Related papers: Mutating ordered $\tau$-rigid modules with applica…

Mutation of compact silting objects is a fundamental operation in the representation theory of finite-dimensional algebras due to its connections to cluster theory and to the lattice of torsion pairs in module or derived categories. In this…

We study universal mapping properties of $(\sigma,\tau)$-derivations over commutative algebras and characterize them over rings of integers of quadratic number fields. As a result we provide extension of some well known results on UFD's of…

Let $\mathcal{D}$ be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with a rigid object $R$. Let $\Lambda=\operatorname{End}_{\mathcal{D}}R$ be the endomorphism algebra of $R$. We introduce the notion of mutation of maximal…

We study (support) $\tau$-tilting modules over the trivial extensions of finite dimensional algebras. More precisely, we construct two classes of (support)$\tau$-tilting modules in terms of the adjoint functors which extend and generalize…

We introduce the new concept of silting modules. These modules generalise tilting modules over an arbitrary ring, as well as support $\tau$-tilting modules over a finite dimensional algebra recently introduced by Adachi, Iyama and Reiten.…

The notion of a $\tau$-exceptional sequence was introduced by Buan and Marsh in 2018 as a generalisation of an exceptional sequence for finite dimensional algebras. We calculate the number of complete $\tau$-exceptional sequences over…

For a finite dimensional hereditary algebra, we consider: exceptional sequences in the category of finite dimensional modules, silting objects in the bounded derived category, and m-cluster tilting objects in the m-cluster category. There…

We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. In particular, we study which $\tau$-rigid C-modules are also $\tau$-rigid B-modules.

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…

We prove a theorem which gives a bijection between the support $\tau$-tilting modules over a given finite-dimensional algebra $A$ and the support $\tau$-tilting modules over $A/I$, where $I$ is the ideal generated by the intersection of the…

Let $k$ be an algebraically closed field. Let $R$ be a local commutative finite dimensional $k$-algebra and let $Q$ be a quiver with no loops or oriented cycles. We show that mutation of $\tau$-exceptional sequences over $\Lambda =…

In representation theory of algebras the notion of `mutation' often plays important roles, and two cases are well known, i.e. `cluster tilting mutation' and `exceptional mutation'. In this paper we focus on `tilting mutation', which has a…

We first introduce the notion of $CM$-$\tau$-tilting free algebras as the generalization of $CM$-free algebras and show the homological properties of $CM$-$\tau$-tilting free algebras. Then we give a bijection between Gorenstein projective…

We introduce weak exceptional sequence of modules which can be viewed as another modification of the standard case, different than the works of Igusa-Todorov \cite{Igusa-Todorov} and Buan-Marsh \cite{Buan-Marsh}. For hereditary algebras it…

We study projective presentations of finite-dimensional modules over finite-dimensional algebras. We discuss if projective presentations of maximal rank behave additively. More precisely, we ask if the direct sum of copies of a projective…

We study the relation between simple-minded systems and two-term tilting complexes for self-injective Nakayama algebras. More precisely, we show that any simple-minded system of a self-injective Nakayama algebra is the image of the set of…

We study the extended module category $m\operatorname{-mod}\Lambda$ of an algebra $\Lambda$, recently introduced by Gupta and Zhou. The existence of a postprojective component of $m\operatorname{-mod}\Lambda$ is shown for certain algebras…

Let $B$ be an one-point extension of a finite dimensional $k$-algebra $A$ by a simple $A$-module at a source point $i$. In this paper, we classify the $\tau$-tilting modules over $B$. Moreover, it is shown that there are equations $$|\tilt…

Motivated by $\tau$-tilting theory developed by Adachi, Iyama and Reiten, for a finite-dimensional algebra $\Lambda$ with action by a finite group $G$, we introduce the notion of $G$-stable support $\tau$-tilting modules. Then we establish…

Tachikawa's second conjecture predicts that a finitely generated, orthogonal module over a finite-dimensional self-injective algebra is projective. This conjecture is an important part of the Nakayama conjecture. Our principal motivation of…