Related papers: On the notion of sequentially Cohen-Macaulay modul…

Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module $M$ over a (commutative) Noetherian ring $R$. Second, in the other direction, we…

The associated primes of an arbitrary lexsegment ideal $I\subset S=K[x_1,...,x_n]$ are determined. As application it is shown that $S/I$ is a pretty clean module, therefore, $S/I$ is sequentially Cohen-Macaulay and satisfies Stanley's…

We give a necessary and sufficient condition for a simplicial complex to be approximately Cohen-Macaulay. Namely it is approximately Cohen-Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear and…

This paper contains two theorems concerning the theory of maximal Cohen--Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen--Macaulay modules $M$ and $N$ must have finite length, provided only finitely…

We study a variety for graded maximal Cohen--Macaulay modules, which was introduced by Dao and Shipman. The main result of this paper asserts that there are only a finite number of isomorphism classes of graded maximal Cohen--Macaulay…

We study a modified version of the classical Ulrich modules, which we call $c$-Ulrich. Unlike the traditional setting, $c$-Ulrich modules always exist. We prove that these modules retain many of the essential properties and applications…

Let $R$ be a commutative, Noetherian, local ring and $M$ an $R$-module. Consider the module of homomorphisms $\operatorname{Hom}_R(R/\mathfrak{a},M/\mathfrak{b} M)$ where $\mathfrak{b}\subseteq\mathfrak{a}$ are parameter ideals of $M$. When…

The Cohen-Macaulay type of idealizations of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings is explored. There are two extremal cases, one of which is closely related to the theory of Ulrich modules \cite{BHU, GOTWY1, GOTWY2,…

We introduce the notion of residual intersections of modules and prove their existence. We show that projective dimension one modules have Cohen-Macaulay residual intersections, namely they satisfy the relevant Artin-Nagata property. We…

The aim of this survey is to discuss invariants of Cohen-Macaulay local rings that admit a canonical module. Attached to each such ring R with a canonical ideal C, there are integers--the type of R, the reduction number of C--that provide…

Let $\mathfrak{q}$ be an ideal of a Noetherian local ring $(A,\mathfrak{m})$ and $M$ a non-zero finitely generated $A$-module. We present a criterion of Cohen-Macaulayness of the form module $G_M(\mathfrak{q})$ in terms of (non-)vanishing…

In this paper, we establish the global analogues of some dualities and equivalences in local algebra by developing the theory of relative Cohen-Macaulay modules. Let R be a commutative Noetherian ring (not necessarily local) with identity…

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 investigate the behavior of Cohen-Macaulay defect undertaking tensor product with a perfect module. Consequently, we study the perfect defect of a module. As an application, we connect to associated prime ideals of tensor products.

Let $(R, \m, k)$ be a complete Cohen-Macaulay local ring. In this paper, we assign a numerical invariant, for any balanced big Cohen-Macaulay module, called $\uh$-length. Among other results, it is proved that, for a given balanced big…

We prove new characterizations based on Gr\"obner bases for the Cohen-Macaulay property of a projective monomial curve.

We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen-Macaulay. As an application, we give a complete characterization of the…

We define three new homological dimensions - Cohen-Macaulay injective, projective, and flat dimension - which inhabit a theory similar to that of classical injective, projective, and flat dimension. Finiteness of the new dimensions…

We give a combinatorial description of local cohomology modules of a graded module over a semigroup ring, with support at the graded maximal ideal. This combinatorial framework yields Hochster-type formulas for the Hilbert series of such…

In order to study quantum measurement theory, sequential product defined for any two quantum effects is introduced. Physically motivated conditions ask the sequential product to be continuous with respect to the strong operator topology. In…