Related papers: Grothendieck groups and Auslander-Reiten (d+2)-ang…

Extriangulated categories were introduced by Nakaoka and Palu to give a unification of properties in exact categories and extension-closed subcategories of triangulated categories. A notion of tilting pairs in an extriangulated category is…

Let $R$ be an isolated Gorenstein singularity with a non-commutative resolution $A=End_R(R\oplus M)$. In this paper, we show that the relative singularity category $\Delta_R(A)$ of $A$ has a number of pleasant properties, such as being…

We prove that the Grothendieck rings of category $\mathcal{C}^{(t)}_Q$ over quantum affine algebras $U_q'(\g^{(t)})$ $(t=1,2)$ associated to each Dynkin quiver $Q$ of finite type $A_{2n-1}$ (resp. $D_{n+1}$) is isomorphic to one of category…

Auslander's formula shows that any abelian category C is equivalent to the category of coherent functors on C modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to…

We show that the category of finite-dimensional modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category is equivalent to the Gabriel-Zisman localisation of the category with respect to a certain class…

Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied.…

Extriangulated categories give a simultaneous generalization of triangulated categories and exact categories. In this paper, we study silting subcategories of an extriangulated category. First, we show that a silting subcategory induces a…

We show that the category of finitely generated free modules over certain local rings is n-angulated for every n at least 3. In fact, we construct several classes of n-angles, parametrized by equivalence classes of units in the local rings.…

We study the properties of rings satisfying Auslander-type conditions. If an artin algebra $\Lambda$ satisfies the Auslander condition (that is, $\Lambda$ is an $\infty$-Gorenstein artin algebra), then we construct two kinds of…

Given an additive category $\mathcal{C}$ and an integer $n\geqslant 2$. We form a new additive category $\mathcal{C}[\epsilon]^n$ consisting of objects $X$ in $\mathcal{C}$ equipped with an endomorphism $\epsilon_X$ satisfying…

We study Auslander-Reiten components of an artin algebra with bounded short cycles, namely, there exists a bound for the depths of maps appearing on short cycles of non-zero non-invertible maps between modules in the given component. First,…

In this paper, we study ideal quotients of triangulated categories by higher cluster tilting subcategories. Koenig and Zhu proved that the ideal quotient by a $2$-cluster tilting subcategory is an abelian category; moreover, by Morita's…

Let $\Lambda$ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of mod$\Lambda$. We show that $\mathcal{M}$ has an additive generator if and only if the n-almost split sequences form a basis for the relations for the…

In this paper, we introduce and study relative Auslander--Gorenstein pairs. This consists of a finite-dimensional Gorenstein algebra together with a self-orthogonal module that provides a further homological feature of the algebra in terms…

We give a new characterization of silting subcategories in the stable category of a Frobenius extriangulated category, generalizing the result of Di et al. (J. Algebra 525 (2019) 42-63) about the Auslander-Reiten type correspondence for…

Given an exact category $\mathcal{C}$, we denote by $\mathcal{C}_l$ the smallest additive subcategory containing injectives and indecomposable objects which appear as the first term of an almost split conflation. We prove that a deflation…

Let $k$ be a field. We show that locally presentable, $k$-linear categories $\mathcal{C}$ dualizable in the sense that the identity functor can be recovered as $\coprod_i x_i\otimes f_i$ for objects $x_i\in \mathcal{C}$ and left adjoints…

We establish a $d$-dimensional Auslander correspondence for $d$-truncated proper connective DG-algebras via $d$-extended module categories. A $d$-truncated proper connective DG-algebra $\Gamma$ is called Auslander if its $d$-extended module…

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…

For the cluster category of a hereditary or a canonical algebra, equivalently for the cluster category of the hereditary category of coherent sheaves on a weighted projective line, we study the Grothendieck group with respect to an…