Serre duality
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 (http://planetmath.org/BilinearMap)
where is any coherent sheaf on and is a sheaf, called the dualizing sheaf. Here “perfect” means that the natural map above induces an isomorphism
In special cases, this reduces to more approachable forms. If is nonsingular (or more generally, Cohen-Macaulay), then is simply , where is the sheaf of differentials on .
If is locally free, then
so that we obtain the somewhat more familiar looking fact that there is a perfect pairing .
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 |
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Canonical name | SerreDuality |
Date of creation | 2013-03-22 13:51:24 |
Last modified on | 2013-03-22 13:51:24 |
Owner | mps (409) |
Last modified by | mps (409) |
Numerical id | 12 |
Author | mps (409) |
Entry type | Definition |
Classification | msc 14F25 |
Related topic | DualityInMathematics |
Defines | dualizing sheaf |