13 - Dualizing Complexes over Commutative Rings

Summary

Let A be a noetherian commutative ring. A complex R ∈ D(A) is called a dualizing complex (DC) if it has bounded finitely generated cohomology, finite injective dimension, and the derived Morita property, which says that the derived homothety morphism : A → RHomA(R, R) in D(A) is an isomorphism.We prove uniqueness of DCs and existence when A is essentially finite type over a regular noetherian ring.A residue complex is a DC that consists of injective modules of the correct multiplicity in each degree. There is a stronger uniqueness property for residue complexes. To understand residue complexes, we review the Matlis classification of injective A-modules. In the last two sections we talk about Van den Bergh rigidity. We prove that if A has a rigid DC R, then it is unique up to a unique rigid isomorphism. Existence of a rigid DC is harder to prove, and we just give a reference to it. Rigid residue complexes always exist, and they are unique in a very strong sense. We end this chapter with remarks that explain how rigid residue complexes allow a new approach to residues and duality on schemes and Deligne--Mumford stacks.