Serre Duality

Carl Lian June 17, 2016

Abstract Notes for a talk on Serre Duality following Vakil’s notes [V] with appeal to [H] and [M] toward the end. A feature of this proof seems to be that it generally pushes the hardest ingredients to the end. In particular, the existence of the dualizing sheaf is relatively cheap, and the work goes into identifying it with the canonical bundle on smooth varieties. Note that there are no assumptions on the ground ﬁeld.

1 Preliminaries on Ext and Ext

Let X be a ringed space and F an OX -module. Recall that we have left-exact functors Hom(F, −) and Hom(F, −); the former lands in Γ(OX )-modules and the latter in OX -modules. The category of OX -modules has enough injectives, so we get the right derived functors Exti (global Ext) and Exti (sheaf Ext) satisfying the usual properties. In particular, they can be computed via injective resolutions. The following facts are not hard; the key points are that the tensor product of an injective sheaf with a locally free sheaf is injective, and the restriction of an injective sheaf to an open subset is injective.

Proposition 1.1 If E is locally free, Exti(F ⊗ E∨, G) =∼ Exti(F, G ⊗ E) =∼ Exti(F, G) ⊗ E and Exti(F ⊗ E∨, G) =∼ Exti(F, G ⊗ E)

i i Proposition 1.2 If U ⊂ X is open, Ext (F, G)(U) = Ext (F|U , G|U ). Hence “sheaf Ext is local.”

(b’) The functors Ext and Ext are genuinely diﬀerent, aside from landing in i i i diﬀerent categories. For instance, Ext (OX , F) = H (X, F), but Ext (OX , F) = 0 0 unless i = 0, in which case Ext (OX , F) = F. However, the two functors are related in the following way.

Theorem 1.3 (Local-to-global spectral sequence) There is a spectral se- p,q j i i+j quence with E2 = H (X, Ext (F, G)) converging to Ext (F, G).

1 This follows (after checking the needed hypothesis) from the Grothendieck spectral sequence: Hom is the composition of Hom and the global sections functor Γ. In the category of abelian groups, Ext can also be computed with projective resolutions, but this is no longer true for sheaves: there are not enough projec- tives (for example, free sheaves may not be projective)! However, Ext can be computed with locally free resolutions: this follows from acyclicity of the trivial sheaf (and hence of locally free sheaves) and a spectral sequence. (Note how this fails for Ext.) A consequence of this is that if X is a locally Noetherian scheme and F, G are coherent, then Exti(F, G) is coherent (not true for quasicoherent sheaves, even for i = 0!).

2 Statements

Theorem 2.1 (Serre Duality) Let X be a Cohen-Macaulay (CM) projective k-scheme of pure dimension n. Then, there is a unique (up to unique isomorphism) coherent sheaf ωX such that: (a) For any vector bundle F, there is a natural perfect pairing Hi(X, F) × n−i ∨ H (X, F ⊗ ωX ) → k. i (b) For any coherent sheaf F, there is a natural perfect pairing Ext (F, ωX )× Hn−i(X, F) → k.

(b’) For any coherent sheaf F, there is a natural perfect pairing Hom(F, ωX )× Hn(X, F) → k. (The CM hypothesis is not needed for this statement.) ∼ Vn (c) If X is smooth, then ωX = KX := ΩX .

The uniqueness of ωX is a formal consequence of Yoneda’s Lemma. Note that (b’) is just (b) for i = 0, but we state it separately because it needs to be proven first. The CM hypothesis will mostly be black-boxed: it is used only to construct a finite flat morphism X → Pn, which is possible by the “Miracle Flatness” theorem. This hypothesis is necessary: when X is the union of two general 2-planes in P4, Serre Duality fails [V, Exercise 30.1.E]. In fact, statement (b) implies X must be Cohen-Macaulay, see [H, Theorem 7.6]. n A consequence of (b’) is that we get a trace map t : H (X, ωX ) → k. Indeed, taking F = ωX , the perfect pairing gives us a distinguished element n ∨ t ∈ H (X, ωX ) corresponding to id ∈ Hom(ωX , ωX ). Then given an element n of Hom(F, ωX ), we get an induced map on H , yielding the map Hom(F, ωX )× n n H (X, F) → H (X, ωX ); one checks that composing with t recovers the original pairing. In other words, we have a diagram

