Serre duality

The most general version of Serre duality states that on certain schemes X of dimensionMathworldPlanetmath n, including all projective varieties over any algebraically closed field k, there is a natural perfectPlanetmathPlanetmath pairing (


where 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 isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


In special cases, this reduces to more approachable forms. If X is nonsingularPlanetmathPlanetmath (or more generally, Cohen-Macaulay), then ω is simply nΩ, where Ω is the sheaf of differentials on X.

If is locally free, then


so that we obtain the somewhat more familiar looking fact that there is a perfect pairing Hi(X,*ω)×Hn-i(X,)k.

While Serre duality is not in a strict sense a generalizationPlanetmathPlanetmath 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