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For each positive integer $n$ we introduce the notion of $n$-exangulated categories as higher dimensional analogues of extriangulated categories defined by Nakaoka-Palu. We characterize which $n$-exangulated categories are $n$-exact in the…
We give a direct proof of the following known result: the Grothendieck group of a triangulated category with a silting subcategory is isomorphic to the split Grothendieck group of the silting subcategory. Moreover, we obtain its…
A ladder determinantal module is an arbitrary direct sum of ideals of maximal minors of a generic ladder matrix. In this article, we give necessary and sufficient conditions for the special fiber ring of such modules to be Gorenstein. These…
We define a zeta-function of a pre-triangulated dg-category and investigate its relationship with the motivic zeta-function in the geometric case.
For an exact category $\mathcal{E}$ with enough projectives and with a $d\mathbb{Z}$-cluster tilting subcategory, we show that the singularity category of $\mathcal{E}$ admits a $d\mathbb{Z}$-cluster tilting subcategory. To do this we…
In this paper, we study properties of maps between fibrant objects in model categories. We give a characterization of weak equivalences between fibrant object. If every object of a model category is fibrant, then we give a simple…
Additive categories play a fundamental role in mathematics and related disciplines. Given an additive category equipped with a biadditive functor, one can construct its category of extensions, which encodes important structural information.…
In this paper, we prove that given a differential graded category C and B a full differential graded subcategory closed under coproducts, there is a canonical recollement of differential graded categories, for which we use enriched…
In this note, I define a notion of a compactly supported object in a triangulated category. I prove a number of propositions relating this to traditional notions of support and give an application to the theory of derived Morita…
In the context of extriangulated categories, we establish the injective version of Schanuel's lemma in homological algebra.
We study the equivalences induced by some special silting objects in the derived category over dg-algebra whose positive cohomologies are all zero.
A notion of support for objects in any Grothendieck category is introduced. This is based on the spectral category of a Grothendieck category and uses its Boolean lattice of localising subcategories. The support provides a classification of…
We study objects in triangulated categories which have a two-dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which the object is spherical. This general…
We give a principle in derived categories, which lies behind the classical Auslander-Reiten duality and its generalized version by Iyama and Wemyss. We apply the principle to show the validity of the Auslander-Reiten conjecture over a…
The purpose of this short note is to understand the Grothendieck ring of the Deligne category $\underline{\text{Rep}}(GL_t)$. We give a formula for the class of an indecomposable object in terms of exterior powers of the tautological object…
In this paper, we give a description of the self-injective dimension of string algebras and obtain a necessary and sufficient condition for a string algebra to be Gorenstein.
We call a triangulated category \emph{hereditary} provided that it is equivalent to the bounded derived category of a hereditary abelian category, where the equivalence is required to commute with the translation functors. If the…
The generalized divided differences are introduced. They are applied to investigate some properties characterizing generalized higher-order convexity. Among others some support-type property is proved.
We introduce the finitistic extension degree of a ring and investigate rings for which it is finite. The Auslander-Reiten Conjecture is proved for rings of finite finitistic extension degree and these rings are also shown to have finite…
We introduce the concept of strict ample sequence in a fibered triangulated category and define the stability of the objects in a triangulated category. Then we construct the moduli space of (semi) stable objects by GIT construction.