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).[1] 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[2]
- .
In particular, if X itself is a Cohen–Macaulay scheme, then the above duality holds for any locally free sheaf.
Dualizing sheaf of projective schemes
As mentioned above, the dualizing sheaf exists for all projective schemes. For X a closed subscheme of Pn of codimension r, its dualizing sheaf can be given as . In other words, one uses the dualizing sheaf on the ambient Pn to construct the dualizing sheaf on X.[1]