n n Hom(F, ωX ) × H (X, F) / H (X, ωX )

t ) k

2 In fact, the perfect pairings of (a) and (b) factor through t as well, but we will not prove this. The strategy of the proof is as follows: first, prove all statements for Pn by direct calculation. Then, prove statement (b) by considering a finite flat map n ! n π : X → P ; the dualizing sheaf ωX will be π ωP . Finally, prove statement (c) by considering a regular embedding i : X → PN ; we will show that the dualizing sheaf (canonically) satisfies the same adjunction formula as the canonical sheaf and hence these sheaves agree. Then, part (a) follows from part (b):

Proposition 2.2 (b) implies (a).

Proof We have (functorially)

i ∨ ∼ n−i H (X, F) = Ext (F, ωX ) ∼ n−i ∨ = Ext (OX , F ⊗ ωX ) ∼ n−i ∨ = H (X, F ⊗ ωX ).

3 Serre Duality holds for projective space

n We prove statement (b) with ωP = O(−n − 1); hence we get statement (c) as well. By Proposition 2.2, we get (a) for free.

n Lemma 3.1 Let X = P and ωX = O(−n − 1). Then (b’) holds.

Proof As discussed in the previous section, there is a canonical map Hom(F, ωX )× n n n ∼ H (X, F) → H (X, ωX ). Then, choose any isomorphism H (X, ωX ) = k. First, if F =∼ O(m) is a line bundle, our pairing is Hom(O(m), O(−n − 1)) × Hn(O(m)) → Hn(O(−n − 1)) =∼ k. But recall that Hom(O(m), O(−n − 1)) = H0(O(−m−n−1)) can be identiﬁed with the space of monomials of degree −m− n−1 in n+1 variables with non-negative exponents, and Hn(O(m)),Hn(O(−n− 1)) with the spaces of monomials of degrees m, −n − 1 in n + 1 variables with negative exponents. Then, one checks that the pairing map is multiplication, which is easily seen to be perfect. This proves (b’) for direct sums of line bundles E. Now, let F be arbitrary. Then there exists a short exact sequence 0 → G → E → F → 0 where E is a direct sum of line bundles. Hence we have a diagram

0 / Hom(F, ωX ) / Hom(E, ωX ) / Hom(G, ωX )

0 / Hn(X, F)∨ / Hn(X, E)∨ / Hn(X, G)∨ where the rows are exact (note that Hn+1(X, G) = 0 because X = Pn has dimension n). The middle vertical arrow is an isomorphism, from which we conclude the left vertical arrow is injective. But then this is true for all F, so the right vertical arrow is injective too. Diagram chasing gives the conclusion.

3 n Proposition 3.2 Let X = P and ωX = O(−n − 1). Then (b) holds. i n−i ∨ Proof We will show that the functors Ext (−, ωX ),H (X, −) are universal δ-functors. Because we have a isomorphism of functors for i = 0 (part (b’)), this is enough. We have left-exactness for i = 0 and we have long exact sequences, so we only need to check coeﬀaceability. Let F be arbitrary; then there is a surjec- tion O(−m)N → F for m >> 0 and some N. Indeed, Hn−i(Pn, O(−m)N ) = 0 i for all i > 0, and Ext (O(−m)N , ω) = Hi(Pn, O(m − n − 1))N = 0 for all i > 0.

4 The dualizing sheaf exists in general

Lemma 4.1 Let X be an equidimensional projective k-scheme. Then, there exists a finite morphism π : X → Pn. If X is CM then π is flat. Proof Embed X in PN , and find a codimension n + 1 linear space H missing X. Projecting away from H yields π, and if X is CM then Miracle Flatness [V, Theorem 26.2.11]. This proof only works when k is infinite, but can be modified for k finite [V, Exercise 30.3.D].

