Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras

Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra A , how to construct all the Gorenstein-projective A -modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ = A M B A 0 B . We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ = A M B A 0 B .

On relative counterpart of Auslander’s conditions

It is now well known that the conditions used by Auslander to define the Gorenstein projective modules on Noetherian rings are independent. Recently, Ringel and Zhang adopted a new approach in investigating Auslander’s conditions. Instead of looking for examples, they investigated rings on which certain implications between Auslander’s conditions hold. In this paper, we investigate the relative counterpart of Auslander’s conditions. So, we extend Ringel and Zhang’s work and introduce other concepts. Namely, for a semidualizing module [Formula: see text], we introduce weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings as rings representing relations between the relative counterpart of Auslander’s conditions. Moreover, we introduce a relative notion of the well-known Frobenius category. We show how useful are [Formula: see text]-Frobenius categories in characterizing weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings.

Gorenstein-Projective Modules over Morita Rings

Let [Formula: see text] be a Morita ring which is an Artin algebra. In this paper we investigate the relations between the Gorenstein-projective modules over a Morita ring [Formula: see text] and the algebras [Formula: see text] and [Formula: see text]. We prove that if [Formula: see text] is a Gorenstein algebra and both [Formula: see text] and [Formula: see text] (resp., both [Formula: see text] and [Formula: see text]) have finite projective dimension, then [Formula: see text] (resp., [Formula: see text]) is a Gorenstein algebra. We also discuss when the CM-freeness and the CM-finiteness of a Morita ring [Formula: see text] is inherited by the algebras [Formula: see text] and [Formula: see text].

From subcategories to the entire module categories

In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the {2\times 2} lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

A note on deformations of Gorenstein-projective modules over finite dimensional algebras

In this note, we present a survey of results concerning universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras.