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

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.