Koszul complexes and fully faithful integral functors
aa r X i v : . [ m a t h . AG ] D ec KOSZUL COMPLEXES AND FULLY FAITHFUL INTEGRALFUNCTORS
FERNANDO SANCHO DE SALAS
Abstract.
We characterise those objects in the derived category of a scheme whichare a sheaf supported on a closed subscheme in terms of Koszul complexes. This isapplied to generalize to arbitrary schemes the fully faithfullness criteria of an integralfunctor.
Contents
Introduction 1Acknowledgements 21. Koszul complexes, depth and support 21.1. System of parameters. Koszul complex 21.2. Depth. Singularity set 22. Fully faithful Integral functors 6References 8
Introduction
Let
X, Y be two proper schemes over a field k and letΦ : D bc ( X ) → D bc ( Y )be an integral functor between their derived categories of complexes of quasi-coherentmodules with bounded and coherent cohomology. Let K • ∈ D bc ( X × Y ) be the kernelof Φ. We want to characterise those kernels K • such that Φ is a fully faithful. Thiswas solved in [1] for smooth projective schemes over a field of zero characteristic. ForGorenstein schemes and zero characteristic it was solved in [3]. For Cohen-Macaulayschemes and arbitrary characteristic it was solved in [2]. Here we remove the Cohen-Macaulay hypothesis and reproduce the fully faithfullness criteria of [2] for arbitraryschemes. The point is to replace the locally complete intersection zero-cycles of [2]by Koszul complexes associated to a system of parameters. These Koszul complexesallow to characterise, for an arbitrary scheme X , those objects in D bc ( X ) consisting ofa sheaf supported on a closed subscheme (Propositions 1.7 and 1.9). This is the mainingredient for the fully faithfullness criteria. Date : July 8, 2018.2000
Mathematics Subject Classification.
Primary: 18E30; Secondary: 14F05, 14J27, 14E30,13D22, 14M05.
Key words and phrases.
Geometric integral functors, Fourier-Mukai, fully faithful, equivalence ofcategories.Work supported by research projects MTM2006-04779 (MEC) and SA001A07 (JCYL).
Acknowledgements.
I would like to thank Leovigildo Alonso, who suggested to methe use of Koszul complexes to deal with the general (non Cohen-Macaulay) case.1.
Koszul complexes, depth and support
We introduce Koszul complexes and use them to characterize those objects in thederived category consisting of a sheaf supported on a closed subscheme.1.1.
System of parameters. Koszul complex. . Let O be a noetherian local ringof dimension n and maximal ideal m . Let x be the closed point. Definition 1.1.
A sequence f = { f , . . . , f n } of n elements in m is called a system ofparameters of O if O / ( f , . . . , f n ) is a zero dimensional ring. In other words, ( f , . . . , f n )is a m -primary ideal. We shall also denote O /f = O / ( f , . . . , f n ). △ It is a basic fact of dimension theory that there always exists a system of parameters.In fact, for any m -primary ideal I , there exist f , . . . , f n in I which are a system ofparameters of O .We shall denote by Kos • ( f ) the Koszul complex associated to a system of parameters f . That is, if we denote L = O ⊕ n and ω : L → O the morphism given by f , . . . , f n ,then the Koszul complex is V i O L in degree − i and the differential V i O L → V i − O L is theinner contraction with ω . It is immediate to see that Hom • (Kos • ( f ) , O ) ≃ Kos • ( f )[ − n ].The cohomology modules H i (Kos • ( f )) are supported at x (indeed they are annihi-lated by ( f , . . . , f n )). Moreover H (Kos • ( f )) = O /f and H i (Kos • ( f )) = 0 for i > i < − n .For any complex M • of O -modules, we shall denoteTor O i (Kos • ( f ) , M • ) = H − i (Kos • ( f ) ⊗ O M • )Ext i O (Kos • ( f ) , M • ) = H i (Hom • O (Kos • ( f ) , M • ))From the isomorphism Hom • (Kos • ( f ) , O ) ≃ Kos • ( f )[ − n ] it follows easily that(1.1) Ext i O (Kos • ( f ) , M • ) ≃ Tor O n − i (Kos • ( f ) , M • ) . Depth. Singularity set.
