Tokuji Araya

The Auslander-Reiten conjecture for Gorenstein rings

The Nakayama conjecture is one of the most important conjectures in ring theory. The Auslander-Reiten conjecture is closely related to it. The purpose of this paper is to show that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring R, then it holds for R.

A homological dimension related to AB rings

There are many homological dimensions which are closely related to ring theoretic properties. The notion of a AB ring has been introduced by Huneke and Jorgensen. It has nice homological properties. In this paper, we shall define a homological dimension which is closely related to a AB ring, and investigate its properties.

On the Left Perpendicular Category of the Modules of Finite Projective Dimension

In this article, we characterize several properties of commutative noetherian local rings in terms of the left perpendicular category of the category of finitely generated modules of finite projective dimension. As an application, we prove that a local ring is regular if (and only if) there exists a strong test module for projectivity having finite projective dimension. We also obtain corresponding results with respect to a semidualizing module.

Remarks on Reflexive Subcategories

In this article, we study reflexive subcategories of the category of finitely generated modules over Cohen–Macaulay local rings R. One of the main results yields a complete classification of the reflexive subcategories for the case when R is a one-dimensional complete local hypersurface of finite Cohen–Macaulay representation type. The other result yields a one-to-one correspondence between the subclasses of a stable category and the reflexive subcategories over a complete local hypersurface. Then, we attempt to extend the result in Takahashi [8, Proposition 4.4] to a reflexive subcategory.