arXiv subjects

Yasuaki Ogawa

Publications and source records attributed to Yasuaki Ogawa.

11 recordsLinked to original sources

arXiv2026

Exact dg structures are classified intrinsically by bi-Serre subcategories

The aim of this article is to leverage Enomoto's classification of exact structures to the dg level. Let $\mathscr{A}$ be a connective additive idempotent complete dg category. We establish an intrinsic Enomoto-type classification of exact dg structures on $\mathscr{A}$ in terms of bi-Serre subcategories in $\mathsf{tr}(\mathscr{A})$. Our proof is a direct dg argument based on dg totalizations of $3$-term complexes and dg duality, rather than a reduction to the greatest exact dg structure and the classification of its substructures. As a consequence, we establish a bijection between exact dg structures on $\mathscr{A}$ and bi-Serre subcategories in $\mathsf{tr}(\mathscr{A})$. Moreover, this bijection is an isomorphism of posets and yields a complete lattice structure on the class of exact dg structures on $\mathscr{A}$. In particular, we provide another construction of Rump--Chen's greatest exact dg structure on $\mathscr{A}$.

math.RT↗

arXiv2026

Extended heart construction (I): The heart of $n$-cotorsion pairs on triangulated categories

The heart of a $t$-structure and the ideal quotient by a cluster tilting subcategory are classical constructions that produce abelian categories from triangulated categories. Their higher analogues, namely $n$-extended hearts and ideal quotient categories by $(n+1)$-cluster tilting subcategories, are generally no longer abelian, but are known to carry both pretriangulated and extriangulated structures when the underlying triangulated category is algebraic. In this article, we introduce the notion of an abelian $n$-truncated category as a common framework for such higher constructions. We extend the heart construction for cotorsion pairs to $n$-cotorsion pairs on arbitrary triangulated categories, and prove that the resulting extended heart naturally carries compatible pretriangulated and extriangulated structures forming an abelian $n$-truncated category. This construction simultaneously generalizes the $n$-extended heart of a $t$-structure and the ideal quotient by an $(n+1)$-cluster tilting subcategory. It may also be regarded as a higher-dimensional generalization of the general heart construction for cotorsion pairs on triangulated categories. Finally, we show that the heart can be realized as an extriangulated localization of a suitable relative extriangulated structure on the ambient triangulated category.

math.CT↗

arXiv2026

Weak Waldhausen categories and a localization theorem

Waldhausen categories were introduced to extend algebraic $K$-theory beyond Quillen's exact categories. In this article, we modify Waldhausen's axioms so that it matches better with the theory of extriangulated categories, introducing a weak Waldhausen category and defining its Grothendieck group. Examples of weak Waldhausen categories include any extriangulated category, hence any exact or triangulated category, and any Waldhausen category. A key feature of this structure is that it allows for "one-sided" extriangulated localization theory, and thus enables us to extract right exact sequences of Grothendieck groups that we cannot obtain from the theory currently available. To demonstrate the utility of our Weak Waldhausen Localization Theorem, we give three applications. First, we give a new proof of the Extriangulated Localization Theorem proven by Enomoto--Saito, which is a generalization at the level of $K_0$ of Quillen's classical Localization Theorem for exact categories. Second, we give a new proof that the index with respect to an $n$-cluster tilting subcategory $\mathscr{X}$ of a triangulated category $\mathscr{C}$ induces an isomorphism between $K_0^{\mathsf{sp}}(\mathscr{X})$ and the Grothendieck group of an extriangulated substructure of $\mathscr{C}$. Last, we produce a weak Waldhausen $K_0$-generalization of a localization construction due to Sarazola that involves cotorsion pairs but allows for non-Serre localizations. We show that the right exact sequences of Grothendieck groups obtained from our Sarazola construction and the Extriangulated Localization Theorem agree under a common setup.

