6 Serre duality
Recall the famous Poincar´eduality, which is a basic result on the structure of singular homology and cohomology groups of manifolds: Let X be an n-dimensional orientable closed manifold. Then for any integer i i,wehaveH (X, R) Hn i(X, R). ' n i Thanks to the universal coe cient theorem, the right-hand-side is isomorphic to H (X, R)_, the dual of the (n i)-th cohomology group. Assume that X is a smooth manifold. Then one may interpret this duality as a perfect pairing as follows. Applying de Rham’s the- i i orem, one may identify H (X, R)=HdR(X, R), and hence, the composition of the cup product map and the integration map Hi(X, R) n i n ⇥ H (X, R) H (X, R) R defined by (⌘,⇠) (⌘ ⇠) gives a perfect ! ! 7! X ^ pairing. R Since the sheaf cohomology generalizes the singular cohomology, we may expect there is an analogous “duality” theorem for varieties/schemes – at least, under mild assumptions on the underlying space. Serre duality is a special case of the duality called the coherent duality in a much general setting. It is based on earlier works in several complex variables, however, we will observe it algebraically. One di↵erence between the Poincar´eduality and the Serre duality is the role of the “dualizing sheaf”. We will see that the dualizing sheaf coincides with the canonical sheaf when the underlying space X is a nonsingular variety over an algebraically closed field. n First, we begin with the duality on the projective space. Let k be a field, let X = Pk be the projective n-space, and let ! = n⌦ = ( n 1) be the canonical sheaf. X ^ X/k OX We have the duality theorem for X as follows.
n Theorem 185 (Duality for Pk ). (1) Hn(X,! ) k. X ' (2) Fix an isomorphism Hn(X,! ) k. For any coherent sheaf on X,thenatural X ' F pairing Hom( ,! ) Hn(X, ) Hn(X,! ) k F X ⇥ F ! X ' is a perfect pairing of finite-dimensional vector spaces over k.
(3) For every i 0, there is a natural functorial isomorphism i n i Ext ( ,! ) ⇠ H (X, )_. F X ! F Proof. (1) is clear from the computations we already made. Note that the pairing in (2) is natural, since any morphism !X induces a natural map on cohomol- i i F! ogy groups H (X, ) H (X,!X ) for each i.If (q) for some q Z,then F ! F'O 2 Hom( ,! ) H0(X,! ( q)), so this pairing comes from the product map between F X ' X
73 Topic 3 – Derived functors and cohomology
Laurent polynomials. In particular, the natural pairing is a perfect pairing if is a finite F direct sum of sheaves of the form (q ). Since any coherent sheaf can be presented O i F as a cokernel 0 of a map of sheaves , where each is a E1 !E0 !F! E1 !E0 Ej finite direct sum of sheaves (q ). Now both Hom( ,! ) and Hn(X, ) are left-exact O i X _ contravariant functors, so we have the following commutative diagram
0 / 0 / Hom( ,! ) / Hom( ,! ) / Hom( ,! ) F X E0 X E1 X
' ' ✏ ✏ ✏ 0 / 0 / Hn(X, ) / Hn(X, ) / Hn(X, ) . F _ E0 _ E1 _ The statement just follows from the 5-lemma. For (3), one may check that both sides are contravariant -functors for Coh(X), F2 indexed by i 0. For i = 0 we have an isomorphism. Since any quasi-coherent sheaf F can be written as a quotient of a sheaf = ( q)withq 0, we have E O Exti( ,! )= LHi(X,! (q)) = 0 E X X for i>0 by Serre vanishing. Similarly, weM have
n i n i H (X, )_ = H (X, ( q)) = 0 E O for i>0 and q>0. Hence, both sides areMcoe↵acable (= any object can be written as F a quotient of an object so that the image of under the given functor is 0) for i>0. E E By Grothendieck, such -functors are universal, hence isomorphic.
In particular, when is locally free of finite rank on a projective n-space X,wehavea F natural isomorphism i i n i Ext ( ,! ) H (X, _ ! ) ⇠ H (X, )_. F X ' F ⌦ X ! F n Remark 186. If we choose the coordinates x0, ,xn of X = P ,wemaypickupa ··· k Cechˇ cocycle xn x x ↵ = 0 d 1 d n x x x ^···^ x 1 ··· n ✓ 0 ◆ ✓ 0 ◆ n which lies in C (U,!X ), where U = D+(xi) . One can check that ↵ gives a generator n { } of H (X,!X ), and is stable under the linear change of coordinates. To generalize this, we take (1) and (2) as our guide. The key point was indeed: to fix an isomorphism t : Hn(X,! ) ⇠ k. X ! Definition 187. Let X be a projective scheme of dimension n over a field k.A dualizing sheaf for X is a coherent sheaf !X on X, together with a trace morphism t : Hn(X,! ) k, such that for all coherent sheaves on X, the natural pairing X ! F n n t Hom( ,! ) H (X, ) H (X,! ) k F X ⇥ F ! X ! gives an isomorphism Hom( ,! ) ⇠ Hn(X, ) . F X ! F _
74 Proposition 188. Let X be a projective scheme over k. Then the pair (!X ,t) of a dualizing sheaf and a trace morphism, if it exists, is unique up to a unique isomor- phism. More precisely, if we have two pairs (! ,t) and (!0,t0), then there is a unique isomorphism ' : ! ⇠ ! such that t = t Hn('). ! 0 0 Proof. Since ! is a dualizing sheaf, there is an isomorphism Hom(! ,!) Hn(X,! ) . 0 0 ' _ In particular, there is a unique morphism ' : ! ! corresponding to the element ! 0 t Hn(X,! ) . Note that the correspondence is given by the natural induced map on 2 _ cohomology, that is, t Hn(')=t. 0 Similarly, using the fact that ! is a dualizing sheaf, there is an isomorphism : ! ! 0 ! such that t Hn( )=t . It follows that t Hn( ) Hn(')=t Hn( ')=t. 0 Since ! is a dualizing sheaf, the morphism ' : ! ! corresponds to the identity ! map; since there is a unique morphism ! ! which preserves t. Similarly, we conclude ! that ' is the identity map on ! . 0 In particular, the pair (! ,t) represents the functor Hn(X, ) from Coh(X)to F7! F _ Mod(k).
