Almost Cohen-Macaulay and almost regular algebras via almost flat extensions

TL;DR

This paper explores the properties of almost Cohen-Macaulay and almost regular algebras using almost flat extensions, providing new insights into their structure and behavior in algebraic settings.

Contribution

It introduces a new approach to almost Cohen-Macaulay modules via a different definition of almost zero modules and studies their behavior under almost flat extensions.

Findings

Almost zero local cohomology modules imply the existence of big Cohen-Macaulay algebras.

Almost faithfully flat extensions preserve the almost Cohen-Macaulay property.

The structure of F-coherent rings in positive characteristic is characterized in terms of almost regularity.

Abstract

The present paper deals with various aspects of the notion of almost Cohen-Macaulay property, which was introduced and studied by Roberts, Singh and Srinivas. We employ the definition of almost zero modules as defined by a value map, which is different from the version of Gabber-Ramero. We prove that, if the local cohomology modules of an algebra $T$ of certain type over a local Noetherian ring are almost zero, $T$ maps to a big Cohen-Macaulay algebra. Then we study how the almost Cohen-Macaulay property behaves under almost faithfully flat extension. As a consequence, we study the structure of $F$-coherent rings of positive characteristic in terms of almost regularity.