arXiv subjects

Yaohua Zhang

Publications and source records attributed to Yaohua Zhang.

7 recordsLinked to original sources

arXiv2026

Triangulated categories with a compact silting object, Brown-Comenetz duality and Brown representability theorems

The paper develops a Brown--Comenetz dual framework for Neeman's representability theorems for triangulated categories with a single compact generator (Invent. math., 244:531-616, 2026). Starting from a locally Hom-finite approximable triangulated category, we use the Brown--Comenetz duals of compact objects to construct a triangulated subcategory $\E$, which plays the role of an injective-side analogue of the compact subcategory $\T^c$. We introduce the intrinsic subcategory $\T_c^+$, dual to Neeman's subcategory $\T_c^-$, and characterize its objects by strong $\E$-coapproximating systems and homotopy inverse limits. Under the compact silting hypothesis, we prove Brown representability theorems identifying $(\T_c^+)^{\op}$ with locally finite $\E$-homological functors and $(\T_c^b)^{\op}$ with finite $\E$-homological functors. We also establish localization results for recollements on the Brown--Comenetz side and derive applications to derived categories of finite-dimensional algebras.

math.RT↗

arXiv2026

Localization theorems for weakly approximable triangulated categories

Weakly approximable triangulated categories, introduced by Neeman, provide a powerful framework for studying localization phenomena in triangulated categories. In this paper, we establish new localization theorems showing that, under mild assumptions, a recollement of weakly approximable triangulated categories induces short exact sequences on several natural triangulated subcategories as well as on the associated (big) singularity categories. As applications, we illustrate our results in the derived categories of rings, DG algebras, and schemes.

math.RT↗

arXiv2025

N-fold module factorizations: triangle equivalences and recollements

As an extension of Eisenbud's matrix factorization into the non-commutative realm, X.W. Chen introduced the concept of module factorizations over an arbitrary ring. A theorem of Chen establishes a triangle equivalence between the stable category of module factorizations with Gorenstein projective components and the stable category of Gorenstein projective modules over a quotient ring. In this paper, we introduce $n$-fold module factorizations, which generalize both the commutative $n$-fold matrix factorizations and the non-commutative module factorizations. To adapt triangle equivalences in module factorizations to $n$-fold module factorizations, we identify suitable subcategories of module factorizations and rings for the $n$-analogue. We further provide the $n$-analogue of Chen's theorem on triangle equivalences. Additionally, we study recollements involving the stable categories of higher-fold module factorizations, revealing intriguing recollements within the stable categories of Gorenstein modules of specific matrix subrings.

math.RA↗

arXiv2025

A note on gluing cosilting objects

Based on the recent works of M. Saorin and A. Zvonoreva on gluing (co)silting objects and of L. Angeler Hugel, R. Laking, J. Stovicek and J. Vitoria on mutating (co)silting objects, we first study further on gluing pure-injective cosilting objects in algebraically compactly generated triangulated categories and gluing cosilting complexes in the derived categories of rings. Then we discuss the compatibility of cosilting gluing and cosilting mutation.

math.RT↗

arXiv2024

Gluing simple-minded collections in triangulated categories

We provide a technique to glue simple-minded collections along a recollement of Hom-finite Krull-Schmidt triangulated categories over a field. This gluing technique for simple-minded collections is shown to be compatible with those for gluing bounded $t$-structures, silting objects, and co-$t$-structures in the literature. Furthermore, it also enjoys the properties of preserving partial order and commuting with the operation of mutation.

math.RT↗

arXiv2024

Support $τ$-tilting subcategories in exact categories

Let $\mathcal{E}=(\mathcal{A},\mathcal{S})$ be an exact category with enough projectives $\mathcal{P}$. We introduce the notion of support $τ$-tilting subcategories of $\mathcal{E}$. It is compatible with existing definitions of support $τ$-tilting modules (subcategories) in various context. It is also a generalization of tilting subcategories of exact categories. We show that there is a bijection between support $τ$-tilting subcategories and certain $τ$-cotorsion pairs. Given a support $τ$-tilting subcategory $\mathcal{T}$, we find a subcategory $\mathcal{E}_{\mathcal{T}}$ of $\mathcal{E}$ which is an exact category and $\mathcal{T}$ is a tilting subcategory of $\mathcal{E}_{\mathcal{T}}$. If $\mathcal{E}$ is Krull-Schmidt, we prove the cardinal $|\mathcal{T}|$ is equal to the number of isomorphism classes of indecomposable projectives $Q$ such that ${\rm Hom}_{\mathcal{E}}(Q,\mathcal{T})\neq 0$. We also show a functorial version of Brenner-Butler's theorem.

math.RT↗

arXiv2024

Ideal mutations in triangulated categories and generalized Auslander-Reiten theory

We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino \cite{iyama2008mutation} by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom-finite Krull-Schmidt triangulated category $\mathcal{T}$ over an algebraically closed field $K$. (1) We generalize a theorem of Jorgensen \cite[Theorem 3.3]{jorgensen2010quotients} to a more general setting; (2) We provide a method to detect whether $\mathcal{T}$ has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical $\mathcal{J}$: (i) $\mathcal{J}$ is functorially finite, (ii) Gh$_{\mathcal{J}}= {\rm CoGh}_{\mathcal{J}}$, and (iii) Gh$_{\mathcal{J}}$-source maps coincide with Gh$_{\mathcal{J}}$-sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations.

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