arXiv subjects

Wuzhong Yang

Publications and source records attributed to Wuzhong Yang.

3 recordsLinked to original sources

arXiv2023

Largest exact structures and almost split sequences on hearts of twin cotorsion pairs

Hearts of cotorsion pairs on extriangulated categories are abelian categories. On the other hand, hearts of twin cotorsion pairs are not always abelian. They were shown to be semi-abelian by Liu and Nakaoka. Moreover, Hassoun and Shah proved that they are quasi-abelian under certain conditions. In this article, we first show that the heart of any twin cotorsion pair has a largest exact category structure and is always quasi-abelian. We also provide a sufficient and necessary condition for the heart of a twin cotorsion pair being abelian. Then by using the results we have got, we investigate the almost split sequences in the hearts of twin cotorsion pairs. Finally, as an application, we show that a Krull-Schmidt, Hom-finite triangulated category has a Serre functor whenever it has a cluster tilting object.

math.RT↗

arXiv2017

Triangulated categories with cluster-tilting subcategories

Let $\C$ be a triangulated category with a cluster tilting subcategory $\T$. We introduce the notion of $\T[1]$-cluster tilting subcategories (also called ghost cluster tilting subcategories) of $\C$, which are a generalization of cluster tilting subcategories. We first develop a basic theory on ghost cluster tilting subcategories. Secondly, we study links between ghost cluster tilting theory and $τ$-tilting theory: Inspired by the work of Iyama, Jørgensen and Yang \cite{ijy}, we introduce the notion of $τ$-tilting subcategories and tilting subcategories of $\mod\T$. We show that there exists a bijection between weak $\T[1]$-cluster tilting subcategories of $\C$ and support $τ$-tilting subcategories of $\mod\T$. Moreover, we figure out the subcategories of $\mod\T$ which correspond to cluster tilting subcategories of $\C$. This generalizes and improves several results by Adachi-Iyama-Reiten \cite{AIR}, Beligiannis \cite{Be2}, and Yang-Zhu \cite{YZ}. Finally, we prove that the definition of ghost cluster tilting objects is equivalent to the definition of relative cluster tilting objects introduced by the first and the third author in \cite{YZ}.

math.RT↗

arXiv2017

Relative cluster tilting objects in triangulated categories

Assume that $\D$ is a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object $T$. We introduce the notion of relative cluster tilting objects, and $T[1]$-cluster tilting objects in $\D$, which are a generalization of cluster-tilting objects. When $\D$ is $2$-Calabi-Yau, the relative cluster tilting objects are cluster-tilting. Let $\la={\rm End}^{op}_{\D}(T)$ be the opposite algebra of the endomorphism algebra of $T$. We show that there exists a bijection between $T[1]$-cluster tilting objects in $\D$ and support $τ$-tilting $\la$-modules, which generalizes a result of Adachi-Iyama-Reiten \cite{AIR}. We develop a basic theory on $T[1]$-cluster tilting objects. In particular, we introduce a partial order on the set of $T[1]$-cluster tilting objects and mutation of $T[1]$-cluster tilting objects, which can be regarded as a generalization of `cluster-tilting mutation'. As an application, we give a partial answer to a question posed in \cite{AIR}.

math.RT↗