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)
Exti(ℱ,ω)×Hn-i(X,ℱ)→k, |
where ℱ is any coherent sheaf on X and ω is a sheaf, called the dualizing sheaf. Here “perfect” means that the natural map above induces an isomorphism
Exti(ℱ,ω)≅Hom(Hn-i(X,ℱ),k). |
In special cases, this reduces to more approachable forms. If X is nonsingular (or more generally, Cohen-Macaulay), then ω is simply ⋀nΩ, where Ω is the sheaf of differentials on X.
If ℱ is locally free, then
Exti(ℱ,ω)≅Exti(𝒪X,ℱ*⊗ω)≅Hi(X,ℱ*⊗ω), |
so that we obtain the somewhat more familiar looking fact that there is a perfect pairing Hi(X,ℱ*⊗ω)×Hn-i(X,ℱ)→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.
Title | Serre duality |
---|---|
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 |