Now, let π : X → Y be a finite morphism and G a quasicoherent OY -module. ! We will define a quasicoherent OX -module π G as follows. First, if Y = Spec B is affine and X = Spec A, G = N, then let M = HomB(A, N), which is a priori a B-module, but in fact an A-module as well. This construction behaves well under localization at an element of B, so it will define a quasicoherent sheaf on ! X. Globally, we thus define the OX -module π G = HomOY (π∗OX , G). This is quasicoherent, and in fact coherent if G is coherent and π is finite. Recall (Frobenius reciprocity!) that we have the adjunction ∼ HomA(M, HomB(A, N)) = HomB(MB,N). ! ∼ This globalizes to a natural isomorphism π∗HomOX (F, π G) = HomOY (π∗F, G), ! ∼ and thus by taking global sections we get the adjunction HomOX (F, π G) =

HomOY (π∗F, G). We can now prove statements (b’) and (b) for X. n ! Lemma 4.2 Let π : X → Y = P be as above, and set ωX = π ωY . Then (b’) holds. Proof Stare at the following diagram; the bottom row is a perfect pairing by assumption, and the left vertical arrow is an isomorphism by the above discussion. ! n n ! HomX (F, π ωY ) × H (X, F) / H (X, π ωY )

n ! H (Y, π∗π ωY )

n n HomY (π∗F, ωY ) × H (Y, π∗F) / H (Y, ωY ) / k

4 Note that ﬂatness of π was not used, so (b’) holds without the CM hypothesis.

n ! Lemma 4.3 Let π : X → Y = P be as above, where X is CM. and ωX = π ωY . Then (b) holds.

i ! n−i ∨ Proof As before, we prove that ExtX (−, π ωY ) and H (X, −) are coefface- N able, hence universal δ-functors. Again, any sheaf F is a quotient of OX (−m) for some m >> 0 and some N. Here, we take the projective embedding of X,Y N ∗ in P in the proof of Lemma 4.1 (so π OY (1) = OX (1)!). i ! ∼ i ! Now, for i > 0, ExtX (OX (−m), π ωY ) = Ext (OX , π ωY (m)) = 0 by Serre vanishing. Also, n−i ∨ ∼ n−i ∗ ∨ H (X, OX (−m)) = H (Y, π∗(OX ⊗ π OY (−m))) ∼ n−i ∨ = H (Y, π∗OX ⊗ OY (−m)) ∼ i ∨ = H (Y, (π∗OX ) ⊗ OY (m) ⊗ ωY ) = 0, by statement (a) for Y = Pn and Serre vanishing. Note that we have used that π is finite flat to conclude π∗OX is a vector bundle.

5 The dualizing sheaf is the canonical bundle for smooth varieties

Let X be a smooth (and hence CM) projective variety of pure dimension n. Then, any embedding X → PN = Y is a regular embedding of codimension N − n = r. The ﬁrst step is to show:

∼ r Lemma 5.1 We have a natural isomorphism ωX = ExtY (π∗OX , ωY ), where the latter is interpreted as a coherent OX -module. There is a slight subtlety here. We saw in the previous section that for any r finite morphism X → Y and quasicoherent sheaf G on Y , HomY (π∗OX , G) is a priori a quasicoherent OY -module, but in fact has the structure of a quasicoher- r ent OX -module. On the other hand, if G is not quasicoherent, HomY (π∗OX , G) will not in general have the structure of a OX -module. In particular, when we r take an injective resolution of ωY and attempt to compute ExtY (π∗OX , ωY ), the resulting cohomology groups will not be ωX -modules, because the terms in our injective resolution may not be quasicoherent. On the other hand, one can show that if X → Y is a closed embedding, it r is again true that HomY (π∗OX , G) is an OX -module, even if G is not quasicoherent. The reason is topological: the open sets of X all come from restrictions of open sets of Y , so we have enough data to define a OX -action. Thus, we ! ! can define the OX -module π G as before, and recover the adjunction (π∗, π ). r Furthermore, we can now make sense of ExtY (π∗OX , ωY ) as a OX -module. It is coherent as an OY -module, and thus coherent as a OX -module. Now, Lemma 5.1 follows from:

5 r ∼ Lemma 5.2 We have a natural isomorphism HomX (F, ExtY (π∗OX , ωY )) = r ExtY (π∗F, ωY ).

