Related papers: Cohen-Macaulay approximations and the $\text{SC}_r…

Let $(R,\fm)$ be a relative Cohen-Macaulay local ring with respect to an ideal $\fa$ of $R$ and set $c:=\h\fa$. In this paper, we investigate some properties of the Matlis dual $\H_{\fa}^c(R)^{\vee}$ of the $R$-module $\H_{\fa}^c(R)$ and we…

Let $R$ be a commutative Noetherian local ring. We characterize when its completion has an isolated singularity, thereby strengthening the Dao-Takahashi refinement of the Auslander-Huneke-Leuschke-Wiegand theorem. We investigate the ascent…

In this paper, we first introduce $\mathcal {W}_F$-Gorenstein modules to establish the following Foxby equivalence: $\xymatrix@C=80pt{\mathcal {G}(\mathcal {F})\cap \mathcal {A}_C(R) \ar@<0.5ex>[r]^{C\otimes_R-} & \mathcal {G}(\mathcal…

It is shown that a module is sequentially Cohen-Macaulay if and only if the index of reducibility for distinguished parameter ideals are eventually constant with special value. As corollaries to the main theorem we given to characterize the…

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring with residue field $k$. If $M$ is a finitely generated $A$-module then set $\text{curv}(M) = \limsup_n\sqrt[n]{\beta_n^A(M)}$. We show that under mild hypotheses the existence of a…

Let $(R, {\frak m})$ be a local ring, $I$ a proper ideal of $R$ and $M$ a finitely generated $R$-module of dimension $d$. We discuss the local homology modules of $H^d_I(M)$. When $M$ is Cohen-Macaulay, it is proved that $H^d_{{\frak…

Inspired by classical work on the depth formula for tensor products of finitely generated $R$-modules, we introduce two conditions which we call $(\mathbf{ldep})$ and $(\mathbf{rdep})$ and their derived variations. We show for…

We establish a criterion for the strong $F$-regularity of a (non-Gorenstein) Cohen-Macaulay reduced complete local ring of dimension at least $2$, containing a perfect field of prime characteristic $p$. We also describe an explicit…

Let R be a Cohen-Macaulay local ring possessing a canonical module. In this paper we consider when the maximal ideal of R is self-dual, i.e. it is isomorphic to its canonical dual as an R-module. local rings satisfying this condition are…

We give a two step method to study certain questions regarding associated graded module of a Cohen-Macaulay (CM) module $M$ w.r.t an $\mathfrak{m}$-primary ideal $\mathfrak{a}$ in a complete Noetherian local ring $(A,\mathfrak{m})$. The…

The purpose of this note is to show that a finitely generated graded module $M$ over $S=k[x_1,\ldots,x_n]$, $k$ a field, is sequentially Cohen-Macaulay if and only if its arithmetic degree ${\rm adeg}(M)$ agrees with ${\rm adeg}(F/{\rm…

By extending some basic results of Grothendieck and Foxby about local cohomology to commutative DG-rings, we prove new amplitude inequalities about finite DG-modules of finite injective dimension over commutative local DG-rings,…

We introduce a new invariant for subcategories X of finitely generated modules over a local ring R which we call the radius of X. We show that if R is a complete intersection and X is resolving, then finiteness of the radius forces X to…

Motivated by a recent result of Yoshino, and the work of Bergh on reducible complexity, we introduce reducing versions of invariants of finitely generated modules over commutative Noetherian local rings. Our main result considers modules…

The aim of this note is to study the class of one dimensional Cohen-Macaulay local rings, $(R, \mathfrak{m})$ say, possessing a canonical ideal $K$ which is a reduction of $\mathfrak{m}$. We call $R$ to have canonical reduction $K$. We show…

In the present article, we investigate the following deformation problem. Let $(R,\mathfrak m)$ be a local (graded local) Noetherian ring with a (homogeneous) regular element $y \in \mathfrak m$ and assume that $R/yR$ is quasi-Gorenstein.…

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…

Let $(R,\fm)$ be a local ring and $C$ be a homologically bounded and finitely generated $R$-complex. Then, we prove that $C$ is a dualizing complex of $R$ if and only if $C$ is a Cohen-Macaulay semidualizing complex of type one or…

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module $\omega$. Attached to each such ring R, when $\omega$ is an ideal, there are integers--the type of R, the reduction number of $\omega$--that provide valuable…

Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module,…