Related papers: Classifying tilting modules over the Auslander alg…

Let $\Lambda$ be a radical square zero algebra of a Dynkin quiver and let $\Gamma$ be the Auslander algebra of $\Lambda$. Then the number of tilting right $\Gamma$-modules is $2^{m-1}$ if $\Lambda$ is of $A_{m}$ type for $m\geq 1$.…

We construct two functors from the submodule category of a self-injective representation-finite algebra $\Lambda$ to the module category of the stable Auslander algebra of $\Lambda$. These functors factor through the module category of the…

For a finite dimensional algebra $\Lambda$ and a non-negative integer $n$, we characterize when the set $\tilt_n\Lambda$ of additive equivalence classes of tilting modules with projective dimension at most $n$ has a minimal (or…

We give a characterization of $\tau$-rigid modules over Auslander algebras in terms of projective dimension of modules. Moreover, we show that for an Auslander algebra $\Lambda$ admitting finite number of non-isomorphic basic tilting…

In this paper, we show that for an algebra $\Lambda$ with radical square zero and an indecomposable $\Lambda$-module $M$ such that $\Lambda$ is Gorenstein of finite type or $\tau M$ is $\tau$-rigid, $M$ is $\tau$-rigid if and only if the…

We show that for a given Nakayama algebra $\Theta$, there exist countably many cyclic Nakayama algebras $\Lambda_i$, where $i \in \mathbb{N}$, such that the syzygy filtered algebra of $\Lambda_i$ is isomorphic to $\Theta$ and we describe…

Let $\Lambda$ be an Auslander's 1-Gorenstein Artinian algebra with global dimension two. If $\Lambda$ admits a trivial maximal 1-orthogonal subcategory of $\mod\Lambda$, then for any indecomposable module $M \in \mod \Lambda$, we have that…

We apply tilting theory over preprojective algebras $Lambda$ to a study of moduli space of $Lambda$-modules. We define the categories of semistable modules and give an equivalence, so-called reflection functors, between them by using…

We give a construction of Gorenstein projective $\tau$-tilting modules in terms of tensor products of modules. As a consequence, we give a class of non-self-injective algebras admitting non-trivial Gorenstein projective $\tau$-tilting…

In the 1980s, Harada introduced a new class of algebras now called Harada algebras. Harada algebras provides us with a rich source of Auslander's 1-Gorenstein algebras. In this paper, we have two main results about Harada algebras. The…

Let $\Lambda$ be a finite dimensional Auslander algebra. For a $\Lambda$-module $M$, we prove that the projective dimension of $M$ is at most one if and only if the projective dimension of its socle soc\,$M$ is at most one. As an…

For an Artinian $(n-1)$-Auslander algebra $\Lambda$ with global dimension $n(\geq 2)$, we show that if $\Lambda$ admits a trivial maximal $(n-1)$-orthogonal subcategory of $\mod\Lambda$, then $\Lambda$ is a Nakayama algebra and the…

Let $\Lambda$ be a $\mathbb{Z}$-graded artin algebra. Two classical results of Gordon and Green state that if $\Lambda$ has only finitely many indecomposable gradable modules, up to isomorphism, then $\Lambda$ has finite representation…

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 study which algebras have tilting modules that are both generated and cogenerated by projective-injective modules. Crawley-Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global…

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…

$n\mathbb{Z}$-cluster tilting subcategories are an ideal setting for higher dimensional Auslander-Reiten theory. We give a complete classification of $n\mathbb{Z}$-cluster tilting subcategories of module categories of Nakayama algebras. In…

Let $\Lambda$ be an $n$-Auslander algebra with global dimension $n+1$. In this paper, we prove that $\Lambda$ is representation-finite if and only if the number of non-isomorphic indecomposable $\Lambda$-modules with projective dimension…

Let A be a finite-dimensional algebra. If A is self-injective, then all modules are reflexive. Marczinzik recently has asked whether A has to be self-injective in case all the simple modules are reflexive. Here, we exhibit an 8-dimensional…

Let $\Lambda $ be an artin algebra and $T$ a $\tau$-tilting $\Lambda$-module. We prove that $T$ is a tilting module if and only if ${\rm Ext}_{\Lambda}^{i}(T,\Fac T)=0$ for all $i\geq 1$, where $\Fac T$ is the full subcategory consisting of…