where A is a differential graded local K -algebras with residue field A π −→ K :MC L ( A ) = (cid:26) x ∈ ( L ⊗ m A ) (cid:12)(cid:12)(cid:12)(cid:12) dx + 12 [ x, x ] = 0 (cid:27) , Def L ( A ) = MC L ( A )exp( L ⊗ m A ) , and, as in the classical case, these two functors have the same obstruction theory. A simplecomputation, see [24] for details, shows that if f : A → B is a surjective morphism of DG-Artin rings such that its kernel is an acyclic complex annihilated by the maximal ideal m A ,then f : Def L ( A ) → Def L ( B ) is a bijective map.If t ∈ A is a homogeneous non-trivial closed element of degree − n annihilated by themaximal ideal m A , we have a small extension:(4.1) 0 → K [ n ] t −→ A p −→ A/ ( t ) → rq − : Def L ( A/ ( t )) → H n ( L ) is defined in the followingway. Consider a formal symbol s of degree − n −
1, the differential graded Artin K -algebra A ⊕ K s, m A s = s = 0 , ds = t, and the surjective morphisms: q : A ⊕ K s → A/ ( t ) , q ( a + bs ) = p ( a ) ,r : A ⊕ K s → K ⊕ K s, r ( a + bs ) = π ( a ) + bs . As observed above, the map q : Def L ( A ⊕ K s ) → Def L ( A/ ( t )) is bijective and it is notdifficult to see [24, pag. 737] that an element x ∈ Def L ( A/ ( t )) lifts to Def L ( A ) if and onlyif rq − ( x ) = 0 ∈ Def L ( K ⊕ K s ) ∼ = H n ( L ) . If L is an abelian DG-Lie algebra, then Def L ( A/ ( t )) = H ( L ⊗ m A / ( t )) and the map rq − : Def L ( A/ ( t )) → H ( L [ n ]) = H n ( L ) is the connecting homomorphism of the shortexact sequence of complexes0 → L [ n ] → L ⊗ m A → L ⊗ m A ( t ) → . Definition 4.2.
We shall say that the morphism rq − : Def L ( A/ ( t )) → H n ( L ) associ-ated to the small extension (4.1) is an obstruction map if the morphism p is surjective incohomology, or equivalently if t is not exact in A .If L is abelian, every obstruction map vanishes by K¨unneth formula. Since the maps rq − depends only on Def L , that in turn depends only on the homotopy class of L , it followsthat if L is homotopy abelian, then every (derived) obstruction map vanishes. Conversely,using the L ∞ -minimal model of L, it is possible to prove that: if all obstructions vanish,then L is homotopy abelian. It is worth to point out, see [19, Thm. 6.3] and [24, Proof ofThm. 7.1], that the homotopy abelianity of L is equivalent to the vanishing of obstructionsarising from the subclass of small extensions (4.1) where A has zero differential. Theorem 4.3.
Let F be a coherent sheaf on a projective scheme X over a field K ofcharacteristic 0, and assume that F is locally unobstructed in the derived sense. Then theobstructions to derived deformations of F are contained in the kernel of the natural map ofgraded vector spaces α : M n Ext n +2 X ( F , F ) → M n H ( X, E xt n +2 X ( F , F )) . In particular, if α is injective then the DG-Lie algebra Tot( L ( U , E • )) controlling deforma-tions of F is homotopy abelian over K .Proof. It the same sitution as in the proof of Theorem 4.1 the assumption implies that everyDG-Lie algebra End O Ui ( E •| U i ) is homotopy abelian over K and the map of graded vectorspaces α is the (derived) obstruction map induced by the morphism of DG-Lie algebrasTot( L ( U , E • )) → L ( U , E • ) = Y i End O Ui ( E •| U i ) . If α is injective, then Tot( L ( U , E • )) is numerically homotopy abelian, and hence homotopyabelian over the field K by Remark 1.4. (cid:3) The case of locally complete intersection ideal sheaves.
The results of Sec-tion 2 imply in particular that the assumption about local unobstructness of F made inTheorems 4.1 and 4.3 is valid when F is a locally complete intersection ideal sheaf I ⊂ O X :this means that I is a coherent sheaf and there exists an affine open cover U = { U i } suchthat on every U i the quotient sheaf O Z = O X / I admits a Koszul resolution K ∗ → O Z ,where K ∗ : 0 → ∧ r O rU i → · · · → ∧ O rU i → O rU i ( f ,...,f r ) −−−−−−→ O U i ,f , . . . , f r ∈ I ( U i ) , H ( K ∗ ) = O U i I | U i , H i ( K ∗ ) = 0 for every i = 0 . We do not require that the integer r is the same for every U i : for instance if U i does notintersect the closed subscheme defined by I we have r = 1 and f an invertible element. Lemma 4.4.
