Almost coherent rings

TL;DR

This paper characterizes almost coherent rings using almost flat and almost absolutely pure modules, extending classical results into the framework of almost mathematics and addressing a key open question.

Contribution

It provides a new characterization of almost coherent rings and demonstrates that almost coherent modules are not almost isomorphic to coherent modules.

Findings

Characterization of almost coherent rings via almost flat modules

Extension of classical results into almost mathematics

Negative answer to a previously posed question about almost coherent modules

Abstract

Inspired from the work of P. Scholze on the finiteness of \(\mathbf{F}_{p}\)-cohomology groups of proper rigid-analytic varieties over \(p\)-adic fields, Zavyalov recently introduced the notion of almost coherent rings, which plays a key role in the almost ring theory. In this paper, we characterize almost coherent rings in terms of almost flat modules and almost absolutely pure modules, integrating numerous classical results into almost mathematics. Besides, we show that every almost coherent $R$-module is not almost isomorphic to a coherent $R$-module, giving a negative answer to a question proposed in [14,B. Zavyalov, {\it Almost coherent modules and almost coherent sheaves}, Memoirs of the European Mathematical Society 19. Berlin: European Mathematical Society (EMS), 2025].