The depth of an O -module M , depth( M ), is the firstinteger i such that either: • Ext i ( O / m , M ) = 0 or • H ix (Spec O , M ) = 0 or • Ext i O ( N, M ) = 0 for some non zero finite O -module N supported at x or • Ext i O ( N, M ) = 0 for any non zero finite O -module N supported at x . Lemma 1.2.
The depth of M is the first integer i such that either: • Ext i O (Kos • ( f ) , M ) = 0 for some system of parameters f of O or • Ext i O (Kos • ( f ) , M ) = 0 for every system of parameters f of O .Proof. It is an easy consequence of the spectral sequence E p,q = Ext p ( H − q (Kos • ( f ) , M ) = ⇒ E p + q ∞ = Ext p + q (Kos • ( f ) , M )Indeed, let d = depth( M ), f a system of parameters of O and r the first integer suchthat Ext i O (Kos • ( f ) , M ) = 0. Let us see that d = r . Since Ext d ( H (Kos • ( f )) , M ) = 0,one obtains, by the spectral sequence, that Ext d O (Kos • ( f ) , M ) = 0. Hence d ≥ r . OSZUL COMPLEXES AND FULLY FAITHFUL INTEGRAL FUNCTORS 3
Assume that r = d . Then Hom r − i ( H − i (Kos • ( f ) , M ) = 0 for any i ≥
0, because H − i (Kos • ( f ) is supported at x and r − i < d . From the exact trianglesKos • ( f ) ≤− i − → Kos • ( f ) ≤− i → H − i (Kos • ( f )[ i ]and taking into account that Hom r (Kos • ( f ) ≤ , M ) = Hom r (Kos • ( f ) , M ) = 0 one ob-tains that Hom r (Kos • ( f ) ≤− i , M ) = 0 for any i ≥
0. This is absurd because Kos • ( f ) ≤− i =0 for i >> (cid:3) Let F be a coherent sheaf on a scheme X of dimension n . We write n x for thedimension of the local ring O x of X at a point x ∈ X , F x for the stalk of F at x and k ( x ) for the residual field of x . F x is a O x -module. The integer number codepth( F x ) = n x − depth( F x ) is called the codepth of F at x . For any integer m ∈ Z , the m -thsingularity set of F is defined to be S m ( F ) = { x ∈ X | codepth( F x ) ≥ n − m } . Then, if X is equidimensional, a closed point x is in S m ( F ) if and only if depth( F x ) ≤ m .Since depth( F x ) is the first integer i such that either • Ext i O x ( k ( x ) , F x ) = 0 or • H ix ( F x ) = 0 or • Ext i O x (Kos • ( f x ) , F x ) = 0 for some system of parameters f x of O x or • Ext i O x (Kos • ( f x ) , F x ) = 0 for every system of parameters f x of O x we have alternative descriptions of S m ( F ):(1.2) S m ( F ) = { x ∈ X | H ix (Spec O X,x , F x ) = 0 for some i ≤ m + n x − n } = { x ∈ X | Ext i O x (Kos • ( f x ) , F x ) = 0 for some i ≤ m + n x − n and some system of parameters f x of O X,x } = { x ∈ X | Ext i O x (Kos • ( f x ) , F x ) = 0 for some i ≤ m + n x − n and any system of parameters f x of O X,x } Lemma 1.3. [3, Lemma 1.10] . If X is smooth, then the m -th singularity set of F canbe described as S m ( F ) = ∪ p ≥ n − m { x ∈ X | Tor O x p ( k ( x ) , F x ) = 0 } , where k ( x ) is the residue field of O x . In the singular case, this characterization of S m ( F ) is not true. There is a similarinterpretation for Cohen-Macaulay schemes replacing k ( x ) by O Z x where Z x is a locallycomplete intersection zero cycle supported on x (see [2, Lemma 3.5]). Now, for arbitraryschemes, the analogous interpretation is the following. Lemma 1.4.
The m -th singularity set S m ( F ) can be described as S m ( F ) = { x ∈ X | there is an integer i ≥ n − m with Tor O x i (Kos • ( f x ) , F ) = 0 for any system of parameters f x of O X,x } . Proof.
