Dualizing Complexes

Abstract

In this chapter we provide a short and self-contained approach to the notion of a dualizing complex for Noetherian rings, to be used in the next chapter. Most of the results are not new, but some proofs are. In particular, we provide a proof of the existence of a dualizing complex for a complete Noetherian local ring independent of the Cohen structure theorem. This is part of an interesting interaction between the notion of a dualizing complex for a Noetherian ring and the notion of a Čech complex. This interaction also appears when we consider a change of rings of the form $R\rightarrow \hat{R}^{\mathfrak {a}}$. In that case, given a dualizing complex of a Noetherian ring R, we provide an explicit construction of a dualizing complex for $\hat{R}^{\mathfrak {a}}$ involving the Čech complex built on a generating set of $\mathfrak {a}$. In the last section we provide some new properties of dualizing complexes related to the completion functor, a recurrent theme in this monograph.