A point-free approach to L-Surjunctivity and Stable Finiteness

TL;DR

This paper introduces quasi frames to analyze L-Surjunctivity and Stable Finiteness Conjectures, generalizing known results and proving new properties for rings, modules, and cellular automata within this framework.

Contribution

It develops a point-free approach using quasi frames to study and extend results related to L-Surjunctivity and Stable Finiteness conjectures.

Findings

Endomorphism rings are stably finite for finitely generated modules with Krull dimension.

Linear cellular automata are surjunctive when modules are Artinian.

Generalization of known conjectures to a broader algebraic context.

Abstract

The category of quasi frames (or qframes) is introduced and studied. In the context of qframes we can jointly study problems related to the L-Surjunctivity and Stable Finiteness Conjectures. As a consequences of our main results, we can generalize some of the known results on these conjectures. In particular, let $R$ be a ring, let $G$ be a sofic group, fix a crossed product $R*G$ and let $N$ be a right $R$-module. It is proved that: (1) the endomorphism ring $End_{R* G}(N\otimes_R R*G)$ is stably finite, provided $N$ is finitely generated and has Krull dimension; (2) any linear cellular automaton $\phi:N^G\to N^G$ is surjunctive, provided $N$ is Artinian.