It follows from (1.1) and (1.2). (cid:3)
Proposition 1.5. [3, Prop 1.13] . Let X be an equidimensional scheme of dimension n and F a coherent sheaf on X . FERNANDO SANCHO DE SALAS (1) S m ( F ) is a closed subscheme of X and codim S m ( F ) ≥ n − m . (2) If Z is an irreducible component of the support of F and c is the codimensionof Z in X , then codim S n − c ( F ) = c and Z is also an irreducible component of S n − c ( F ) . Corollary 1.6. [3, Cor. 1.14] . Let X be a scheme and let F be a coherent O X -module.Let h : Y ֒ → X be an irreducible component of the support of F and c the codimensionof Y in X . There is a non-empty open subset U of Y such that for any x ∈ U and anysystem of parameters f x of O X,x one has
Tor O x c (Kos • ( f x ) , F x ) = 0Tor O x c + i (Kos • ( f x ) , F x ) = 0 , for every i > .Proof. By Lemma 1.4 the locus of the points that verify the conditions is U = Y ∩ ( S n − c ( F ) − S n − c − ( F )), which is open in Y by Proposition 1.5. Proving that U is notempty is a local question, and we can then assume that Y is the support of F . Now Y = S n − c ( F ) by (2) of Proposition 1.5 and U = S n − c ( F ) − S n − c − ( F ) is non-emptybecause the codimension of S n − c − ( F ) in X is greater or equal than c + 1 again byProposition 1.5. (cid:3) For any scheme X we denote by D ( X ) the derived category of complexes of quasi-coherent O X -modules and by D bc ( X ) the faithful subcategory consisting of those com-plexes with bounded and coherent cohomology sheaves.The following proposition characterises objects of the derived category supported ona closed subscheme. Proposition 1.7. [1, Prop. 1.5][3, Prop. 1.15] . Let j : Y ֒ → X be a closed immersionof codimension d of irreducible schemes and K • an object of D bc ( X ) . Assume that (1) If x ∈ X − Y is a closed point, then there exists a system of parameters f x of O x such that Tor O x i (Kos • ( f x ) , K • x ) = 0 for every i . (2) If x ∈ Y is a closed point, then there exists a system of parameters f x of O x such that Tor O x i (Kos • ( f x ) , K • x ) = 0 when either i < or i > d .Then there is a sheaf K on X whose topological support is contained in Y and such that K • ≃ K in D bc ( X ) . Moreover, this topological support coincides with Y unless K • = 0 .Proof. We just reproduce the proof of [3, Prop. 1.15], with the corresponding changes.Let us write H q = H q ( K • ). For every system of parameters f x of O x there is a spectralsequence E − p,q = Tor O x p (Kos • ( f x ) , H qx ) = ⇒ E − p + q ∞ = Tor O x p − q (Kos • ( f x ) , K • x )Let q be the maximum of the q ’s with H q = 0. If x ∈ supp( H q ), one has thatTor O x (Kos • ( f x ) , H q x ) ≃ H (Kos • ( f x )) ⊗ O x H q x = 0 for every system of parameters f x of O x . A nonzero element in Tor O x (Kos • ( f x ) , H q x ) survives up to infinity in thespectral sequence. Since there is a system of parameters f x of O x such that E q ∞ =Tor O x − q (Kos • ( f x ) , K • ) = 0 for every q > q ≤
0. A similarargument shows that the topological support of all the sheaves H q is contained in Y :assume that this is not true and let us consider the maximum q of the q ’s such that H qx = 0 for a certain point x ∈ X − Y ; then Tor O x (Kos • ( f x ) , H q x ) = 0 and a nonzero OSZUL COMPLEXES AND FULLY FAITHFUL INTEGRAL FUNCTORS 5 element in Tor O x (Kos • ( f x ) , H q x ) survives up to infinity in the spectral sequence, whichis impossible since Tor O x i (Kos • ( f x ) , K • ) = 0 for every i .Let q ≤ q be the minimum of the q ’s with H q = 0. We know that H q is topologi-cally supported on a closed subset of Y . Take a component Y ′ ⊆ Y of the support. If c ≥ d is the codimension of Y ′ , then there is a non-empty open subset U of Y ′ such thatTor O x c (Kos • ( f x ) , H q x ) = 0 for any closed point x ∈ U and any system of parameters f x of O x , by Corollary 1.6. Elements in Tor O x c (Kos • ( f x ) , H q x ) would be