News about Koszul and \v{C}ech complexes: Another view at local cohomology and completion-I

Abstract: In the first part of the talk, we present some elementary new facts about Koszul and \v{C}ech complexes with respect to a single element. We construct free resolutions of the \v{C}ech complex for a system of elements in a commutative ring.

This is used in order to construct quasi-isomorphisms between the \v{C}ech complexes and certain Koszul complexes. The free resolution of the \v{Cech} complex is applied in order to find relations to the left derived functors of the completion as a certain Koszul homology.

This material provides an elementary introduction to some of the results of the speakers joint work with A.-M. Simon (see "Completion, \v{C}ech and local homology and cohomology. Interactions between them. Springer Monograph, 2018") as well as some further developments.

One focus is the study of weakly proregular sequences and of proregular sequences which provides a new insight for the local cohomology as well as the left derived functors of the completion. Finally we shall present an application to prisms in the sense of Bhatt and Scholze.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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