Almost coherent rings

TL;DR

This work develops a comprehensive framework for almost coherent rings by translating classical coherence criteria into the almost mathematics setting. It establishes that $M$ is almost flat iff $\mathrm{Tor}_1^R(N,M)$ is almost zero for all finitely presented $N$, and proves that almost coherence is equivalent to the stability of almost flat modules under products, among other criteria, while also introducing and characterizing almost absolutely pure modules. The paper further provides a counterexample showing an almost coherent ring need not be almost isomorphic to a coherent module, answering a question of Zavyalov in the negative. By deriving numerous equivalences for almost pure exact sequences and precovers/covers, the results integrate traditional coherence theory with almost mathematics, yielding practical tools for identifying and working with almost coherent rings in p-adic and almost ring contexts.

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].

Theorems & Definitions (41)

• Lemma 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Definition 2.5
• Proposition 2.6
• proof