killed in thespectral sequence by Tor O x p (Kos • ( f x ) , H q +1 x ) with p ≥ c + 2. By Lemma 1.4 the set { x ∈ X | Tor O x i (Kos • ( f x ) , H q +1 x ) = 0 for some i ≥ c + 2 and any parameters f x of O x } is equal to S n − ( c +2) ( H q +1 ) and then has codimension greater or equal than c + 2 byProposition 1.5. Thus there is a point x ∈ Y ′ such that any nonzero element inTor O x c (Kos • ( f x ) , H q x ) survives up to the infinity in the spectral sequence. Therefore,Tor O x c − q (Kos • ( f x ) , K • x ) = 0 for any system of parameters f x of O x . Thus c − q ≤ d which leads to q ≥ c − d ≥ q = q = 0. So K • = H in D b ( X ) and thetopological support of K = H is contained in Y . Actually, if K • = 0, then this supportis the whole of Y : if this was not true, since Y is irreducible, the support would havea component Y ′ ⊂ Y of codimension c > d and one could find, reasoning as above, anon-empty subset U of Y ′ such that Tor O x c (Kos • ( f x ) , K • x ) = 0 for all x ∈ U and allsystem of parameters f x of O x . This would imply that c ≤ d , which is impossible. (cid:3) Assume now that X is separated. Let x be a closed point of X and φ x : Spec O x → X the natural morphism. Let f x be a system of parameters of O x . We shall still denoteby Kos • ( f x ) the direct image by φ x of the Koszul complex Kos • ( f x ). Let U be an affineopen subset containing x . Then φ x is the composition of φ ′ x : Spec O x → U with theopen embedding i U : U ֒ → X . Since X is separated, i U is an affine morphism, and then φ x ∗ ≃ R φ x ∗ .One has that Lemma 1.8.
For any K • ∈ D ( X ) one has Hom iD ( X ) (Kos • ( f x ) , K • ) ≃ Ext i O x (Kos • ( f x ) , K • x ) Proof.
Let C be the cone of K • → φ x ∗ φ ∗ x K • . It is clear that x / ∈ supp( C ). On the otherhand φ x ∗ Kos • ( f x ) is supported at x . Then Hom i ( φ x ∗ Kos • ( f x ) , C ) = 0 andHom iD ( X ) ( φ x ∗ Kos • ( f x ) , K • ) ≃ Hom iD ( X ) ( φ x ∗ Kos • ( f x ) , φ x ∗ φ ∗ x K • )and one concludes because φ ∗ x φ x ∗ Kos • ( f x ) ≃ Kos • ( f x ). (cid:3) Taking into account the equation (1.1), Proposition 1.7 may be reformulated asfollows:
Proposition 1.9.
Let j : Y ֒ → X be a closed immersion of codimension d of irreducibleschemes of dimensions m and n respectively, and let K • be an object of D bc ( X ) . Assumethat for any closed point x ∈ X there is a system of parameters f x of O x such that Hom iD ( X ) (Kos • ( f x ) , K • ) = 0 , unless x ∈ Y and m ≤ i ≤ n . Then there is a sheaf K on X whose topological supportis contained in Y and such that K • ≃ K in D bc ( X ) . Moreover, the topological supportis Y unless K • = 0 . (cid:3) FERNANDO SANCHO DE SALAS
Spanning classes.
Lemma 1.10.
For each closed point x ∈ X choose a system of parameters f x of O x .The set Ω = { Kos • ( f x ) for all closed points x ∈ X } is a spanning class for D bc ( X ) .Proof. Take a non-zero object E • in D bc ( X ). Let q be the maximum of the q ’s suchthat H q ( E • ) = 0, x a closed point of the support of H q ( E • ) and − l the minimum of the p ’s such that H p (Kos • ( f x )) = 0. ThenHom − l − q D ( X ) ( E • , Kos • ( f x )) ≃ Hom O X ( H q ( E • ) , H − l (Kos • ( f x )) ≃ Hom O x ( H q ( E • ) x , H − l (Kos • ( f x )) = 0 . On the other hand, by Proposition 1.9 with Y = ∅ , if Hom iD ( X ) (Kos • ( f x ) , E • ) = 0 forevery i and every x , then E • = 0. (cid:3) Fully faithful Integral functors
In this section scheme means a separated scheme of finite type over an algebraicallyclosed field k .Let X and Y be proper schemes, K • an object in D bc ( X × Y ) andΦ K • X → Y : D ( X ) → D ( Y )the integral functor associated to K • . If X is projective and K • has finite homologicaldimension over both X and Y , then Φ K • X → Y maps D bc ( X ) to D bc ( Y ) and it has an integralright adjoint (see [2, Def. 2.1], [2, Prop. 2.7] and [2, Prop. 2.9]).The notion of strong simplicity is the following. Definition 2.1.