Let I ⊂ O X be a locally complete intersection ideal sheaf on a projectivescheme over a field K of characteristic 0, and denote by O Z = O X / I the structure sheaf onthe closed subscheme Z defined by I .Let F be either a line bundle over Z or F = I ⊗ L for a line bundle L on X . Then forevery open affine subset U ⊂ X the sheaf F | U has unobstructed derived deformations.Proof. Since Ext nU ( F , F ) = H ( U, E xt nU ( F , F )), by Theorem 4.3 it is sufficient to prove that F | U is locally unobstructed in the derived sense, i.e., that every point belongs to an openaffine set V ⊂ U such that the DG-Lie algebra controlling deformations of F | V is homotopyabelian. Choose V sufficiently small such that I is generated by a regular sequence andeither F | U ≃ O Z ∩ V or F | U ≃ I V .Since the homotopy class of the DG-Lie algebra controlling deformations is independentof the choice of the resolution we can consider the Koszul resolutions K ∗ q-iso −−−→ O Z ∩ V , K < −−−→ I | V , where K ∗ : 0 → ∧ r O rV → · · · → ∧ O rV → O rV ( f ,...,f r ) −−−−−−→ O V . According to Theorems 2.2 and 2.4 both the DG-Lie algebras End ∗ O V ( K ∗ ) and End ∗ O V ( K < )are numerically homotopy abelian, and hence homotopy abelian over the field K . (cid:3) Corollary 4.5.
Let I ⊂ O X be a locally complete intersection ideal sheaf on a projectivescheme over a field K of characteristic 0, and denote by O Z = O X / I the structure sheaf onthe closed subscheme Z defined by I . Let F be either a line bundle over Z or F = I ⊗ L fora line bundle L on X . Then: (1) the obstructions to deforming F are contained in the kernel of the natural map α : Ext X ( F , F ) → H ( X, E xt X ( F , F )) . In particular, if α is injective then F has unobstructed deformations; NDOMORPHISMS OF KOSZUL COMPLEXES 15 (2) the obstructions to derived deformations of F are contained in the kernel of thenatural map of graded vector spaces α : M n Ext n +2 X ( F , F ) → M n H ( X, E xt n +2 X ( F , F )) . In particular, if α is injective then the DG-Lie algebra Tot( L ( U , E • )) controllingdeformations of F is homotopy abelian over K .Proof. Immediate from the above results. (cid:3)
Example 4.6.
Let Z be a locally complete intersection subvariety of a smooth projec-tive manifold X , with normal sheaf N Z | X . Then E xt nX ( O Z , O Z ) ≃ ∧ n N Z | X and then theobstruction to deformations of the coherent sheaf O Z are contained in the kernel of themap Ext X ( O Z , O Z ) → H ( Z, ∧ N Z | X ) . If H i ( Z, ∧ k N Z | X ) = 0 for every k ≥ i >
0, then the coherent sheaf O Z isunobstructed in the derived sense: in fact by the local-to-global Ext spectral sequence weget that the map α of Corollary 4.5 is injective. Example 4.7.
Let Z be a locally complete intersection subvariety of codimension p ≥ X , with ideal sheaf I . Then E xt kX ( I , I ) = 0 for k ≥ p and(4.2) E xt kX ( I , I ) ∼ = E xt kX ( O Z , O Z ) ∼ = ∧ k N Z | X for 0 ≤ k < p . The first part is clear since, locally, the coherent sheaf I has projective dimension < p as O X -module. In view of the natural isomorphism H om X ( O Z , O Z ) ≃ −→ H om X ( O X , O Z ), thefunctor H om X ( − , O Z ) applied to the short exact sequence 0 → I j −→ O X π −→ O Z →
0, give asequence of natural isomorphisms E xt kX ( I , O Z ) ∼ = −→ E xt k +1 X ( O Z , O Z ) , k ≥ . Then we observe that for every i = 0 , p − E xt iX ( I , O X ) ∼ = E xt i +1 X ( O Z , O X ) = 0,while the natural map E xt p − X ( I , O X ) → E xt p − ( I , O Z )is an isomorphism: this follows easily by a direct computation involving the Koszul complexas locally free resolution of I . Moreover, since E xt X ( O Z , O X ) = E xt X ( O Z , O X ) = 0 we have H om X ( I , O X ) ≃ H om X ( O X , O X ) ≃ O X and in particular the morphism H om X ( I , O X ) −→ H om X ( I , O Z )is trivial.Therefore, the functor H om X ( I , − ) applied to the short exact sequence 0 → I j −→ O X π −→ O Z → H om X ( I , I ) ∼ = H om X ( I , O X ) ∼ = O X , E xt k − X ( I , O Z ) ∼ = −→ E xt kX ( I , I ) 0 < k < p . In conclusion, we can rephrase the result of Corollary 4.5 as follows: if p > I are contained in the kernel of the mapExt X ( I , I ) → H ( Z, ∧ N Z | X ) . If H i ( Z, ∧ k N Z | X ) = 0 for every 0 ≤ k < p and every i >
0, then the local to globalmorphism α is injective and the coherent sheaf I is unobstructed in the derived sense. Acknowledgement.
M.M. wishes to acknowledge the support by Italian MIUR under PRIN project2015ZWST2C “Moduli spaces and Lie theory”. F.C. thanks the second named author who proposedthis project for her master thesis in the far 2014 [2]. F.C. was supported by the Engineering andPhysical Sciences Research Council [EP/L015234/1]. The EPSRC Centre for Doctoral Training inGeometry and Number Theory (The London School of Geometry and Number Theory), UniversityCollege London
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University of Edinburgh,School of Mathematics “James Clerk Maxwell building”,The King’s Buildings, EH9 3FD, United Kingdom.
E-mail address : [email protected] Universit`a degli studi di Roma “La Sapienza”,Dipartimento di Matematica “Guido Castelnuovo”,P.le Aldo Moro 5, I-00185 Roma, Italy.
E-mail address : [email protected] URL ::