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 , 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. 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 (by definition) 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 . For a smooth curve C, its dualizing sheaf can be given by the canonical sheaf . For a nodal curve C with a node p, we may consider the normalization with two points x, y identified. Let be the sheaf of rational 1-forms on with possible simple poles at x and y, and let be the subsheaf consisting of rational 1-forms with the sum of residues at x and y equal to zero.