An object K • in D bc ( X × Y ) is strongly simple over X if it satisfiesthe following conditions:(1) For every closed point x ∈ X there is a system of parameters f x of O x suchthat Hom iD ( Y ) (Φ K • X → Y (Kos • ( f x ) , Φ K • X → Y ( k ( x ))) = 0unless x = x and 0 ≤ i ≤ dim X .(2) Hom D ( Y ) (Φ K • X → Y ( k ( x )) , Φ K • X → Y ( k ( x ))) = k for every closed point x ∈ X . △ Theorem 2.2.
Let X and Y be proper schemes over an algebraically closed field ofcharacteristic zero, and let K • be an object in D bc ( X × Y ) of finite homological dimensionover both X and Y . Assume also that X is projective and integral. Then the functor Φ K • X → Y : D bc ( X ) → D bc ( Y ) is fully faithful if and only if the kernel K • is strongly simpleover X .Proof. The same proof as [2, Thm. 3.6] works, replacing the use of Proposition 3.1 of[2] by its analogous result (Proposition 1.9). (cid:3)
Definition 2.3.
An object K • of D bc ( X × Y ) satisfies the orthonormality conditionsover X if it has the following properties: OSZUL COMPLEXES AND FULLY FAITHFUL INTEGRAL FUNCTORS 7 (1) For every closed point x ∈ X there is a system of parameters f x of O x suchthat Hom iD ( Y ) (Φ K • X → Y (Kos • ( f x ) , Φ K • X → Y ( k ( x ))) = 0unless x = x and 0 ≤ i ≤ dim X .(2) There exists a closed point x such that at least one of the following conditionsis fulfilled:(2.1) Hom D ( Y ) (Φ K • X → Y ( O X ) , Φ K • X → Y ( k ( x ))) ≃ k .(2.2) Hom D ( Y ) (Φ K • X → Y (Kos • ( f x )) , Φ K • X → Y ( k ( x ))) ≃ k for any system of parameters f x of O x .(2.2 ∗ ) Hom D ( Y ) (Φ K • X → Y ( O x /f x ) , Φ K • X → Y ( k ( x ))) ≃ k for any system of parameters f x of O x .(2.3) 1 ≤ dim Hom D ( Y ) (Φ K • X → Y (Kos • ( f x )) , Φ K • X → Y ( O x /f x )) ≤ l ( O x /f x ) for any sys-tem of parameters f x of O x , where l ( O x /f x ) is the length of O x /f x .(2.3 ∗ ) 1 ≤ dim Hom D ( Y ) (Φ K • X → Y ( O x /f x ) , Φ K • X → Y ( O x /f x )) ≤ l ( O x /f x ) for any systemof parameters f x of O x . △ Theorem 2.4.
