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Dualizing Complexes

  • Peter Schenzel
  • Anne-Marie Simon
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

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.

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für InformatikMartin-Luther-Universität Halle-WittenbergHalleGermany
  2. 2.Service de Geometrie DifferentielleUniversité Libre de BruxellesBrusselsBelgium

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