Xiao and Zhu have shown that if C is a locally finite triangulated category, then the Auslander-Reiten triangles generate relations for Grothendieck group of . The notion ( d + 2 ) -angulated categories “higher dimensional” analogue categories. In this article, we show category with odd only -angles This extends result Zhu, gives converse Zhu's also true.

We carefully define and study C∗-algebras over topological spaces, possibly non-Hausdorff, and review some relevant results from point-set topology along the way. We explain the triangulated category structure on the bivariant Kasparov theory over a topological space and study the analogue of the bootstrap class for C∗-algebras over a finite topological space.

Introduction Let S be a Noetherian separated scheme of finite dimension, and let SH(S) denote the motivic stable homotopy category of Morel and Voevodsky. In order to get a motivic version of the Postnikov tower, Voevodsky [23] constructs a filtered family of triangulated subcategories of SH(S):

In this paper, we introduce and investigate the notions of ξ-strongly copure projective objects in a triangulated category. This extends Asadollahi’s notion of ξ-Gorenstein projective objects. Then we study the ξ-strongly copure projective dimension and investigate the existence of ξ-strongly copure projective precover.

Abstract We consider the smallest triangulated subcategory of unbounded derived module category a ring that contains injective modules and is closed under set indexed coproducts. If this entire category, then we say injectives generate for ring. In particular, ask whether, if collection rings, do related constructions, vice versa. provide sufficient conditions statement to hold various construc...

Let $\mathcal{C}$ be a $(d+2)$-angulated category. In this note, we show that if is locally finite, then has Auslander-Reiten $(d+2)$-angles. This extends result of Xiao and Zhu for triangulated categories.

We consider a homology theory h : T ! A on a triangulated category T with values in a graded abelian category A . If the functor h re ects isomorphisms, is full and is such that for any object x in A there is an object X in T with an isomorphism between h(X) and x, we prove that A is a hereditary abelian category, all idempotents in T split and the kernel of h is a square zero ideal which as a ...

In this thesis the global structure of three classes of algebraic triangulated categories is investigated by describing their thick, localizing and smashing subcategories and by analyzing the Smashing Conjecture. We show that the Smashing Conjecture for the stable module category of a self-injective artin algebra A is equivalent to the statement that a class of model categories associated with ...

Let k be an algebraically closed field and let T be the k-linear algebraic triangulated category generated by a w-spherical object for an integer w. For certain values of w this category is classical. For instance, if w = 0 then it is the compact derived category of the dual numbers over k. As main results of the paper we show that for w ≤ 0, the category T has no non-trivial t-structures, but ...

Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p-group due to Benson, Carlson and Rickard, we consider classifying thick subcategories of th...