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 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 Cohen–Macaulay sheaf 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.Relative dualizing sheaf
Given a proper finitely presented morphism of schemes, defines the relative dualizing sheaf or as the sheaf such that for each open subset and a quasi-coherent sheaf on, there is a canonical isomorphism
which is functorial in and commutes with open restrictions.
Example:
If is a local complete intersection morphism between schemes of finite type over a field, then each point of has an open neighborhood and a factorization, a regular embedding of codimension followed by a smooth morphism of relative dimension. Then
where is the sheaf of relative Kähler differentials and is the normal bundle to.
See also: Hodge bundle.
