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 ${\displaystyle H^{i}}$ is the dual space of another one, ${\displaystyle H^{n-i}}$. 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 ${\displaystyle K_{X}}$ to be the bundle of n-forms on X, the top exterior power of the cotangent bundle:

${\displaystyle K_{X}=\Omega _{X}^{n}={\bigwedge }^{n}(T^{*}X).}$

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

${\displaystyle H^{i}(X,E)\cong H^{n-i}(X,K_{X}\otimes E^{\ast })^{\ast }}$

of finite-dimensional k-vector spaces. Here ${\displaystyle \otimes }$ denotes the tensor product of vector bundles. It follows that the dimensions of the two cohomology groups are equal:

${\displaystyle h^{i}(X,E)=h^{n-i}(X,K_{X}\otimes E^{\ast }).}$

Serre also proved the same duality statement for X a compact complex manifold and E a holomorphic vector bundle.[1]

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 ${\displaystyle H^{n}(X,K_{X})}$ is a perfect pairing:

${\displaystyle H^{i}(X,E)\times H^{n-i}(X,K_{X}\otimes E^{\ast })\to H^{n}(X,K_{X})\to k.}$

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.[2]

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 ${\displaystyle H^{0}(X,L)}$ and ${\displaystyle H^{1}(X,L)}$. Serre duality describes the ${\displaystyle H^{1}}$ group in terms of an ${\displaystyle H^{0}}$ group (for a different line bundle).[3] That is more concrete, since ${\displaystyle H^{0}}$ 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

${\displaystyle h^{0}(X,L)-h^{1}(X,L)=d-g+1.}$

Using Serre duality, this can be restated in more elementary terms:

${\displaystyle h^{0}(X,L)-h^{0}(X,K_{X}\otimes L^{*})=d-g+1.}$

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 ${\displaystyle 2g-2}$. Therefore, Riemann-Roch implies that for a line bundle L of degree ${\displaystyle d>2g-2}$, ${\displaystyle h^{0}(X,L)}$ is equal to ${\displaystyle d-g+1}$. When the genus g is at least 2, it follows by Serre duality that ${\displaystyle h^{1}(X,TX)=h^{0}(X,K_{X}^{\otimes 2})=3g-3}$. Here ${\displaystyle H^{1}(X,TX)}$ 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 ${\displaystyle 3g-3}$.

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 ${\displaystyle \omega _{X}}$ on X called the dualizing sheaf. (Some authors call this sheaf ${\displaystyle K_{X}}$.) 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

${\displaystyle \operatorname {Ext} _{X}^{i}(E,\omega _{X})\cong H^{n-i}(X,E)^{*}}$

of finite-dimensional k-vector spaces.[4] Here the Ext group is taken in the abelian category of ${\displaystyle O_{X}}$-modules. This includes the previous statement, since ${\displaystyle \operatorname {Ext} _{X}^{i}(E,\omega _{X})}$ is isomorphic to ${\displaystyle H^{i}(X,E^{*}\otimes \omega _{X})}$ 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, ${\displaystyle \omega _{X}}$ is the canonical line bundle ${\displaystyle K_{X}}$ 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:[5]

${\displaystyle \omega _{X}\cong {\mathcal {Ext}}_{O_{Y}}^{r}(O_{X},K_{Y}).}$

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[6]

${\displaystyle \omega _{X}\cong K_{Y}|_{X}\otimes {\bigwedge }^{r}(N_{X/Y}).}$

In this case, X is a Cohen–Macaulay scheme with ${\displaystyle \omega _{X}}$ a line bundle, which says that X is Gorenstein.

Example: Let X be a complete intersection in projective space ${\displaystyle {\mathbf {P} }^{n}}$ over a field k, defined by homogeneous polynomials ${\displaystyle f_{1},\ldots ,f_{r}}$ of degrees ${\displaystyle d_{1},\ldots ,d_{r}}$. (To say that this is a complete intersection means that X has dimension ${\displaystyle n-r}$.) There are line bundles O(d) on ${\displaystyle {\mathbf {P} }^{n}}$ 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

${\displaystyle \omega _{X}=O(d_{1}+\cdots +d_{r}-n-1)|_{X},}$

by the adjunction formula. For example, the dualizing sheaf of a plane curve X of degree d is ${\displaystyle O(d-3)|_{X}}$.

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 ${\displaystyle \omega _{X}^{\bullet }}$ of the bounded derived category of coherent sheaves on X, ${\displaystyle D_{\text{coh}}^{b}(X)}$, called the dualizing complex of X over k. Formally, ${\displaystyle \omega _{X}^{\bullet }}$ is the exceptional inverse image ${\displaystyle f^{!}O_{Y}}$, where f is the given morphism ${\displaystyle X\to Y=\operatorname {Spec} (k)}$. When X is Cohen–Macaulay of pure dimension n, ${\displaystyle \omega _{X}^{\bullet }}$ is ${\displaystyle \omega _{X}[n]}$; 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, ${\displaystyle \omega _{X}^{\bullet }}$ 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

${\displaystyle \operatorname {Hom} _{X}(E,\omega _{X}^{\bullet })\cong \operatorname {Hom} _{X}(O_{X},E)^{*}}$

for any object E in ${\displaystyle D_{\text{coh}}^{b}(X)}$.[7]

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

${\displaystyle \operatorname {Hom} _{X}(E,F\otimes \omega _{X}^{\bullet })\cong \operatorname {Hom} _{X}(F,E)^{*}.}$

Here the tensor product means the derived tensor product, as is natural in derived categories. (To compare with previous formulations, note that ${\displaystyle \operatorname {Ext} _{X}^{i}(E,\omega _{X})}$ can be viewed as ${\displaystyle \operatorname {Hom} _{X}(E,\omega _{X}[i])}$.) When X is also smooth over k, every object in ${\displaystyle D_{\text{coh}}^{b}(X)}$ is a perfect complex, and so this duality applies to all E and F in ${\displaystyle D_{\text{coh}}^{b}(X)}$. The statement above is then summarized by saying that ${\displaystyle F\mapsto F\otimes \omega _{X}^{\bullet }}$ is a Serre functor on ${\displaystyle D_{\text{coh}}^{b}(X)}$ for X smooth and proper over k.[8]

Serre duality holds more generally for proper algebraic spaces over a field.[9]

Notes

1. Serre (1955); Huybrechts (2005), Proposition 4.1.15.
2. Huybrechts (2005), exercise 3.2.3.
3. For a curve, Serre duality is simpler but still nontrivial. One proof is given in Tate (1968).
4. Hartshorne (1977), Theorem III.7.6.
5. Hartshorne (1977), proof of Proposition III.7.5; Stacks Project, Tag 0A9X.
6. Hartshorne (1977), Theorem III.7.11; Stacks Project, Tag 0BQZ.
7. Hartshorne (1966), Corollary VII.3.4(c); Stacks Project, Tag 0B6I; Stacks Project, Tag 0B6S.
8. Huybrechts (2006), Definition 1.28, Theorem 3.12.