Related papers: Contravariantly finite resolving subcategories ove…
Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher-dimensional version of the classification theorem of thick subcategories of the…
We introduce the notion of totally reflexive extension of rings. It unifies Gorenstein orders and Frobenius extensions. We prove that for a totally reflexive extension, a module over the extension ring is totally reflexive if and only if…
A notion of rigidity with respect to an arbitrary semidualizing complex C over a commutative noetherian ring R is introduced and studied. One of the main result characterizes C-rigid complexes. Specialized to the case when C is the relative…
We characterize Gorenstein modules over those local rings that admit a finite contracting endomorphism.
Let $k$ be a commutative ring, let $\mathcal{C}$ be a small, $k$-linear, Hom-finite, locally bounded category, and let $\mathcal{B}$ be a $k$-linear abelian category. We construct a Frobenius exact subcategory…
Let $R$ be a domain that is a complete local $\mathbb{k}$ algebra in dimension one. In an effort to address the Berger's conjecture, a crucial invariant reduced type $s(R)$ was introduced by Huneke et. al. In this article, we study this…
In this note, it is proved that over a commutative noetherian henselian non-Gorenstein local ring there are infinitely many isomorphism classes of indecomposable totally reflexive modules, if there is a nonfree cyclic totally reflexive…
We study totally acyclic complexes of projective modules over triangular matrix rings and then use it to classify Gorenstein projective modules over such rings. We also use this classification to obtain some information concerning…
Let $R$ be a commutative Noetherian Henselian local ring. Denote by $\mathrm{mod} R$ the category of finitely generated $R$-modules, and by ${\mathcal G}$ the full subcategory of $\mathrm{mod} R$ consisting of all G-projective $R$-modules.…
We will study the resolution dimension of functorially finite subcategories. The subcategories with the resolution dimension zero correspond to ring epimorphisms, and rejective subcategories correspond to surjective ring morphisms. We will…
Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical…
Let $A$ be a virtually Gorenstein algebra of finite CM-type. We establish a duality between the subcategory of compact objects in the homotopy category of Gorenstein projective left $A$-modules and the bounded Gorenstein derived category of…
In Commutative Algebra structure results on minimal free resolutions of Gorenstein modules are of classical interest. We define Gorenstein modules of finite length over the weighted polynomial ring via symmetric matrices in divided powers.…
Let $\X$ be a resolving subcategory of an abelian category. In this paper we investigate the singularity category $\ds(\underline\X)=\db(\mod\underline\X)/\kb(\proj(\mod\underline\X))$ of the stable category $\underline\X$ of $\X$. We…
It is a well-known result of Auslander and Reiten that contravariant finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition…
Let R be a commutative noetherian ring. Denote by mod R the category of finitely generated R-modules. In this paper, we investigate the finiteness of the radii of resolving subcategories of mod R with respect to a fixed semidualizing…
We develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the…
We use the theory of Auslander Buchweitz approximations to classify certain resolving subcategories containing a semidualizing or a dualizing module. In particular, we show that if the ring has a dualizing module, then the resolving…
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…
In this paper, we study the resolving of $\mathcal{SGP}(-)$ and $\mathcal{SGF}(-)$, the classes of all strongly Gorenstein projective and flat modules respectively, over a direct product of commutative rings.