About proregular sequences and an application to prisms

TL;DR

This paper explores proregular sequences in commutative algebra, proving their equivalence under different definitions, providing a cohomological characterization, and applying these results to the theory of prisms in algebraic geometry.

Contribution

It establishes the equivalence of two notions of proregularity, offers a cohomological criterion, and applies these concepts to the study of prisms.

Findings

Proregularity notions by Greenlees-May and Lipman are equivalent.

Cohomological characterization of proregularity via Čech homology.

Application of proregularity results to prisms in algebraic geometry.

Abstract

Let $\underline{x} = x_1,\ldots,x_k$ denote an ordered sequence of elements of a commutative ring $R$. Let $M$ be an $R$-module. We recall the two notions that $\underline{x}$ is $M$-proregular given by Greenlees and May (see \cite{[5]}) and Lipman (see \cite{[1]}) and show that both notions are equivalent. As a main result we prove a cohomological characterization for $\underline{x}$ to be $M$-proregular in terms of \v{C}ech homology. This implies also that $\underline{x}$ 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 (see \cite{[3]}).