arXiv subjects

Yuya Otake

Publications and source records attributed to Yuya Otake.

11 recordsLinked to original sources

arXiv2026

Auslander-type grade conditions and subcategories related to syzygy modules

The question of when a syzygy module or a module satisfying a Serre condition satisfies the cohomological vanishing conditions known as higher torsionfreeness has been extensively studied. In this paper, we address these questions using grade conditions introduced and studied by Auslander-Bridger and Auslander-Reiten. We also investigate homological conditions on Ext modules and, as an application, give a partial answer over commutative rings to a question of Auslander-Bridger concerning whether higher reflexive modules are higher torsionfree.

math.RT↗

arXiv2026

On strongly G-regular rings

A noetherian ring is called G-regular when all finitely generated Gorenstein projective modules are projective. In this paper, we study rings satisfying the stronger condition that all Gorenstein projective modules are projective, which we call strongly G-regular. We show that the notion of strongly G-regular rings is closely related to that of quasi-dominant rings introduced by Takahashi and to the covariant/contravariant finiteness of a certain thick subcategory. We also answer a series of questions due to Chen in the negative, showing that the Gorenstein projective analogue of the Auslander-Ringel-Tachikawa theorem fails even for commutative local artin algebras which are weakly Gorenstein in the sense of Ringel and Zhang.

math.AC↗

arXiv2025

On local rings of finite syzygy representation type

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 and descent of finite and countable syzygy representation type along the canonical map from $R$ to its completion. One consequence is a complete affirmative answer to Schreyer's conjecture. We explore analogues of Chen's questions in the context of finite Cohen-Macaulay representation type over Cohen-Macaulay rings. The main result in this direction shows that if $R$ is Cohen-Macaulay and there are only finitely many non-isomorphic indecomposable maximal Cohen-Macaulay modules that are locally free on the punctured spectrum, then either $R$ is a hypersurface or every Gorenstein projective module is projective; moreover, every Gorenstein projective module over the completion of $R$ is a direct sum of finite generated ones. Finally, we study dominant local rings, introduced by Takahashi, under certain finite representation type conditions, and identify a new class of virtually Gorenstein rings.

math.AC↗

arXiv2025

Generation of singularity categories and infinite injective dimension locus via annihilation of cohomologies

Let R be a commutative Noetherian ring. We establish a close relationship between the strong generation of the singularity category of R and the nonvanishing of the annihilator of the singularity category of R. As an application, we prove that the singularity category of R has a strong generator if and only if the annihilator of the singularity category of R is nonzero when R is a Noetherian domain with Krull dimension at most one. We introduce the notion of the co-cohomological annihilator of modules. If the category of finitely generated R-modules has a strong generator, we show that the infinite injective dimension locus of a finitely generated R-module M is closed, with the defining ideal given by the co-cohomological annihilator of M. Finally, we provide a connection between the existence of an extension generator of the category of finitely generated R-modules and the finiteness of the Krull dimension of R.

math.AC↗

arXiv2025

An approach to Martsinkovsky's invariant via Auslander's approximation theory

Auslander developed a theory of the $δ$-invariant for finitely generated modules over commutative Gorenstein local rings, and Martsinkovsky extended this theory to the $ξ$-invariant for finitely generated modules over general commutative noetherian local rings. In this paper, we approach Martsinkovsky$'$s $ξ$-invariant by considering a non-decreasing sequence of integers that converges to it. We investigate Auslander$'$s approximation theory and provide methods for computing this non-decreasing sequence using the approximation.

math.AC↗

arXiv2023

On the Auslander-Bridger-Yoshino theory for complexes of finitely generated projective modules

Let R be a two-sided noetherian ring. Auslander and Bridger developed a theory of projective stabilization of the category of finitely generated R-modules, which is called the stable module theory. Recently, Yoshino established a stable ''complex'' theory, i.e., a theory of a certain stabilization of the homotopy category of complexes of finitely generated projective R-modules. We introduce higher versions of several notions introduced by Yoshino, such as *torsionfreeness and *reflexivity. Also, we prove the Auslander-Bridger approximation theorem for complexes of finitely generated projective R-modules.

math.AC↗

arXiv2023

On the vanishing of Ext modules over a local unique factorization domain with an isolated singularity

This paper provides a method to get a noetherian equicharacteristic local UFD with an isolated singularity from a given noetherian complete equicharacteristic local ring, preserving certain properties. This is applied to invesitgate the (non)vanishing of Ext modules. It is proved that there exist a Gorenstein local UFD $A$ having an isolated singularity such that $\operatorname{Ext}_A^{\gg0}(M,N)=0$ does not imply $\operatorname{Ext}_A^{\gg0}(N,M)=0$, a Gorenstein local UFD $B$ having an isolated singularity such that $\operatorname{Tor}_{>0}^B(M,N)=0$ does not imply $\operatorname{depth}(M\otimes_B N)=\operatorname{depth} M+\operatorname{depth} N-\operatorname{depth} B$, and a Cohen-Macaulay local UFD $C$ having an isolated singularity such that $\operatorname{Ext}_C^{>0}(M,C)=0$ does not imply the total reflexivity of $M$.

math.AC↗

arXiv2023

Ext modules related to syzygies of the residue field

Let R be a commutative noetherian ring. In this paper, we find out close relationships between the module M being embedded in a module of projective dimension at most n and the (n+1)-torsionfreeness of the nth syzygy of M. As an application, when R is local with residue field k, we compute the dimensions as k-vector spaces of Ext modules related to syzygies of k.

math.AC↗

arXiv2022

Maximal Cohen-Macaulay tensor products and vanishing of Ext modules

In this paper, we investigate the maximal Cohen-Macaulay property of tensor products of modules, and then give criteria for projectivity of modules in terms of vanishing of Ext modules. One of the applications shows that the Auslander-Reiten conjecture holds for Cohen-Macaulay normal rings.

math.AC↗

arXiv2022

Stable categories of spherical modules and torsionfree modules

Auslander and Bridger introduced the notions of n-spherical modules and n-torsionfree modules. In this paper, we construct an equivalence between the stable category of n-spherical modules and the category of modules of grade at least n, and provide its Gorenstein analogue. As an application, we prove that if R is a Gorenstein local ring of Krull dimension d>0, then there exists a stable equivalence between the category of (d-1)-torsionfree R-modules and the category of d-spherical modules relative to the local cohomology functor.

math.RT↗

arXiv2022

Morphisms represented by monomorphisms with n-torsionfree cokernel

We introduce and study a new class of morphisms which includes morphisms represented by monomorphisms in the sense of Auslander and Bridger. As an application, we give not only an extension of Kato's theorem on morphisms represented by monomorphisms, but also a common generalization of several results due to Auslander and Bridger that describe relationships between torsionfreeness and the grades of Ext modules.

math.RT↗