Dualizing sheaf

In algebraic geometry, the dualizing sheaf on a proper scheme "X" of dimension "n" over a field "k" is a coherent sheaf formula_1 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 formula_4 is called a trace morphism.
A pair formula_5, if it is exists, is unique up to a natural isomorphism. In fact, in the language of category theory, formula_1 is an object representing the contravariant functor formula_7 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: formula_8 where formula_9 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 formula_10 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 formula_12, defines the relative dualizing sheaf formula_13 or formula_14 as the sheaf such that for each open subset formula_15 and a quasi-coherent sheaf formula_16 on formula_17, there is a canonical isomorphism 
which is functorial in formula_16 and commutes with open restrictions.
Example:
If formula_20 is a local complete intersection morphism between schemes of finite type over a field, then (by definition) each point of formula_21 has an open neighborhood formula_17 and a factorization formula_23, a regular embedding of codimension formula_24 followed by a smooth morphism of relative dimension formula_25. Then
where formula_27 is the sheaf of relative Kähler differentials and formula_28 is the normal bundle to formula_29.
Examples.
Dualizing sheaf of a nodal curve.
For a smooth curve "C", its dualizing sheaf formula_30 can be given by the canonical sheaf formula_31.
For a nodal curve "C" with a node "p", we may consider the normalization formula_32 with two points "x", "y" identified. Let formula_33 be the sheaf of rational 1-forms on formula_34 with possible simple poles at "x" and "y", and let formula_35 be the subsheaf consisting of rational 1-forms with the sum of residues at "x" and "y" equal to zero. Then the direct image formula_36 defines a dualizing sheaf for the nodal curve "C". The construction can be easily generalized to nodal curves with multiple nodes.
This is used in the construction of the Hodge bundle on the compactified moduli space of curves: it allows us to extend the relative canonical sheaf over the boundary which parametrizes nodal curves. The Hodge bundle is then defined as the direct image of a relative dualizing sheaf.
Dualizing sheaf of projective schemes.
As mentioned above, the dualizing sheaf exists for all projective schemes. For "X" a closed subscheme of P"n" of codimension "r", its dualizing sheaf can be given as formula_37. In other words, one uses the dualizing sheaf on the ambient P"n" to construct the dualizing sheaf on "X".