arXiv subjects

Kaito Kimura

Publications and source records attributed to Kaito Kimura.

17 recordsLinked to original sources

arXiv2026

On injective dimension of the conormal module

Let $Q$ be a Gorenstein local ring, let $I$ be a proper ideal of $Q$, and set $R=Q/I$. We investigate when finite injective dimension of the conormal module $I/I^2$ forces $I$ to be generated by a regular sequence. We prove this implication for every ideal in the linkage class of a complete intersection. When $Q$ is regular, we also establish it under several numerical conditions in terms of the number of generators of $I$ and the type of $R$. For normal domain quotients, we show that finite injective dimension of $I/I^2$ forces $(\mathrm{ht}(I)+1)[ω_R]=0$ in $\operatorname{Cl}(R)$, yielding the desired conclusion when the divisor class group is torsionfree. In the graded setting, we further prove the implication for all monomial ideals in polynomial rings. We also obtain a converse to Kunz's description of the first Koszul homology of an almost complete intersection.

math.AC↗

arXiv2026

Thick subcategories over weakly symmetric algebras with radical cube zero

In this paper, we study thick subcategories of the (stable) module categories of finite-dimensional weakly symmetric algebras with radical cube zero. When the spectral radius of the Ext matrix is two, thick subcategories containing the algebra, equivalently thick subcategories of the stable category, are classified by Serre subcategories of a certain abelian category; when it is not two, only the trivial thick subcategories occur. As an application, this yields, to the best of our knowledge, the first examples of commutative Noetherian Gorenstein dominant local rings that are not complete intersections.

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↗

arXiv2026

Complexes of finite Gorenstein flat and injective dimensions

In this paper, we consider a Gorenstein-dimensional analogue of Foxby's characterization of Gorenstein rings. We prove that a commutative Noetherian local ring is Gorenstein if it admits a complex whose depth, Gorenstein flat dimension, and Gorenstein injective dimension are all finite. This gives an affirmative answer to the original question of Christensen, Foxby, and Holm, which had remained open in this generality even for modules, and at the same time establishes its natural extension to complexes.

math.AC↗

arXiv2026

Trace ideals of canonical modules over Schubert cycles and determinantal rings

In this paper, we study the canonical trace of Schubert cycles and determinantal rings. As an application, we give an explicit description of the non-Gorenstein locus and show that its structure is compatible with the known representations of the singular locus and the canonical module. Furthermore, for the CTR property recently introduced by Miyazaki, we establish its stability under base change and provide a characterization in the case of determinantal rings.

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

Finiteness of homological dimensions of Ext modules

Let $R$ be a commutative Noetherian local ring and let $M$ and $N$ be nonzero finitely generated $R$-modules. In this paper, we investigate how the finiteness of the homological dimension of Ext modules between $M$ and $N$ affects that of $M$ and $N$. One of our main result states that if $\operatorname{Ext}^i_R(M,N)$ has finite projective dimension for any $0\le i\le \operatorname{Rfd}_R M$, where $\operatorname{Rfd}_R M$ is the (large) restricted flat dimension of $M$, then $M$ has finite projective or injective dimension if and only if $N$ does.

math.AC↗

arXiv2025

On the depth of tensor products over Cohen-Macaulay rings

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 Cohen-Macaulay local rings that derived $(\mathbf{ldep})$ is equivalent to $\dim(R)$ being a uniform Auslander bound for $R$, and if $\dim(R)>0$ that both are equivalent to $(\mathbf{ldep})$. We introduce an analogous condition we call the \emph{uniform Buchweitz condition} and provide a corresponding theorem for the $(\mathbf{rdep})$ condition. As a consequence of these results, we show $(\mathbf{ldep})$ implies $(\mathbf{rdep})$ when $R$ is Gorenstein and that the $(\mathbf{ldep})$ and $(\mathbf{rdep})$ conditions behave well under modding out by regular sequences and completion, but we give a concrete example showing they need not localize. Using our methods, we extend work of Jorgensen by calculating the value $q_R(M,N):=\sup\{i \mid \operatorname{Tor}^R_i(M,N) \ne 0\}$ under certain conditions.

math.AC↗

arXiv2025

Trace ideals, conductors, and ideals of finite (phantom) projective dimension

In this paper, we consider whether parameter test ideals, conductors, $F$-ideals, and trace ideals are contained in an ideal whose quotient ring has finite phantom projective dimension (for example, ideals generated by a system of parameters or ideals with finite projective dimension). One of the main results asserts that such inclusions do not exist in quasi-Gorenstein complete local domains. We also provide examples of Cohen-Macaulay local rings with good properties where such inclusions occur, thus answering negatively a question of Huneke-Swanson.

math.AC↗

arXiv2024

Stability of annihilators of cohomology and closed subsets defined by Jacobian ideals

Let $R$ be a commutative Noetherian ring of dimension $d$. In this paper, we first show that some power of the cohomology annihilator annihilates the $(d+1)$-th Ext modules for all finitely generated modules when either $R$ admits a dualizing complex or $R$ is local. Next, we study the Jacobian ideal of affine algebras over a field and equicharacteristic complete local rings, and characterize the equidimensionality of the ring in terms of the singular locus and the closed subsets defined by the cohomology annihilator and the Jacobian ideal.

math.AC↗

arXiv2024

Compactness of the Alexandrov topology of maximal Cohen-Macaulay modules

Let $R$ be a Cohen-Macaulay local ring. In this paper, we first describe the radicals of annihilators of stable categories of maximal Cohen-Macaulay $R$-modules. We then prove that the Alexandrov topology of the stable category of maximal Cohen-Macaulay $R$-modules is compact provided that the completion of $R$ has an isolated singularity. Finally, we consider the case of a hypersurface of countable CM-representation type.

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

Asymptotic behavior of homological invariants of localizations of modules

Let $R$ be a commutative noetherian ring, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. We consider the asymptotic injective dimensions, projective dimensions, Bass numbers, and Betti numbers of localizations of $M/I^n M$ at prime ideals of $R$ and prove that these invariants are stable or have polynomial growth for large integers $n$ that do not depend on the prime ideals.

math.AC↗

arXiv2023

Auslander--Reiten conjecture for normal rings

In this paper, sufficient conditions for finitely generated modules over a commutative noetherian ring to be projective are given in terms of vanishing of Ext modules. One of the main results of this paper asserts that the Auslander--Reiten conjecture holds true for every normal ring.

math.AC↗

arXiv2022

Asymptotic stability of depths of localizations of modules

Let R be a commutative noetherian ring, I an ideal of R, and M a finitely generated R-module. The asymptotic behavior of the quotient modules M/I^n M of M is an actively studied subject in commutative algebra. The main result of this paper asserts that the depth of the localization of M/I^n M at any prime ideal of R is stable for large integers n that do not depend on the prime ideal, if the module M or M/I^n M is Cohen-Macaulay for some n>0, or the ring R is one of the following: a homomorphic image of a Cohen-Macaulay ring, a semi-local ring, an excellent ring, a quasi-excellent and catenary ring, and an acceptable ring.

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

Openness of various loci over Noetherian rings

In this paper, we consider the openness of the P-locus of a finitely generated module over a commutative noetherian ring in the case where P is each of the properties FID, Gor, CM, MCM, (S_n), and (T_n). One of the main results asserts that FID-loci over an acceptable ring are open. We give a module version of the Nagata criterion, and prove that it holds for all of the aforementioned properties.

math.AC↗