Related papers: Higher Auslander correspondence for exact categori…

We introduce a notion of generalized Auslander-Reiten duality on a Hom-finite Krull-Schmidt exact category $\mathcal{C}$. This duality induces the generalized Auslander-Reiten translation functors $\tau$ and $\tau^-$. They are mutually…

This work is the sequel to Continuous Quivers of Type A (I). In this paper we define the Auslander-Reiten space of a continuous type $A$ quiver, which generalizes the Auslander-Reiten quiver of type $A_n$ quivers. We prove that extensions,…

Recently, Wang, Wei and Zhang introduced the notion of recollements of extriangulated categories. In this paper, let $(\mathcal{A},\mathcal{B},\mathcal{C})$ be a recollement of extriangulated categories. We provide some methods to construct…

In this paper, we prove a higher dimensional version of Auslander-Iyama-Solberg correspondence. Iyama and Solberg have shown a bijection between $n$-minimal Auslander-Gorenstein algebras and $n$-precluster tilting modules. If $A$ is an…

We define the Grothendieck group of an $n$-exangulated category. For $n$ odd, we show that this group shares many properties with the Grothendieck group of an exact or a triangulated category. In particular, we classify dense complete…

Recently, Wang, Wei and Zhang define the recollement of extriangulated categories, which is a generalization of both recollement of abelian categories and recollement of triangulated categories. For a recollement $(\mathcal A ,\mathcal…

We introduce higher gentle algebras. Our definition allows us to determine the singularity categories and subsequently show that higher gentle algebras are Iwanaga-Gorenstein. Under extra assumptions, we show that cluster-tilted algebras…

The category of Cohen-Macaulay modules of an algebra $B_{k,n}$ is used [JKS16] to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this…

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…

Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor…

We provide several equivalent descriptions of a highest weight category using recollements of abelian categories. Also, we explain the connection between sequences of standard and exceptional objects.

We shall show that the stable categories of graded Cohen-Macaulay modules over quotient singularities have tilting objects. In particular, these categories are triangle equivalent to derived categories of finite dimensional algebras. Our…

Let $\mathscr{C}$ be a Krull-Schmidt $n$-exangulated category and $\mathscr{A}$ be an $n$-extension closed subcategory of $\mathscr{C}$. Then $\mathscr{A}$ inherits the $n$-exangulated structure from the given $n$-exangulated category in a…

Several important types of categories have been shown to be both exact and coexact (in the sense of Barr). The first type consists of abelian categories, which due to their self-dual definition, can be seen to be both exact and coexact by…

We define admissible and weakly admissible subcategories in exact categories and prove that the former induce semi-orthogonal decompositions on the derived categories. We develop the theory of thin exact categories, an exact-category…

We give new improved bounds for the dominant dimension of Nakayama algebras and use those bounds to give a classification of Nakayama algebras with $n$ simple modules that are higher Auslander algebras with global dimension at least $n$.…

Everyone knows that if you have a bivariant homology theory satisfying a base change formula, you get an representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual…

Higher homological algebra was introduced by Iyama. It is also known as $n$-homological algebra where $n \geq 2$ is a fixed integer, and it deals with $n$-cluster tilting subcategories of abelian categories. All short exact sequences in…

In 1996, Edelman and Reiner defined the two higher Stasheff--Tamari orders on triangulations of cyclic polytopes and conjectured them to coincide. We open up an algebraic angle for approaching this conjecture by showing how these orders…

In this paper we show that both the homotopy category of strict $n$-categories, $1\leqslant n \leqslant \infty$, and the homotopy category of Steiner's augmented directed complexes are equivalent to the category of homotopy types. In order…