About proregular sequences and an application to prisms

Abstract

Let $\underset{¯}{x}=x_1,\dots ,x_k$ denote an ordered sequence of elements of a commutative ring R. Let M be an R-module. We recall the two notions that $\underset{¯}{x}$ 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 $\underset{¯}{x}$ to be M-proregular in terms of Čech cohomology. This implies also that $\underset{¯}{x}$ 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.

Keywords:

• Injective module
• non-Noetherian commutative ring
• proregular sequences
• torsion

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13C12
• Secondary: 13C11
• 13D07

Acknowledgement

The author thanks Anne-Marie Simon (Université Libre de Bruxelles) for various discussions about the subject and a careful reading of the manuscript. He also thanks the reviewer for the comments improving some arguments and avoiding several misprints.