The existence of a dualizing sheaf is much more tricky. A standard way of approaches is passing through Grothendieck’s six operations. Suppose we have a morphism f : X Y ! of schemes. Then we have an adjoint property
HomX (f ⇤ , ) = HomY ( ,f ) G F G ⇤F for any Mod(X) and Mod(Y ). In other words, the pair (f ⇤,f ) is an adjoint F2 G2 ⇤ pair; the pullback f ⇤ is a left adjoint of f , and the pushforward f is a right adjoint of ⇤ ⇤ f ⇤. Similarly, we have a bijection of sets
Hom( om( , ), ) = Hom( , ) H E F G F E⌦G for any -modules , and a locally free sheaf of finite rank on X. Of course, this OX F G E is “weak”, since it makes sense only if is too “good” (we want to extend it at least E for coherent sheaves). But anyway, the above bijection claims that the pair of functors ( om( , ), ) plays a role of an adjoint pair. This pair has a nicer property; we H E E⌦ have Hom( , ) = Hom( , om( , )), E⌦F G F H E G under the same assumption. In particular, ( , om( , )) is also an adjoint pair, E⌦ H E whereas f is NOT a left adjoint of f ⇤. ⇤ Nevertheless, it is worthwhile to consider a “right adjoint” of f . Let us have a look ⇤ what happens locally. Let B A be a ring homomorphism, M be an A-module, and ! let N be a B-module. Note that the pushforward f (M) of a quasi-coherent sheaf M is ⇤ (BM)⇠,whereBM means M considered as a B-module. We have a map f f Hom (M,Hom (A, N)) Hom ( M,N) A B ! B B
75 Topic 3 – Derived functors and cohomology defined by: given m M and an element Hom (M,Hom (A, N)), we send m 2 2 A B 7! (m)(1 ). One can show that this is a bijection. The functor Hom (A, ) is a “right A A B adjoint” of ( ), which corresponds to the pushforward functor for ring spectra. One B can also show that this bijection behaves nicely with respect to the localization at an element in B. In particular, this correspondence can be naturally “sheafify”, at least for a morphism which looks like a morphism of ring spectra and modules over them – namely, an a ne morphism and quasi-coherent sheaves. Suppose that f : X Y be an a ne morphism between two schemes. The above discus- ! 1 sion claims that the functor A HomB(A, ) globalizes to a functor f omY (f X , ), G7! H ⇤O G Qco(Y ). The image of is also quasi-coherent, this gives a functor f ! : Qco(Y ) G2 G ps ! Qco(X). Moreover, we have a local isomorphism
! f omX ( ,fps ) ⇠ omY (f , ) ⇤ H F G !H ⇤F G and a global isomorphism
! HomX ( ,fps ) ⇠ HomY (f , ) F G ! ⇤F G for any quasi-coherent sheaves Qco(X) and Qco(Y ). We may regard f ! as a F2 G2 ps right adjoint of f in this manner. ⇤ Caution. This is not a standard way of description; to provide a natural description for the exceptional inverse image functor f !, we need to consider the derived categories. However, both notions coincide when f is finite and flat.
