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.
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Schenzel, P., Simon, AM. (2018). Dualizing Complexes. In: Completion, Čech and Local Homology and Cohomology. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-96517-8_11
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DOI: https://doi.org/10.1007/978-3-319-96517-8_11
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-96516-1
Online ISBN: 978-3-319-96517-8
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