Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings

Abstract

Abstract. For any ring R the category of monomorphisms is a full subcategory of the morphsim category over R, where the latter is equivalent to the module category of the triangular matrix ring with entries the ring R. In this work, we consider the monomorphism category as a full subcategory of the module category over the Morita ring with all entries the ring R and zero bimodule homomorphisms. This approach provides an interesting link between Morita rings and monomorphism categories. The aim of this paper is two-fold. First, we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work which is strongly connected with monomorphism categories. In the second part, we investigate monomorphisms where the domain has finite projective dimension. In particular, we show that the latter category is a Gorenstein subcategory of the monomorphism category over a Gorenstein algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory and show that it carries a structure of a Gorenstein abelian category.

Ladders of recollements of abelian categories

“…Example 3.1. Let R be a ring and consider the Morita context ring ∆ (0,0) = R R R R (see [20,24]). Its modules are tuples of the form Proposition 4.4], the module category Mod-∆ (0,0) admits a recollement of module categories with an infinite ladder (of period three).…”

Recollements of abelian categories and ideals in heredity chains—a recursive approach to quasi-hereditary algebras

“…Since f Ae = 0, it follows that N ⊗ A N = 0. Then by [7,Example 4.16], Λ (0,0) is a Morita context ring, whose addition is componentwise, and multiplication is given as follows :…”

Ladders of Compactly Generated Triangulated Categories and Preprojective Algebras

“…The module category of this class of Morita rings was investigated in [15] in connection with aspects of Gorenstein homological algebra. In particular, the module category Mod-∆ (0,0) is equivalent to the double morphism category DMor (Mod-Λ) of Mod-Λ, introduced in [15, subsection 2.2].…”