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 $$ \Ext^{i}(\mathcal{F},\omega)\times H^{n-i}(X,\mathcal{F})\to k, $$ where $\mathcal{F}$ is any coherent sheaf on $X$ and $\omega$ is a fixed sheaf, called the dualizing sheaf. Here ``perfect'' means that the natural map above induces an isomorphism $$ \Ext^{i}(\mathcal{F},\omega)\cong \Hom{H^{n-i}(X,\mathcal{F})}{k}. $$

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 $$\Ext^i(\mathcal{F},\omega)\cong\Ext^i(\O_X,\mathcal{F}^*\otimes\omega)\cong H^i(X,\mathcal{F}^*\otimes\omega),$$ 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.