math.KT↗

arXiv2025

An enhanced extriangulated subquotient

Bondal-Kapranov's notion of enhanced triangulated categories behaves well in the framework of localization theory, in the sense that the Verdier quotient of triangulated categories can be lifted to the Drinfeld dg quotient of pretriangulated dg categories. In this paper, we develop a parallel enhancement for Nakaoka-Palu's notion of extriangulated categories, which unifies exact and triangulated categories. The enhancement of extriangulated categories was recently initiated by Xiaofa Chen under the name exact dg categories. Moreover, it is known that certain ideal quotients of extriangulated categories remain extriangulated, and that such ideal quotients admit dg enhancements via the dg quotient of the corresponding connective exact dg category. Motivated by Chen's framework of enhanced extriangulated categories, we introduce the concept of a cohomological envelope of an exact dg category and generalize his construction of the enhanced ideal quotient. We show that the dg quotient of exact dg categories, when passing to cohomological envelopes and their substructures -- referred to as exact dg subquotients -- is compatible with a broad class of extriangulated quotients in the sense of Nakaoka-Ogawa-Sakai. To further clarify the scope of our approach, we formulate the notion of an extriangulated subquotient, which enables the localization of any extriangulated category by extension-closed subcategories. This construction encompasses not only ideal and Verdier quotients, but also the quotient of an exact category by a biresolving subcategory. Notably, the extriangulated subquotient admits a natural lifting to the exact dg subquotient.

math.CT↗

arXiv2024

A resolution theorem for extriangulated categories with applications to the index

Quillen's Resolution Theorem in algebraic $K$-theory provides a powerful computational tool for calculating $K$-groups of exact categories. At the level of $K_0$, this result goes back to Grothendieck. In this article, we first establish an extriangulated version of Grothendieck's Resolution Theorem. Second, we use this Extriangulated Resolution Theorem to gain new insight into the index theory of triangulated categories. Indeed, we propose an index with respect to an extension-closed subcategory $\mathscr{N}$ of a triangulated category $\mathscr{C}$ and we prove an additivity formula with error term. Our index recovers the index with respect to a contravariantly finite, rigid subcategory $\mathscr{X}$ defined by Jørgensen and the second author, as well as an isomorphism between $K_0^{\mathsf{sp}}(\mathscr{X})$ and the Grothendieck group of a relative extriangulated structure $\mathscr{C}_{R}^{\mathscr{X}}$ on $\mathscr{C}$ when $\mathscr{X}$ is $n$-cluster tilting. In addition, we generalize and enhance some results of Fedele. Our perspective allows us to remove certain restrictions and simplify some arguments. Third, as another application of our Extriangulated Resolution Theorem, we show that if $\mathscr{X}$ is $n$-cluster tilting in an abelian category, then the index introduced by Reid gives an isomorphism $K_0(\mathscr{C}_R^{\mathscr{X}}) \cong K_0^{\mathsf{sp}}(\mathscr{X})$.

math.KT↗

arXiv2024

Localization of triangulated categories with respect to extension-closed subcategories

The aim of this paper is to develop a framework for localization theory of triangulated categories $\mathcal{C}$, that is, from a given extension-closed subcategory $\mathcal{N}$ of $\mathcal{C}$, we construct a natural extriangulated structure on $\mathcal{C}$ together with an exact functor $Q:\mathcal{C}\to\widetilde{\mathcal{C}}_\mathcal{N}$ satisfying a suitable universality, which unifies several phenomena. Precisely, a given subcategory $\mathcal{N}$ is thick if and only if the localization $\widetilde{\mathcal{C}}_\mathcal{N}$ corresponds to a triangulated category. In this case, $Q$ is nothing other than the usual Verdier quotient. Furthermore, it is revealed that $\widetilde{\mathcal{C}}_\mathcal{N}$ is an exact category if and only if $\mathcal{N}$ satisfies a generating condition $\mathsf{cone}(\mathcal{N},\mathcal{N})=\mathcal{C}$. Such an (abelian) exact localization $\widetilde{\mathcal{C}}_\mathcal{N}$ provides a good understanding of some cohomological functors $\mathcal{C}\to\mathsf{Ab}$, e.g., the heart of $t$-structures on $\mathcal{C}$ and the abelian quotient of $\mathcal{C}$ by a cluster-tilting subcategory $\mathcal{N}$.

