PlanetMath: Serre duality

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Serre duality

(Definition)

The most general version of Serre duality states that on certain schemes of dimension , including all projective varieties over any algebraically closed field , there is a natural perfect pairing

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

where $\mathcal{F}$ is any coherent sheaf on

and $\omega$ is a fixed sheaf, called the dualizing sheaf. Here “perfect” means that the natural map above induces an isomorphism

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

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

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

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

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.

"Serre duality" is owned by mps. [ full author list (3) | owner history (5) ]

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