Let (R, 𝔪) be a commutative noetherian local ring. In this paper, we prove that if 𝔪 is
decomposable, then for any finitely generated R-module M of infinite projective dimension 𝔪 …

We construct a local Cohen–Macaulay ring R with a prime ideal p∈ Spec (R) such that R
satisfies the uniform Auslander condition (UAC), but the localization R p does not satisfy …

A Cohen–Macaulay local ring $ R $ satisfies trivial vanishing if $\operatorname {Tor} _i^ R
(M, N)= 0$ for all large $ i $ implies that $ M $ or $ N $ has finite projective dimension. If $ R …

The purpose of this paper is, as part of the stratification of Cohen–Macaulay rings, to
investigate the question of when the fiber products are almost Gorenstein rings. We show …

For a finitely generated module M over a commutative Noetherian ring R, we settle the
Auslander–Reiten conjecture when at least one of Hom R (M, R) and Hom R (M, M) has …

We consider a local (or standard graded) ring $ R $ with ideals $\mathcal {I}'$, $\mathcal {I}
$, $\mathcal {J}'$, and $\mathcal {J} $ satisfying certain Tor-vanishing constraints. We …

This paper deals with the subject of singularities of a germ of an analytic space given by the
gluing of germs of complex analytic spaces and some invariants such as degree of a finite …

Let R and S be polynomial rings of positive dimensions over a field k. Let I⊆ R, J⊆ S be non-
zero homogeneous ideals none of which contains a linear form. Denote by F the fiber …

In this paper, we show that the gluing of germs of formal complex spaces is also a formal
complex space. Moreover, we study the Betti numbers of the gluing of formal complex …

For finitely generated modules $ M $ and $ N $ over a commutative Noetherian local ring $
R $, we give various sufficient criteria for detecting freeness of $ M $ or $ N $ via vanishing …