# Serre duality

The most general version of Serre duality states that on certain schemes $X$ of dimension $n$, including all projective varieties over any algebraically closed field $k$, there is a natural perfect pairing (http://planetmath.org/BilinearMap)

 $\mathrm{Ext}^{i}(\mathcal{F},\omega)\times H^{n-i}(X,\mathcal{F})\to k,$

where $\mathcal{F}$ is any coherent sheaf on $X$ and $\omega$ is a sheaf, called the dualizing sheaf. Here “perfect” means that the natural map above induces an isomorphism

 $\mathrm{Ext}^{i}(\mathcal{F},\omega)\cong\mathrm{Hom}(H^{n-i}(X,\mathcal{F}),k).$

In special cases, this reduces to more approachable forms. If $X$ is nonsingular (or more generally, Cohen-Macaulay), then $\omega$ is simply $\bigwedge^{n}\Omega$, where $\Omega$ is the sheaf of differentials on $X$.

If $\mathcal{F}$ is locally free, then

 $\mathrm{Ext}^{i}(\mathcal{F},\omega)\cong\mathrm{Ext}^{i}(\mathcal{O}_{X},% \mathcal{F}^{*}\otimes\omega)\cong H^{i}(X,\mathcal{F}^{*}\otimes\omega),$

so that we obtain the somewhat more familiar looking fact that there is a perfect pairing $H^{i}(X,\mathcal{F}^{*}\otimes\omega)\times H^{n-i}(X,\mathcal{F})\to k$.

While Serre duality is not in a strict sense a generalization of Poincaré duality, they are philosophically similar, and both fit into a larger pattern on duality results.

Title Serre duality SerreDuality 2013-03-22 13:51:24 2013-03-22 13:51:24 mps (409) mps (409) 12 mps (409) Definition msc 14F25 DualityInMathematics dualizing sheaf