Let X and Y be proper schemes over an algebraically closed field ofarbitrary characteristic, and let K • be an object in D bc ( X × Y ) of finite homologicaldimension over both X and Y . Assume also that X is projective, Cohen-Macaulay,equidimensional and connected. Then the functor Φ K • X → Y : D bc ( X ) → D bc ( Y ) is fully faith-ful if and only if the kernel K • satisfy the orthonormality conditions over X (Definition2.3).Proof. The proof is essentially the same as [2, Thm. 3.8]. We give the details.The direct is immediate. Let us see the converse. Let us denote Φ = Φ K • X → Y . Oneknows that Φ has a right adjoint H and that H ◦ Φ ≃ Φ M X → X . Using condition (1) ofDefinition 2.3, one sees that M is a sheaf whose support is contained in the diagonaland π ∗ M is locally free. Since X is connected, we can consider the rank r of π ∗ M ,which is nonzero by condition (2) of Definition 2.3; thus the support of M is thediagonal. To conclude, we have only to prove that r = 1.Since M is a sheaf topologically supported on the diagonal and π ∗ M is locally free,it follows that if F is a sheaf, then Φ M X → X ( F ) is also a sheaf.Now assume that K • satisfies (2.1) of Definition 2.3. ThenHom D ( X ) ( O X , Φ M X → X ( k ( x ))) ≃ Hom D ( Y ) (Φ K • X → Y ( O X ) , Φ K • X → Y ( k ( x ))) ≃ k. Hence Φ M X → X ( k ( x )) ≃ k ( x ); that is, j ∗ x M ≃ k ( x ), where j x : { x } ֒ → X is the inclusion,and r = 1.If K • satisfies (2.2) of Definition 2.3, thenHom O X ( O x /f x , j ∗ x M ) ≃ Hom D ( X ) (Kos • ( f x ) , j ∗ x M ) ≃ Hom D ( X ) (Kos • ( f x ) , Φ M X → X ( k ( x ))) ≃ Hom D ( Y ) (Φ K • X → Y (Kos • ( f x )) , Φ K • X → Y ( k ( x ))) ≃ k for any system of parameters f x of O x . Hence j ∗ x M ≃ k ( x ) and r = 1. FERNANDO SANCHO DE SALAS (2.2 ∗ ) is equivalent to (2.2), becauseHom D ( Y ) (Φ K • X → Y (Kos • ( f x )) , Φ K • X → Y ( k ( x ))) ≃ Hom D ( X ) (Kos • ( f x ) , Φ M X → X ( k ( x ))) ≃ Hom D ( X ) ( O x /f x , Φ M X → X ( k ( x ))) ≃ Hom D ( Y ) (Φ K • X → Y ( O x /f x ) , Φ K • X → Y ( k ( x )))where the second isomorphism is due to the fact that Φ M X → X ( k ( x )) is a sheaf and to H (Kos • ( f x )) = O x /f x .Finally, assume that K • satisfies (2.3) of Definition 2.3 (which is equivalent to (2.3 ∗ )by similar arguments), and let us prove that then condition (2.2 ∗ ) of Definition 2.3holds as well.We already know that if F is a sheaf supported at a point x , then φ ( F ) = Φ M X → X ( F )is also a sheaf supported at x . Moreover φ is exact and it has a left adjoint G (see theproof of [2, Thm. 3.8]). Let us denote B = O x /f x .First notice thatHom D ( Y ) (Φ K • X → Y ( B ) , Φ K • X → Y ( B )) ≃ Hom O X ( B, Φ M X → X ( B )) ≃ Hom O X ( G ( B ) , B )Hence, condition (2.3 ∗ ) means that( ∗ ) 1 ≤ dim Hom O X ( G ( B ) , B ) ≤ l ( B ) . Analogously, condition (2.2 ∗ ) means that Hom O X ( G ( B ) , k ( x )) ≃ k .Using the exactness of φ , one proves by induction on the length ℓ ( F ) that the unitmap F → φ ( F ) is injective for any sheaf F supported on x . It follows easily (see theproof of [2, Thm. 3.8] for details) that the morphism G ( F ) → F is an epimorphism.In particular η : G ( B ) → B is surjective, and dim Hom O X ( G ( B ) , B ) ≥ ℓ ( B ). By( ∗ ), dim Hom O X ( G ( B ) , B ) = ℓ ( B ). Now the proof follows as in [2, Thm. 3.8]: Let j : Spec B ֒ → X be the inclusion. The exact sequence of B -modules0 → N → j ∗ G ( B ) j ∗ ( η ) −−−→ B → → Hom B ( B, B ) → Hom B ( j ∗ G ( B ) , B ) → Hom B ( N , B ) → B ( N , B ) = 0 because the two first terms have thesame dimension. Let us see that this implies N = 0. If k ( x ) → B is a nonzero, andthen injective, morphism, we have Hom B ( N , k ( x )) = 0 so that N = 0 by Nakayama’slemma. In conclusion, j ∗ G ( B ) ≃ B , and then Hom O X ( G ( B ) , k ( x )) ≃ k . (cid:3) References [1]
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