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

Main article: Coherent 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.

References

• Hartshorne, Robin (1977), Algebraic geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052
• Hartshorne, Robin (1966), Residues and duality, Lecture Notes in Mathematics, 20, Berlin, New York: Springer-Verlag, ISBN 978-3-540-03603-6, MR 0222093
• Hazewinkel, Michiel, ed. (2001) [1994], "Duality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Huybrechts, Daniel (2005), Complex geometry, Berlin: Springer-Verlag, ISBN 3-540-21290-6, MR 2093043
• Huybrechts, Daniel (2006), Fourier–Mukai transforms in algebraic geometry, Oxford University Press, ISBN 978-0199296866, MR 2244106
• Serre, Jean-Pierre (1955), "Un théorème de dualité", Commentarii Mathematici Helvetici, 29: 9–26, doi:10.1007/BF02564268, MR 0067489
• Tate, John (1968), "Residues of differentials on curves" (PDF), Annales Scientifiques de l'École Normale Supérieure, Série 4, 1: 149–159, doi:10.24033/asens.1162, ISSN 0012-9593, MR 0227171

External links

• The Stacks Project Authors, The Stacks Project