Indeed, assuming Lemma 5.2 we get natural isomorphisms (applying statements (b) and (b’))

r ∼ r HomX (F, ExtY (π∗OX , ωY )) = ExtY (π∗F, ωY ) ∼ n ∨ = H (Y, π∗F) =∼ Hn(X, F)∨ ∼ = HomX (F, ωX ), so we conclude Lemma 5.1 by Yoneda’s Lemma. To prove Lemma 5.2, we ﬁrst need:

i Lemma 5.3 For i < r, ExtY (π∗OX , ωY ) = 0.

i Proof Because ExtY (π∗OX , ωY ) is coherent on Y , it suﬃces to show that 0 = 0 i 0 i H (Y, ExtY (π∗OX , ωY )(m)) = H (Y, ExtY (π∗OX , ωY (m))) for m >> 0. But j i j i 0 = H (Y, ExtY (π∗OX , ωY )(m)) = H (Y, ExtY (π∗OX , ωY (m))) for j > 0 by Serre vanishing. Thus, the local-to-global spectral sequence degenerates on the E2-page if m is suﬃciently large, hence

0 i ∼ 0 i H (Y, ExtY (π∗OX , ωY )(m)) = H (Y, ExtY (π∗OX , ωY (m))) ∼ i = ExtY (π∗OX , ωY (m)) ∼ i = ExtY (π∗OX (−m), ωY ) ∼ N−i ∨ = H (Y, π∗OX (−m)) , by statement (b). If i < r, then N − i > n = dim Supp π∗OX (−m), so N−i H (Y, π∗OX (−m)) = 0.

Proof of Lemma 5.2 Run the Grothendieck spectral sequence for the functors HomY (π∗OX , −) and HomX (F, −): it applies since if G is injective, then ! HomY (π∗OX , G) = π G is acyclic for HomX (F, −). Now, by Lemma 5.3, we i j have ExtX (F, ExtY (π∗OX , ωY )) = 0 for all j < r, so at the desired term, the Grothendieck spectral sequence degenerates on the E2 page, yielding the conclusion.

The ﬁnal step is:

Lemma 5.4 There is a natural isomorphism

r ∼ ExtY (π∗OX , ωY ) = det NX/Y ⊗OX ωY |X .

This will prove part (c) of Serre Duality, since canonical sheaves satisfy the same adjunction formula [V, Exercise 21.5.B]. Note that up to now we have not used smoothness of X.

6 Sketch of proof of Lemma 5.4 See also [H, Theorem 7.11]. Recall that we can compute Ext locally. We give a locally free resolution of π∗OX : namely, locally on Y , X is cut out by a regular sequence x1, . . . , xr of length r. Then, take the Koszul complex and sheaﬁfy. Then, taking Hom(−, ωY ) and looking locally, we ﬁnd that the cohomology is ωY /(x1, . . . , xr)ωY = ωY |X in dimension r (and zero everywhere else), by [M, Theorem 43]. On the other hand, we pick up transition functions according to the ambiguity in the choice of x1, . . . , xr, exactly corresponding to those of det NX/Y , which we understand locally as 2 ∨ ((x1, . . . , xr)/(x1, . . . , xr) ) .

6 Acknowledgments

I thank Remy van Dobben de Bruyn and Qixiao Ma for enlightening con- versations, and Qixiao in particular for pointing out the easy proof of Lemma 5.2. I also thank everyone who attended the original talk for inspiration.

References

[H] R. Hartshorne, Algebraic Geometry. Grad. Texts in Math. 52, Springer- Verlag. New York-Heidelberg, 1977. [M] H. Matsumura, Commutative Algebra, 2nd ed. Math. Lecture Note Series 56, Benjamin/Cummings Publ. Co. Inc. Reading, MA 1980. [V] R. Vakil, The Rising Sea: Foundations of Algebraic Geometry, preprint 2015.