Related papers: Support $\tau$-tilting modules over one-point exte…

Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E, and let M be a ${\tau}$-rigid C-module with U its Bongartz ${\tau}$-complement. If the induced module, $M{\otimes_C}B$, is ${\tau}$-rigid as a…

Let $A$ be a hereditary algebra over an algebraically closed field $k$ and $A^{(m)}$ be the $m$-replicated algebra of $A$. Given an $A^{(m)}$-module $T$, we denote by $\delta (T)$ the number of non isomorphic indecomposable summands of $T$.…

Let $(\mathcal{B},\mathcal{A}, i, e, l)$ be a cleft extension of abelian categories. We prove that the functor $l$ preserves and reflects (Wakamatsu) tilting pairs of subcategories under certain conditions, unifying an abundance of known…

We study completely positive module maps on $C^{*}$-algebras which are $C^*$-module over another $C^*$-algebra with compatible actions. We extend several well known dilation and extension results to this setup, including the Stinespring…

Mutation of {\tau}-tilting modules is a basic operation to construct a new support {\tau}-tilting module from a given one by replacing a direct summand. The aim of this paper is to give a positive answer to the question posed in [AIR,…

Let $A$ and $B$ be Banach algebras and let $B$ be an algebraic Banach $A-$bimodule. Then the $\ell^1-$direct sum $A\times B$ equipped with the multiplication $$(a_1,b_1)(a_2,b_2)=(a_1a_2,a_1\cdot b_2+b_1\cdot a_2+b_1b_2),~~ (a_1, a_2\in A,…

Let $\tilde{G}$ be a finite group and $G$ a normal subgroup of $\tilde{G}$. In this paper, we give a necessary and sufficient condition for $\mathrm{Ind}_G^{\tilde{G}}M$ to be a support $\tau$-tilting $k\tilde{G}$-module for a $kG$-module…

Every cluster-tilted algebra $B$ is the relation extension $C\ltimes \text{Ext}^2_C(DC,C)$ of a tilted algebra $C$. A $B$-module is called induced if it is of the form $M\otimes_C B$ for some $C$-module $M$. We study the relation between…

We introduce the notions of Gorenstein projective $\tau$-rigid modules, Gorenstein projective support $\tau$-tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi-Iyama-Reiten's bijection theorem on support…

First, we study recollement of a derived category of unbounded complexes of modules induced by a partial tilting complex. Second, we give equivalent conditions for P^{centerdot} to be a recollement tilting complex, that is, a tilting…

We find a relationship between the global dimension of an algebra $A$ and the global dimension of the endomorphism algebra of a $\tau$-tilting module, when $A$ is of finite global dimension. We show that, in general, the global dimension of…

The notion of (semi)bricks, regarded as a generalization of (semi)simple modules, appeared in a paper of Ringel in 1976. In recent years, there has been several new developments motivated by links to {\tau}-tilting theory studied by…

We prove that if $B\subseteq A$ is an extension of finite dimensional algebras such that the projective dimension of $A/B$ as a $B$-bimodule is finite, if $A$ has finite finitistic dimension, then so does $B$. We exhibit examples…

Let A be an augmented algebra over a semi-simple algebra S. We show that the Ext algebra of S as an A-module, enriched with its natural A-infinity structure, can be used to reconstruct the completion of A at the augmentation ideal. We use…

Let $A$ be an amenable separable \CA and $B$ be a non-unital but $\sigma$-unital simple \CA with continuous scale. We show that two essential extensions $\tau_1$ and $\tau_2$ of $A$ by $B$ are approximately unitarily equivalent if and only…

In this paper, we explain a strategy on $g$-vectors to discover some new minimal $\tau$-tilting infinite two-point algebras. Consequently, the $\tau$-tilting finiteness of various two-point monomial algebras, including all radical cube zero…

We give several examples of tilting-discrete symmetric algebras; in particular, one explores which algebra has tilting-discrete trivial extension. We provide a counter example of the conjecture stating any {\tau} -tilting finite symmetric…

In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras.…

Let $\Lambda^r_n$ be the path algebra of the linearly oriented quiver of type $\mathbb{A}$ with $n$ vertices modulo the $r$-th power of the radical, and let $\widetilde{\Lambda}^r_n$ be the path algebra of the cyclically oriented quiver of…

For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for Hom($X,\tau Y$). We use this to describe support $\tau$-tilting modules for $A$. We give a combinatorial realization of maps in both directions…