About proregular sequences and an application to prisms

Abstract

Let denote an ordered sequence of elements of a commutative ring R. Let M be an R-module. We recall the two notions that is M-proregular given by Greenlees and May and Lipman and show that both notions are equivalent. As a main result we prove a cohomological characterization for to be M-proregular in terms of Čech cohomology. This implies also that is M-weakly proregular if it is M-proregular. A local-global principle for proregularity and weakly proregularity is proved. This is used for a result about prisms as introduced by Bhatt and Scholze.