Specifying The Auslander transpose in submodule category and its applications

TL;DR

This paper explicitly describes the Auslander transpose in the submodule category over a commutative noetherian local ring and applies this to linkage, Auslander-Reiten theory, and computations within module categories.

Contribution

It provides an explicit description of the Auslander transpose in the submodule category and applies it to linkage and Auslander-Reiten theories, extending computational techniques.

Findings

Explicit description of Auslander transpose in submodule category

Characterization of horizontally linked morphisms via module category

Computation of Auslander-Reiten translations within module categories

Abstract

Let $(R, \m)$ be a $d$-dimensional commutative noetherian local ring. Let $\M$ denote the morphism category of finitely generated $R$-modules and let $\Sc$ be the submodule category of $\M$. In this paper, we specify the Auslander transpose in submodule category $\Sc$. It will turn out that the Auslander transpose in this category can be described explicitly within ${\rm mod}R$, the category of finitely generated $R$-modules. This result is exploited to study the linkage theory as well as the Auslander-Reiten theory in $\Sc$. Indeed, a characterization of horizontally linked morphisms in terms of module category is given. In addition, motivated by a result of Ringel and Schmidmeier, we show that the Auslander-Reiten translations in the subcategories $\HH$ and $\G$, consisting of all morphisms which are maximal Cohen-Macaulay $R$-modules and Gorenstein projective morphisms, respectively, may be computed within ${\rm mod}R$ via $\G$-covers. Corresponding result for subcategory of epimorphisms in $\HH$ is also obtained.