3. Properties of Rigid Complexes 4. Rigid Dualizing Complexes 5. Rigid Complexes and CM Homomorphisms

Let A be a noetherian commutative ring. Denote by Dbf (ModA) the derived category of bounded complexes of A-modules with finitely generated cohomology modules. Definition 1. (Grothendieck [RD]) A dualizing complex over A is a complex R ∈ Dbf (ModA) satisfying the two conditions: (i) R has finite injective dimension. (ii) The canonical morphism A → RHomA(R,R) is an isomorphism. Condition (i) means that there is an integer d such that ExtiA(M,R) = 0 for all i> d and all modules M. Example 2. If K is a regular noetherian ring of finite Krull dimension (say a field, or the ring of integers Z) then R: = K ∈ Dbf (ModK) is a dualizing complex. Dualizing complexes over commutative rings are part of Grothendieck’s duality theory in algebraic geometry, which was developed in [RD]. This duality theory deals with dualizing complexes on schemes and relations between them. In this lecture I will explain a new approach to dualizing complexes over com-mutative rings, due to James Zhang and myself (see [YZ4] and [YZ5]). Specifically, I’ll talk about existence and uniqueness of rigid dualizing complexes. The purpose of rigidity is to eliminate automorphisms, and to make the dualizing complexes functorial