# Serre duality

In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties. On an n-dimensional variety, the theorem says that a cohomology group $H^{i}$ is the dual space of another one, $H^{n-i}$ . Serre duality is the analog for coherent sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientation sheaf.

## Serre duality for vector bundles

Let X be a smooth variety of dimension n over a field k. Define the canonical line bundle $K_{X}$ to be the bundle of n-forms on X, the top exterior power of the cotangent bundle:

$K_{X}=\Omega _{X}^{n}={\bigwedge }^{n}(T^{*}X).$ Suppose in addition that X is proper (for example, projective) over k. Then Serre duality says: for an algebraic vector bundle E on X and an integer i, there is a natural isomorphism

$H^{i}(X,E)\cong H^{n-i}(X,K_{X}\otimes E^{\ast })^{\ast }$ of finite-dimensional k-vector spaces. Here $\otimes$ denotes the tensor product of vector bundles. It follows that the dimensions of the two cohomology groups are equal:

$h^{i}(X,E)=h^{n-i}(X,K_{X}\otimes E^{\ast }).$ Serre also proved the same duality statement for X a compact complex manifold and E a holomorphic vector bundle.

As in Poincaré duality, the isomorphism in Serre duality comes from the cup product in sheaf cohomology. Namely, the composition of the cup product with a natural trace map on $H^{n}(X,K_{X})$ is a perfect pairing:

$H^{i}(X,E)\times H^{n-i}(X,K_{X}\otimes E^{\ast })\to H^{n}(X,K_{X})\to k.$ The trace map is the analog for coherent sheaf cohomology of integration in de Rham cohomology. There is also a direct connection between Serre duality and Poincaré duality, via Hodge theory.

## Algebraic curves

A fundamental application of Serre duality is to algebraic curves. (Over the complex numbers, it is equivalent to consider compact Riemann surfaces.) For a line bundle L on a smooth projective curve X over a field k, the only possibly nonzero cohomology groups are $H^{0}(X,L)$ and $H^{1}(X,L)$ . Serre duality describes the $H^{1}$ group in terms of an $H^{0}$ group (for a different line bundle). That is more concrete, since $H^{0}$ of a line bundle is simply its space of sections.

Serre duality is especially relevant to the Riemann-Roch theorem for curves. For a line bundle L of degree d on a curve X of genus g, the Riemann-Roch theorem says that

$h^{0}(X,L)-h^{1}(X,L)=d-g+1.$ Using Serre duality, this can be restated in more elementary terms:

$h^{0}(X,L)-h^{0}(X,K_{X}\otimes L^{*})=d-g+1.$ The latter statement (expressed in terms of divisors) is in fact the original version of the theorem from the 19th century. This is the main tool used to analyze how a given curve can be embedded into projective space and hence to classify algebraic curves.

Example: Every global section of a line bundle of negative degree is zero. Moreover, the degree of the canonical bundle is $2g-2$ . Therefore, Riemann-Roch implies that for a line bundle L of degree $d>2g-2$ , $h^{0}(X,L)$ is equal to $d-g+1$ . When the genus g is at least 2, it follows by Serre duality that $h^{1}(X,TX)=h^{0}(X,K_{X}^{\otimes 2})=3g-3$ . Here $H^{1}(X,TX)$ is the first-order deformation space of X. This is the basic calculation needed to show that the moduli space of curves of genus g has dimension $3g-3$ .

## Serre duality for coherent sheaves

Another formulation of Serre duality holds for all coherent sheaves, not just vector bundles. As a first step in generalizing Serre duality, Grothendieck showed that this version works for schemes with mild singularities, Cohen–Macaulay schemes, not just smooth schemes.

Namely, for a Cohen-Macaulay scheme X of pure dimension n over a field k, Grothendieck defined a coherent sheaf $\omega _{X}$ on X called the dualizing sheaf. (Some authors call this sheaf $K_{X}$ .) Suppose in addition that X is proper over k. For a coherent sheaf E on X and an integer i, Serre duality says that there is a natural isomorphism

$\operatorname {Ext} _{X}^{i}(E,\omega _{X})\cong H^{n-i}(X,E)^{*}$ of finite-dimensional k-vector spaces. Here the Ext group is taken in the abelian category of $O_{X}$ -modules. This includes the previous statement, since $\operatorname {Ext} _{X}^{i}(E,\omega _{X})$ is isomorphic to $H^{i}(X,E^{*}\otimes \omega _{X})$ when E is a vector bundle.

