Related papers: When is there a nontrivial extension-closed subcat…

Let $\mathsf{mod} R$ denote the category of finitely generated $R$-modules for a commutative noetherian ring $R$. In this paper, we investigate KE-closed subcategories of $\mathsf{mod} R$ as a continuation of our previous work. We associate…

An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category \cite{Mit}. A tilting object in a abelian category is a natural generalization of a small projective generator. Moreover, any…

Let $\mathcal{P}$ be the class of rings for which every indecomposable right module is pure-projective or pure-injective. When $R$ is a Noetherian local commutative ring of maximal ideal $P$, it is proven that $R\in\mathcal{P}$ if and only…

Given two rings $R \subseteq S$, $S$ is said to be a minimal ring extension of $R$ if $R$ is a maximal subring of $S$. In this article, we study minimal extensions of an arbitrary ring $R$, with particular focus on those possessing nonzero…

We classify all tilting and cotilting classes over commutative noetherian rings in terms of descending sequences of specialization closed subsets of the Zariski spectrum. Consequently, all resolving subcategories of finitely generated…

Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$ and $M$ an $R$-module with Cosupport in $\mathrm{V}(\mathfrak{a})$. We show that $M$ is $\mathfrak{a}$-coartinian if and only if $\mathrm{Ext}_{R}^{i}(R/\mathfrak{a},M)$ is…

We show that bounded type implies finite type for a constructible subcategory of the module category of a finitely generated algebra over a field, which is a variant of the first Brauer-Thrall conjecture. A full subcategory is constructible…

Let $R$ be a commutative Noetherian ring. It is shown that $R$ is Artinian if and only if every $R$-module is good, if and only if every $R$-module is representable. As a result, it follows that every nonzero submodule of any representable…

For a commutative noetherian ring A, we compare the support of a complex of A-modules with the support of its cohomology. This leads to a classification of all full subcategories of A-modules which are thick (that is, closed under taking…

Let R be a regular ring of characteristic p. Hochster showed that the category of Lyubeznik's F-modules has enough injectives, so that every F-module has an injective resolution in this category. We show that under mild conditions on R, for…

Let $\frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\V(\fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{\fa}_i(M)\neq 0$. We show that $M$ is $\fa$-cofinite if and…

Let $\hat{R}$ be the $I$-adic completion of a commutative ring $R$ with respect to a finitely generated ideal $I$. We give a necessary and sufficient criterion for the category of perfect complexes over $\hat{R}$ to be equivalent to the…

For an $(n-1)$-Auslander algebra $\Lambda$ with global dimension $n$, we give some necessary conditions for $\Lambda$ admitting a maximal $(n-1)$-orthogonal subcategory in terms of the properties of simple $\Lambda$-modules with projective…

Let k be an algebraically closed uncountable field of characteristic zero. Let R be a complete local hypersurface over k. Denote by CM(R) the category of maximal Cohen-Macaulay R-modules and by D^{sg}(R) the singularity category of R.…

We say that a commutative ring R satisfies the restricted minimum (RM) condition if for all essential ideals I in R, factor R/I is an Artinian ring. We will focus on Noetherian reduced rings because in this setting known results for RM…

In this paper, by using functor rings and functor categories, we study finiteness and purity of subcategories of the module categories. We give a characterisation of contravariantly finite resolving subcategories of the module category of…

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ a finitely generated $R$--module. Let $t$ be a non-negative integer such that $\H^i_\fa(M)$ is $\fa$--cofinite for all $i<t$. It is well--known that…

In this paper we describe the categories $\mathbb{L}_R$ , [$\mathbb{R}_R$] whose objects are left [right] ideals of a Noetherian ring $R$ with unity and morphisms are appropriate $R$-linear transformations. Further it is shown that these…

In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely…

Let $R$ be a commutative noetherian local ring and denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules. In this paper, we give some evaluations of the singular locus of $R$ and annihilators of Tor and Ext from a…