Speaker: Peter Schenzel, University of Halle, Leipzig, Germany
Date/Time: 25 June 2021, 5:30pm IST/ 12:00pm GMT / 8:00am EDT (joining
time 5:15pm IST).
Google meet link:
https://meet.google.com/uru-znrx-xmk
Title: News about Koszul and \v{C}ech complexes: Another view at local
cohomology and completion.
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.
Chairperson: Le Tuan Hoa, Institute of Mathematics, Hanoi