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Yuki Mifune

Publications and source records attributed to Yuki Mifune.

13 recordsLinked to original sources
arXiv2026

Ranks of Verdier quotients of derived categories over local rings

Let $R$ be a commutative noetherian local ring with residue field $k$. Denote by $\operatorname{\mathsf{D^b}}(R)$ the bounded derived category of finitely generated $R$-modules. In this paper, we introduce the notion of ranks of triangulated categories and study Verdier quotients of $\operatorname{\mathsf{D^b}}(R)$ from the viewpoint of this invariant. Our main result determines the rank of the Verdier quotient $\operatorname{\mathsf{D^b}}(R)/\operatorname{\mathsf{thick}}(R\oplus k)$ in terms of the ranks of the categories of maximal Cohen-Macaulay modules on the punctured spectrum.

math.AC↗
arXiv2026

Contravariantly finite resolving subcategories over commutative local rings are trivial ones

We show that a contravariantly finite resolving subcategory over a henselian local ring must be the category of free modules or the whole module category or the subcategory of maximal Cohen--Macaulay modules. This removes the Gorenstein assumption from a theorem of Takahashi. We also show that a resolving subcategory of finite type other than the subcategory of free modules must be the subcategory of maximal Cohen--Macaulay modules. As a consequence, every Cohen--Macaulay local ring of finite CM type is uniformly dominant, thereby establishing a stronger form of a conjecture of Takahashi.

math.AC↗
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

A characterization of Cohen-Macaulay rings in terms of levels of perfect complexes

Let $R$ be a commutative noetherian ring, and let $C$ be a semidualizing $R$-module. In this paper, we study levels of bounded complexes of finitely generated $R$-modules with respect to the full subcategory $\mathsf{G}_{C}(R)$ consisting of Gorenstein $C$-projective $R$-modules. Our main result provides a characterization of the Cohen-Macaulayness of $R$ in terms of the finiteness of levels of perfect complexes with respect to $\mathsf{G}_{C}(R)$. This recovers a recent theorem of Christensen, Kekkou, Lyle and Soto Levins on the Gorensteinness of $R$.

math.AC↗
arXiv2025

Structure of modules stably annihilated by a fixed ideal

Let $R$ be a commutative noetherian ring, and denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules. In this paper, for an ideal $I$ of $R$, we introduce the full subcategory $\operatorname{mod}_{I}(R)$ of $\operatorname{mod} R$ consisting of modules whose stable annihilators contain $I$, and we investigate its structure. As an application, we explore the syzygy category of maximal Cohen--Macaulay $R$-modules, extending a theorem of Dey and Liu from the Gorenstein case to the Cohen--Macaulay case.

math.AC↗
arXiv2025

Criteria for finite injective dimension of modules over a local ring

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 concerning multiplicity and type, together with the vanishing of finitely many Ext modules. As applications, we recover a result of Rahmani and Taherizadeh and provide sufficient conditions for a finitely generated $R$-module to have finite injective dimension.

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

Lower bounds for levels of complexes by resolution dimensions

Let $\mathcal{A}$ be an abelian category. Denote by $\mathrm{D}^{b}(\mathcal{A})$ the bounded derived category of $\mathcal{A}$. In this paper, we investigate the lower bounds for the levels of objects in $\mathrm{D}^{b}(\mathcal{A})$ with respect to a (co)resolving subcategory satisfying a certain condition. As an application, we not only recover the results of Altmann--Grifo--Monta\~{n}o--Sanders--Vu, and Awadalla--Marley but also extend them to establish lower bounds for levels with respect to some other subcategories in an abelian category.

math.AC↗
arXiv2024

Upper bounds for dimensions of singularity categories and their annihilators

Let $R$ be a commutative noetherian ring. Denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules and by $\operatorname{D^b}(R)$ the bounded derived category of $\operatorname{mod} R$. In this paper, we first investigate localizations and annihilators of Verdier quotients of $\operatorname{D^b}(R)$. After that, we explore upper bounds for the dimension of the singularity category of $R$ and its (strong) generators. We extend a theorem of Liu to the case where $R$ is neither an isolated singularity nor even a local ring. Some of our results are more generally stated in terms of Spanier--Whitehead category of a resolving subcategory.

math.AC↗
arXiv2024

A generalization of the dimension and radius of a subcategory of modules and its applications

Let $R$ be a commutative noetherian local ring and denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules. In this paper, we give some evaluations of the singular locus of $R$ and annihilators of Tor and Ext from a viewpoint of the finiteness of dimensions/radii of full subcategories of $\operatorname{mod} R$. As an application, we recover a theorem of Dey and Takahashi when $R$ is Cohen--Macaulay. Moreover, we obtain the divergence of the dimensions of specific full subcategories of $\operatorname{mod} R$ in non-Cohen--Macaulay case.

math.AC↗
arXiv2023

On a Verdier quotient of a derived category of a local ring

Let $R$ be a commutative noetherian local ring with residue field $k$. Denote by $\mathsf{D^b}(R)$ the bounded derived category of finitely generated $R$-modules. In this paper, we study the structure of the Verdier quotient $\mathsf{D^b}(R)/\mathsf{thick}(R\oplus k)$. We give necessary and sufficient conditions for it to admit an additive generator.

math.AC↗
arXiv2023

On the finiteness of radii of resolving subcategories

Let R be a commutative noetherian ring. Denote by mod R the category of finitely generated R-modules. In this paper, we investigate the finiteness of the radii of resolving subcategories of mod R with respect to a fixed semidualizing module. As an application, we give a partial positive answer to a conjecture of Dao and Takahashi: we prove that for a Cohen-Macaulay local ring R, a resolving subcategory of mod R has infinite radius whenever it contains a canonical module and a non-MCM module of finite injective dimension.

math.AC↗