Algebraic geometry

Dualizing sheaf

In algebraic geometry, the dualizing sheaf on a proper scheme X of dimension n over a field k is a coherent sheaf together with a linear functional that induces a natural isomorphism of vector spaces for each coherent sheaf F on X (the superscript * refers to a dual vector space). The linear functional is called a trace morphism. A pair , if it is exists, is unique up to a natural isomorphism. In fact, in the language of category theory, is an object representing the contravariant functor from the category of coherent sheaves on X to the category of k-vector spaces. For a normal projective variety X, the dualizing sheaf exists and it is in fact the canonical sheaf: where is a canonical divisor. More generally, the dualizing sheaf exists for any projective scheme. There is the following variant of Serre's duality theorem: for a projective scheme X of pure dimension n and a F on X such that is of pure dimension n, there is a natural isomorphism . In particular, if X itself is a Cohen–Macaulay scheme, then the above duality holds for any locally free sheaf. (Wikipedia).

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Related pages

Reflexive sheaf | Smooth morphism | Normal bundle | Coherent duality | Coherent sheaf | Hodge bundle | Regular embedding | Gorenstein ring | Dualizing module | Category theory