Related papers: Classifying Resolving Subcategories

Let R be a commutative Noetherian local ring with residue field k. Let X be a resolving subcategory of finitely generated R-modules. This paper mainly studies when X contains k or consists of totally reflexive modules. It is proved that X…

Auslander-Reiten duality for module categories is generalized to some sufficiently nice subcategories. In particular, our consideration works for $\mathcal{P}^{<\infty}(\Lambda)$, the subcategory consisting of finitely generated modules…

Let R be a commutative noetherian local ring. As an analogue of the notion of the dimension of a triangulated category defined by Rouquier, the notion of the dimension of a subcategory of finitely generated R-modules is introduced in this…

Holm (H. Holm, Modules with cosupport and injective functors, Algebr. Represent. Theor., 13 (2010), 543-560) considers categories of right modules dual to those with support in a set of finitely presented modules. We extend some of his…

The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ of admissibly finitely presented functors and use it to give a version…

Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor…

We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the…

Auslander and Buchweitz have proved that every finitely generated module over a Cohen-Macaulay (CM) ring with a dualizing module admits a so-called maximal CM approximation. In terms of relative homological algebra, this means that every…

We introduce and study "quasidualizing" modules. An artinian R-module T is quasidualizing if the homothety map \hat R\rightarrow Hom(T,T) is an isomorphism and Ext_R^i(T,T)=0 for each integer i>0. Quasidualizing modules are associated to…

We introduce the notion of mutation of $n$-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen-Macaulay modules…

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…

It is shown that the idempotent completion of the additive hull of the tensor product of the residue category of the category of paths of a locally finite quiver modulo an admissible ideal and a dualizing category is dualizing. Furthermore,…

For a finite dimensional algebra $A$, we prove that the bounded homotopy category of projective $A$-modules and the bounded derived category of $A$-modules are dual to each other via certain categories of locally-finite cohomological…

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 and study a class of objects that encompasses Christensen and Foxby's semidualizing modules and complexes and Kubik's quasi-dualizing modules: the class of $\mathfrak{a}$-adic semidualizing modules and complexes. We give…

We construct two categorifications of the Lusztig--Vogan module associated to a real reductive algebraic group. The first categorification is given by semisimple complexes in an equivariant derived category, and the second is constructed as…

We classify module categories over the category of representations of quantum $SL(2)$ in a case when $q$ is not a root of unity. In a case when $q$ is a root of unity we classify module categories over the semisimple subquotient of the same…

We classify 1-tilting classes over an arbitrary commutative ring. As a consequence, we classify all resolving subcategories of finitely presented modules of projective dimension at most 1. Both these collections are in 1-1 correspondence…

The category of Cohen-Macaulay modules of an algebra $B_{k,n}$ is used [JKS16] to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this…

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…