math.CT↗

arXiv2022

Abelian categories from triangulated categories via Nakaoka-Palu's localization

The aim of this paper is to provide an expansion to Abe-Nakaoka's heart construction of the following two different realizations of the module category over the endomorphism ring of a rigid object in a triangulated category: Buan-Marsh's localization and Iyama-Yoshino's subfactor. Our method depends on a modification of Nakaoka-Palu's HTCP localization, a Gabriel-Zisman localization of extriangulated categories which is also realized as a subfactor of the original ones. Besides of the heart construction, our generalized HTCP localization involves the following phenomena: (1) stable category with respect to a class of objects; (2) recollement of triangulated categories; (3) recollement of abelian categories under a mild assumption.

math.CT↗

arXiv2022

Auslander's defects over extriangulated categories: an application for the General Heart Construction

The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization of exact categories and triangulated categories. Our first aim is to provide an extension to extriangulated categories of Auslander's formula: for some extriangulated category $\mathcal{C}$, there exists a localization sequence $\operatorname{\mathsf{def}}\mathcal{C}\to\operatorname{\mathsf{mod}}\mathcal{C}\to\operatorname{\mathsf{lex}}\mathcal{C}$, where $\operatorname{\mathsf{lex}}\mathcal{C}$ denotes the full subcategory of finitely presented left exact functors and $\operatorname{\mathsf{def}}\mathcal{C}$ the full subcategory of Auslander's defects. Moreover we provide a connection between the above localization sequence and the Gabriel-Quillen embedding theorem. As an application, we show that the general heart construction of a cotorsion pair $(\mathcal{U},\mathcal{V})$ in a triangulated category, which was provided by Abe and Nakaoka, is same as the construction of a localization sequence $\operatorname{\mathsf{def}}\mathcal{U}\to\operatorname{\mathsf{mod}}\mathcal{U}\to\operatorname{\mathsf{lex}}\mathcal{U}$.

math.RT↗

arXiv2021

Localization of extriangulated categories

In this article, we show that the localization of an extriangulated category by a multiplicative system satisfying mild assumptions can be equipped with a natural, universal structure of an extriangulated category. This construction unifies the Serre quotient of abelian categories and the Verdier quotient of triangulated categories. Indeed we give such a construction for a bit wider class of morphisms, so that it covers several other localizations appeared in the literature, such as Rump's localization of exact categories by biresolving subcategories, localizations of extriangulated categories by means of Hovey twin cotorsion pairs, and the localization of exact categories by two-sided admissibly percolating subcategories.

math.CT↗

arXiv2019

Singular equivalences of functor categories via Auslander-Buchweitz approximations

The aim of this paper is to construct singular equivalences between functor categories. As a special case, we show that there exists a singular equivalence arising from a cotilting module $T$, namely, the singularity category of $(^\perp T)/[T]$ and that of $(\mod A)/[T]$ are triangle equivalent. In particular, the canonical module $ω$ over a commutative Noetherian ring induces a singular equivalence between $(\mathsf{CM}R)/[ω]$ and $(\mod R)/[ω]$, which generalizes Matsui-Takahashi's theorem. Our result is based on a sufficient condition for an additive category $\mathcal{A}$ and its subcategory $\mathcal{X}$ so that the canonical inclusion $\mathcal{X}\hookrightarrow\mathcal{A}$ induces a singular equivalence $\mathsf{D_{sg}}(\mathcal{A})\simeq \mathsf{D_{sg}}(\mathcal{X})$, which is a functor category version of Xiao-Wu Chen's theorem.

math.CT↗

arXiv2017

Recollements for dualizing $k$-varieties and Auslander's formulas

Given the pair of a dualizing $k$-variety and its functorially finite subcategory, we show that there exists a recollement consisting of their functor categories of finitely presented objects. We provide several applications for Auslander's formulas: The first one realizes a module category as a Serre quotient of a suitable functor category. The second one shows a close connection between Auslander-Bridger sequences and recollements. The third one gives a new proof of the higher defect formula which includes the higher Auslander-Reiten duality as a special case.

math.CT↗