In order to use this result, one has to determine the dualizing sheaf explicitly, at least in special cases. When X is smooth over k, $\omega _{X}$ is the canonical line bundle $K_{X}$ defined above. More generally, if X is a Cohen–Macaulay subscheme of codimension r in a smooth scheme Y over k, then the dualizing sheaf can be described as an Ext sheaf:

$\omega _{X}\cong {\mathcal {Ext}}_{O_{Y}}^{r}(O_{X},K_{Y}).$ When X is a local complete intersection of codimension r in a smooth scheme Y, there is a more elementary description: the normal bundle of X in Y is a vector bundle of rank r, and the dualizing sheaf of X is given by

$\omega _{X}\cong K_{Y}|_{X}\otimes {\bigwedge }^{r}(N_{X/Y}).$ In this case, X is a Cohen–Macaulay scheme with $\omega _{X}$ a line bundle, which says that X is Gorenstein.

Example: Let X be a complete intersection in projective space ${\mathbf {P} }^{n}$ over a field k, defined by homogeneous polynomials $f_{1},\ldots ,f_{r}$ of degrees $d_{1},\ldots ,d_{r}$ . (To say that this is a complete intersection means that X has dimension $n-r$ .) There are line bundles O(d) on ${\mathbf {P} }^{n}$ for integers d, with the property that homogeneous polynomials of degree d can be viewed as sections of O(d). Then the dualizing sheaf of X is the line bundle

$\omega _{X}=O(d_{1}+\cdots +d_{r}-n-1)|_{X},$ by the adjunction formula. For example, the dualizing sheaf of a plane curve X of degree d is $O(d-3)|_{X}$ .

## Grothendieck duality

Grothendieck's theory of coherent duality is a broad generalization of Serre duality, using the language of derived categories. For any scheme X of finite type over a field k, there is an object $\omega _{X}^{\bullet }$ of the bounded derived category of coherent sheaves on X, $D_{\text{coh}}^{b}(X)$ , called the dualizing complex of X over k. Formally, $\omega _{X}^{\bullet }$ is the exceptional inverse image $f^{!}O_{Y}$ , where f is the given morphism $X\to Y=\operatorname {Spec} (k)$ . When X is Cohen–Macaulay of pure dimension n, $\omega _{X}^{\bullet }$ is $\omega _{X}[n]$ ; that is, it is the dualizing sheaf discussed above, viewed as a complex in (cohomological) degree −n. In particular, when X is smooth over k, $\omega _{X}^{\bullet }$ is the canonical line bundle placed in degree −n.

Using the dualizing complex, Serre duality generalizes to any proper scheme X over k. Namely, there is a natural isomorphism of finite-dimensional k-vector spaces

$\operatorname {Hom} _{X}(E,\omega _{X}^{\bullet })\cong \operatorname {Hom} _{X}(O_{X},E)^{*}$ for any object E in $D_{\text{coh}}^{b}(X)$ .

More generally, for a proper scheme X over k, an object E in $D_{\text{coh}}^{b}(X)$ , and F a perfect complex in $D_{\text{perf}}(X)$ , one has the elegant statement:

$\operatorname {Hom} _{X}(E,F\otimes \omega _{X}^{\bullet })\cong \operatorname {Hom} _{X}(F,E)^{*}.$ Here the tensor product means the derived tensor product, as is natural in derived categories. (To compare with previous formulations, note that $\operatorname {Ext} _{X}^{i}(E,\omega _{X})$ can be viewed as $\operatorname {Hom} _{X}(E,\omega _{X}[i])$ .) When X is also smooth over k, every object in $D_{\text{coh}}^{b}(X)$ is a perfect complex, and so this duality applies to all E and F in $D_{\text{coh}}^{b}(X)$ . The statement above is then summarized by saying that $F\mapsto F\otimes \omega _{X}^{\bullet }$ is a Serre functor on $D_{\text{coh}}^{b}(X)$ for X smooth and proper over k.

Serre duality holds more generally for proper algebraic spaces over a field.