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For any $d\ge 4$, by deformation theory of schemes, we present examples of (complete or excellent) $d$-dimensional mixed characteristic normal local domains admitting no small Cohen-Macaulay algebra, but admitting instances of small…

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…

This paper shows that if $R$ is a homomorphic image of a Cohen-Macaulay local ring, then $R$-module $M$ is sequentially generalized Cohen-Macaulay if and only if the difference between Hilbert coefficients and arithmetic degrees for all…

The aim of this paper is to extend the main result of C. Huneke and G. Lyubeznik in [Adv. Math. 210 (2007), 498--504] to the class of rings that are images of Cohen-Macaulay local rings. Namely, let $R$ be a local Noetherian domain of…

Let R be a polynomial ring and M a finitely generated graded R-module of maximal grade (which means that the ideal I_t(\cA) generated by the maximal minors of a homogeneous presentation matrix, \cA, of M has maximal codimension in R).…

Let $\fa$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\fd_R X$, $\Gfd_R X$ and $\GCfd_RX$ by $\T(X)$. Let $M$ be an $R$-module such that…

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…

By finding a p-adic obstruction, we construct many examples of positive characteristic complete noetherian local rings which do not admit any module-finite Cohen-Macaulay extension. These examples should be contrasted with a result of…

Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological…

We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective…

Let $R$ be a commutative Noetherian local ring. We prove that the finiteness of the injective dimension of a finitely generated $R$-module $C$ is determined by the existence of a Cohen--Macaulay module $M$ that satisfies an inequality…

Let $M$ denote a finitely generated module over a Noetherian ring $R$. For an ideal $I \subset R$ there is a study of the endomorphisms of the local cohomology module $H^g_I(M), g = \operatorname{grade} (I,M),$ and related results. Another…

Let R be a commutative noetherian local ring. A finitely generated R-module C is semidualizing if it is self-orthogonal and satisfies the condition Hom_R(C,C) \cong R. We prove that a Cohen-Macaulay ring R with dualizing module D admits a…

In this article, we show test properties, in the sense of finitely many vanishing of Ext or Tor, of CM (Cohen-Macaulay) modules whose multiplicity and number of generators (resp., type) are related by certain inequalities. We apply these…

Huneke and Wiegand conjectured that, if $M$ is a finitely generated, non-free, torsion-free module with rank over a one-dimensional Cohen-Macaulay local ring $R$, then the tensor product of $M$ with its algebraic dual has torsion. This…

We study the relation between MCM approximations and FID hulls of modules over a Cohen-Macaulay local ring $R$ with canonical module, specifically when $R$ is generically Gorenstein. We then generalize a result of Kato, who proved that a…

Let $M$ be an $R$-module over a Noetherian ring $R$ and $\mathfrak{a}$ be an ideal of $R$ with $c={\rm cd}(\mathfrak{a},M)$. First, we prove that $M$ is finite $\mathfrak{a}$-relative Cohen-Macaulay if and only if ${\rm…

Over a commutative local Cohen--Macaulay ring, we view and study the category of maximal Cohen--Macaulay modules as a ring with several objects. We compute the global dimension of this category and thereby extend a result of Leuschke to the…

We utilize recent results of Andr\'e and Gabber on the existence of weakly functorial integral perfectoid big Cohen-Macaulay (BCM) algebras to study singularities of local rings in mixed characteristic. In particular, we introduce a mixed…

This paper considers the following conjecture: If $R$ is an unmixed, equidimensional local ring that is a homomorphic image of a Cohen-Macaulay local ring, then for any ideal $J$ generated by a system of parameters, the Chern coefficient…