Related papers: Maximal Cohen-Macaulay approximations and Serre's …
Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive answer to the Huneke$^,$s conjecture is given for a noetherian ring $R$ and minimax…
The aim of this paper is to extend Cohen structure theorem beyond local rings. Both Cohen structure theorem and Nagata's generalization of it are special cases of our results. We investigate for which rings $R$ there exists a maximal ideal…
In this paper, we introduce a topology on the set of isomorphism classes of finitely generated modules over an associative algebra. Then we focus on the relative topology on the set of isomorphism classes of maximal Cohen--Macaulay modules…
We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of such degenerations. It is proved that such degenerations over an even-dimensional simple hypersurface singularity of type $(A_n)$ are given by…
Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \mathfrak{m}^q$ in a Noetherian local ring $(A, \mathfrak{m})$ with the Gorenstein tangent cone $\mathrm{G}(\mathfrak{m}) = \bigoplus_{n \geq…
In this paper, we introduce an invariant of Cohen-Macaulay local rings in terms of the reduction number of canonical ideals. The invariant can be defined in arbitrary Cohen-Macaulay rings and it measures how close to being Gorenstein.…
A ladder determinantal module is an arbitrary direct sum of ideals of maximal minors of a generic ladder matrix. In this article, we give necessary and sufficient conditions for the special fiber ring of such modules to be Gorenstein. These…
We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monomial minimal reduction. When this is the case, we study the Cohen-Macaulay and Gorenstein properties of the associated graded ring and provide…
We introduce the notions of sequential sequence and sequential f-sequence in order to characterize sequentially Cohen-Macaulay modules and sequentially generalized Cohen-Macaulay modules. Let R be a Noetherian local ring and M a finitely…
Let $(R,m)$ be a Noetherian local ring and $I$ an ideal with finite projective dimension. If $R/I$ satisfies some property $\mathcal{P}$, it is natural to ask whether $R$ would also satisfy this property $\mathcal{P}$. This is called the…
We study Gorenstein dimension and grade of a module $M$ over a filtered ring whose assosiated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded…
In this paper several quasi-Gorenstein counterparts to some known properties of Gorenstein rings are given. We, furthermore, give an explicit description of the attach prime ideals of certain local cohomology modules.
Let $(A,\mathfrak{m})$ be a complete Cohen-Macaulay local ring. Assume $A$ is not Gorenstein. We say $A$ is a Teter ring if there exists a complete Gorenstein ring $(B,\mathfrak{n})$ with $\dim B = \dim A$ and a surjective map $B…
We provide a simplified proof of a theorem proved by Tavanfar and Shimomoto which states that a quasi-Gorenstein deformation of a $3$-dimensional quasi-Gorenstein local ring $(A,m,k)$ with $H^2_m(A)=k$ admits a small Cohen-Macaulay module.
Buchweitz related the singularity category of a (strongly) Gorenstein ring and the stable category of maximal Cohen-Macaulay modules by a triangle equivalence. We phrase his result in a relative categorical setting based on N-complexes…
This paper studies the structure of special preenvelopes and envelopes with respect to maximal Cohen-Macaulay modules. We investigate the structure of them in terms of their kernels and cokernels. Moreover, using this result, we also study…
Let $R$ be a commutative noetherian ring, let $\frak a$ and $\frak b$ be two ideals of $R$; and let $\Ss$ be a Serre subcategory of $R$-modules. We give a necessary and sufficient condition by which $\Ss$ satisfies $C_{\frak a}$ and…
The so-called 'change-of-ring' results are well-known expressions which present several connections between projective, injective and flat dimensions over the various base rings. In this note we extend these results to the Gorenstein…
The aim of this paper is to study the theory of cohomology annihilators over commutative Gorenstein rings. We adopt a triangulated category point of view and study the annihilation of stable category of maximal Cohen-Macaulay modules. We…
We study homological properties of test modules that are, in principle, modules that detect finite homological dimensions. The main outcome of our results is a generalization of a classical theorem of Auslander and Bridger: we prove that,…