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Serre duality (Definition)

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

$\displaystyle \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 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 $ 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

$\displaystyle \mathrm{Ext}^i(\mathcal{F},\omega)\cong\mathrm{Ext}^i(\O _X,\mathcal{F}^*\otimes\omega)\cong H^i(X,\mathcal{F}^*\otimes\omega),$
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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See Also: duality in mathematics

Also defines:  dualizing sheaf
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Cross-references: duality, similar, Poincaré duality, strict, locally free, nonsingular, isomorphism, induces, sheaf, coherent sheaf, field, algebraically closed, projective varieties, dimension, schemes
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This is version 9 of Serre duality, born on 2003-08-15, modified 2007-01-14.
Object id is 4595, canonical name is SerreDuality.
Accessed 5030 times total.

Classification:
AMS MSC14F25 (Algebraic geometry :: homology theory :: Classical real and complex cohomology)

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