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)

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

where $ℱ$ 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

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

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

If $ℱ$ is locally free, then

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

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

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.