Related papers: From Submodule Categories to the Stable Auslander …

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…

The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ of admissibly finitely presented functors and use it to give a version…

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 build a bijection between the set $\sttilt\Lambda$ of isomorphism classes of basic support $\tau$-tilting modules over the Auslander algebra $\Lambda$ of $K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is an…

The Nakayama conjecture states that an algebra of infinite dominant dimension should be self-injective. Motivated by understanding this conjecture in the context of derived categories, we study dominant dimensions of algebras under derived…

Let $\Lambda$ be an arbitrary monomial algebra. We investigate the stable category $\underline{\operatorname{Gproj}}^{\mathbb{Z}}\Lambda$ of graded Gorenstein-projective $\Lambda$-modules and the orbit category…

We introduce higher dimensional analogues of the Nakayama algebras from the viewpoint of Iyama's higher Auslander--Reiten theory. More precisely, for each Nakayama algebra $A$ and each positive integer $d$, we construct a finite dimensional…

We show that every higher Auslander algebra $A_{n+1}^d$ of type $\mathbb{A}$ such that $\gcd(n,d)=1$ is derived equivalent to a certain replicated algebra $B=B_0^{(n+d)}$. Moreover ${\rm{gldim}} B = nd$ and $B$ admits an $nd$-cluster…

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra,…

We construct an isomorphism between the partially ordered set of tilting modules for the Auslander algebra of $K[x]/(x^n)$ and the interval of rational permutation braids in the braid group on $n$ strands. Hence, there are only finitely…

For a hereditary, finite-dimensional algebra $A$ the Coxeter transformation extends the action of the Auslander--Reiten translation on the non-projective indecomposable modules to a linear endomorphism of the Grothendieck group of the…

We give a complete description of finite dimensional selfinjective algebras over an algebraically closed field whose Auslander-Reiten quiver admits a generalized standard family of quasi-tubes maximally saturated by simple and projective…

Auslander-Reiten duality for module categories is generalized to some sufficiently nice subcategories. In particular, our consideration works for $\mathcal{P}^{<\infty}(\Lambda)$, the subcategory consisting of finitely generated modules…

For a nice-enough category $\mathcal{C}$, we construct both the morphism category ${\rm H}(\mathcal{C})$ of $\mathcal{C}$ and the category ${\rm mod}\mbox{-}\mathcal{C}$ of all finitely presented contravariant additive functors over…

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…

A quasi-hereditary algebra is an algebra equipped with a certain partial order $\unlhd$ on its simple modules. Such a partial order -- called a quasi-hereditary structure -- gives rise to a characteristic tilting module $T_{\unlhd}$ by a…

In the derived category of mod-KQ for Dynkin quiver Q, we construct a full subcategory in a canonical way, so that its endomorphism algebra is a higher Auslander algebra of global dimension $3k+2$ for any $k\geq 1$. Furthermore, we extend…

A non-unital algebra in a closed monoidal category is called self-induced if the multiplication induces an isomorphism between A\otimes_A A and A. For such an algebra, we define smoothening and roughening functors that retract the category…

We show that a properly stratified algebra is Gorenstein if and only if the characteristic tilting module coincides with the characteristic cotilting module. We further show that properly stratified Gorenstein algebras $A$ enjoy strong…

For an $(n-1)$-Auslander algebra $\Lambda$ with global dimension $n$, we give some necessary conditions for $\Lambda$ admitting a maximal $(n-1)$-orthogonal subcategory in terms of the properties of simple $\Lambda$-modules with projective…