Specifying the Auslander transpose in submodule category and its applications

Abstract

Let $(R,\mathfrak{m})$ be a $d$-dimensional commutative Noetherian local ring. Let $\mathcal{M}$ denote the morphism category of finitely generated $R$-modules, and let $\mathcal{S}$ be the full subcategory of $\mathcal{M}$ consisting of monomorphisms, known as the submodule category. This article reveals that the Auslander transpose in the category $\mathcal{S}$ can be described explicitly within $modR$, the category of finitely generated $R$-modules. This result is exploited to study the linkage theory as well as the Auslander–Reiten theory in $\mathcal{S}$. In addition, motivated by a result of Ringel and Schmidmeier, we show that the Auslander–Reiten translations in the subcategories $\mathcal{H}$ and $\mathcal{G}$, consisting of all morphisms which are maximal Cohen–Macaulay $R$-modules and Gorenstein projective morphisms, respectively, may be computed within $modR$ via $\mathcal{G}$-covers. The corresponding result for the subcategory of epimorphisms in $\mathcal{H}$ is also obtained.