We will briefly have a look how this construction helps us to find a dualizing sheaf. Let X PN be a locally Cohen-Macaulay (= all the local rings are Cohen-Macaulay) ✓ k equidimensional projective scheme of dimension n. By taking a linear projection at a linear subspace ⇤of codimension (n + 1) with X ⇤= , we have a finite and flat \ ; morphism ⇡ : X Pn. We have the duality theorem for such an X and a locally free ! sheaf on X of finite rank: F Proposition 189. There is a dualizing sheaf ! which gives an isomorphism Exti( ,! ) X F X ' Hn i(X, ) for every i 0, a locally free sheaf of finite rank. F _ F ! Proof. Since ⇡ is finite, it is a ne. We take !X := ⇡ps!Pn . We have isomorphisms
i i ExtX ( ,!X ) H (X, _ !X ) F ' i n F ⌦ H (P ,⇡ ( _ !X )) (an analogue of Leray spectral sequence) ' i n ⇤ F ⌦ H (P ,⇡ ( omX ( ,!X ))) ' i n ⇤ H F H (P , omPn (⇡ ,!Pn )) (the local isomorphism above) ' i n H ⇤F H (P , (⇡ )_ !Pn )((⇡ ) is locally free) ' i ⇤F ⌦ ⇤F n ExtPn (⇡ ,!P ) ' n i ⇤F n H (X, )_ (Serre duality for P ) ' F
76 Now we go back to the arguments in Hartshorne’s book. At the moment, it is still quite mysterious that how the dualizing sheaf on X looks like unless X is a projective space. We begin with some preliminaries.
N i Lemma 190. Let X be a closed subscheme of codimension r of P . Then xt N ( X ,! N )= k E P O P 0 for all i i i N Proof. Notice that := xt N ( X ,! N ) is a coherent sheaf on P for each i.BySerre F E P O P vanishing, it will be globally generated after twisting by a su ciently large number q. Hence, to conclude that i = 0, it is su cient to show that i(q) does not have a F F nonzero global section for all q 0. Since 0 i i i H (X, (q)) = Ext ( ,! N (q)) = Ext ( ( q),! N ) F PN OX P PN OX P N i N N for q 0, it is isomorphic to H (P , X ( q))_ by Serre duality for P . For i Proof. Let 0 !PN • be an injective resolution of !PN . Note that the group i ! !I i Ext ( ,! N )isthei-th cohomology group h (Hom N ( , )). Since is an - PN P P • X F i i F I i F O module, any morphism factors through := omPN ( X , ). For any F!Ii i J H O I F2 Mod(X), we have Hom ( , ) = Hom N ( , ). We have X F J P F I i i Ext N ( ,! N )=h (Hom ( , •)). P F P X F J i i Also note that HomX ( , ) is exact, and hence, is an injective X -module. Consider 0 J1 J O the complex 0 (this is not an injective resolution of an X -module). !J !J !··· i i O Thanks to the above lemma, we have h ( )= xt ( ,! N ) = 0 for i 77 Topic 3 – Derived functors and cohomology Proposition 192. Let X be a projective scheme over a field k. Then X has a dualizing sheaf !X . N Proof. Embed X as a closed subscheme of Pk for some N,letr be its codimension, and r let ! := xt ( ,! N ) as above. For any -module , we have an isomorphism X E PN OX P OX F r Hom ( ,! ) Ext N ( ,! N ). X F X ' P F P When is coherent, the duality theorem for PN gives an isomorphism F r N r N Ext N ( ,! N ) H (P , )_. P F P ' F Since N r = n =dimX, and is a sheaf on X, we have a functorial isomorphism F n Hom ( ,! ) H (X, )_ X F X ' F for any coherent sheaf Coh(X). In particular, if we take = ! ,theidentity F2 F X element id Hom (! ,! ) gives a trace homomorphism t : Hn(X,! ) k.Byits 2 X X X X ! functoriality, the pair (!X ,t) is a dualizing sheaf for X. This leads to the following duality theorem for a projective scheme. Theorem 193 (Duality for a projective scheme). Let X be a projective scheme of di- mension n over an algebraically closed field k, let !X be a dualizing sheaf on X, and let (1) be a very ample invertible sheaf on X. Then: OX (1) for each i 0 and coherent sheaf on X, there is a natural functorial map F i i n i ✓ :Ext ( ,! ) H (X, )_ X F X ! F such that ✓0 is the map given in the definition of the dualizing sheaf above; (2) the following are equivalent: (i) X is locally Cohen-Macaulay and equidimensional; (ii) for any locally free sheaf on X, we have Hi(X, ( q)) = 0 for i 78 (i) (ii) Embed X as a closed subscheme of PN . For any locally free sheaf on X, ) k F and any closed point x X,wehavedepth =dim = n,sinceX is locally 2 Fx OX,x Cohen-Macaulay and equidimensional of dimension n. Let A = N be the local OP ,x ring of PN at x.Sincek is an algebraically closed field, it is a regular local ring of dimension N. The depth of computed over is same as the one over A. Thanks Fx OX,x to Auslander-Buchsbaum formula pd ( )+depth =depthA = N, A Fx Fx the projective dimension of (= the least length of a projective A-resolution of the mod- Fx ule )isN n. Since the stalk xti ( , ) coincides with the module Exti ( , ) Fx E PN F G x A Fx Gx for any coherent sheaf , the stalk becomes zero for i>N n and for any closed point F x X. We conclude that 2 xti ( , )=0 E PN F for i>N n. On the other hand, by Serre duality for PN , the cohomology group i N i H (X, ( q)) is isomorphic to Ext N ( ,! N (q))_. For a su ciently large q 0, this P P F N N i F Ext is isomorphic to (P , xt N ( ,! N (q))), which is 0 when N i>N n. In other E P F P words, Hi(X, ( q)) = 0 for i 79