Specifying the Auslander transpose in submodule category and its applications

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 S be the full subcategory of M consisting of monomorphisms, known as the submodule category. This article reveals that the Auslander transpose in the category 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 S. In addition, motivated by a result of Ringel and Schmidmeier, we show that the Auslander–Reiten translations in the subcategories H and G, consisting of all morphisms which are maximal Cohen–Macaulay R-modules and Gorenstein projective morphisms, respectively, may be computed within modR via G-covers. The corresponding result for the subcategory of epimorphisms in H is also obtained.