Related papers: On $\tau$-tilting theory
From the viewpoint of mutation, we will give a brief survey of tilting theory and cluster-tilting theory together with a motivation from cluster algebras. Then we will give an introdution to \tau-tilting theory which was recently developed…
We give an introduction to the theory of cluster categories and cluster tilted algebras. We include some background on the theory of cluster algebras, and discuss the interplay with cluster categories and cluster tilted algebras.
The notion of $\tau$-tilting theory was introduced by Adachi, Iyama and Reiten at the beginning of the last decade and quickly became one of the most active areas of research in the representation theory of finite dimensional algebras. The…
A self-contained introduction to the basics of Tau-tilting theory. We assume that the reader is familiar with Auslander-Reiten theory, but circumvent the need for the Brenner-Butler tilting theorem completely.
The aim of this paper is to introduce tau-tilting theory, which completes (classical) tilting theory from the viewpoint of mutation. It is well-known in tilting theory that an almost complete tilting module for any finite dimensional…
We introduce the notion of AIR tilting subcategories of extended hearts of $t$-structures on a triangulated category associated with silting subcategories. This notion generalizes $\tau_{[d]}$-tilting pairs of extended finitely generated…
Comparing the module categories of an algebra and of the endomorphism algebra of a given support $\tau$-tilting module, we give a generalization of the Brenner-Butler's tilting theorem in the framework of $\tau$-tilting theory. Afterwards…
Let $\mathcal C$ be a Krull-Schmidt triangulated category with shift functor $[1]$ and $\mathcal R$ be a rigid subcategory of $\mathcal C$. We are concerned with the mutation of two-term weak $\mathcal R[1]$-cluster tilting subcategories.…
We introduce the notion of (strongly) $n$-AIR-tilting modules, which is a high dimension version of support $\tau$-tilting modules. The relations between them and $n$-silting modules and $n$-quasi-tilting modules, as well as generalized…
The theory of $\tau$-tilting was introduced by Adachi--Iyama--Reiten; one of the main results is a bijection between support $\tau$-tilting modules and torsion classes. We are able to generalise this result in the context of the higher…
$\tau$-tilting theory can be thought of as a generalization of the classical tilting theory which allows mutations at any indecomposable summand of a support $\tau$-tilting pair. Indeed, for any algebra $\Lambda$ its tilting modules…
We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation…
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…
The main theme of this paper is to study $\tau$-tilting subcategories in an abelian category $\mathscr{A}$ with enough projective objects. We introduce the notion of $\tau$-cotorsion torsion triples and show a bijection between the…
As the first attempt to classify $\tau$-tilting finite two-point algebras, we have determined the $\tau$-tilting finiteness for minimal wild two-point algebras and some tame two-point algebras.
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…
In this short note, we explain how the main results in "$\tau$-tilting theory" by Adachi-Iyama-Reiten follow from the results in Section 5 of "General presentations of algebras" by Derksen-Fei.
We completely determine the $\tau$-tilting finiteness of Borel-Schur algebras. To achieve this, we use two recently introduced techniques in silting theory: sign decomposition as introduced by Aoki, Higashitani, Iyama, Kase and Mizuno…
Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type $(2, 2, 2, 2)$. More precisely, a tilting object consisting of rank-two bundles is…
Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support {\tau}-tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given…