Vanishing of (co)homology of Burch and related submodules

Abstract

We introduce the notion of Burch submodules and weakly $\mathfrak{m}$-full submodules of modules over a local ring $(R,\mathfrak{m})$ and study their properties. One of our main results shows that Burch submodules satisfy 2-Tor rigid and test properties. We also show that over a local ring $(R,\mathfrak{m})$, a submodule M of a finitely generated R-module X, such that either $M=\mathfrak{m}X$ or $M(\subseteq \mathfrak{m}X$) is weakly $\mathfrak{m}$-full in X, is 1-Tor rigid, and a test module provided that X is faithful (and $X∕M$ has finite length when M is weakly $\mathfrak{m}$-full). As an application, we give some new class of modules and a new class of rings such that a conjecture of Huneke and Wiegand is affirmative for them.