The Extension Dimension of Abelian Categories

Abstract

Let A be an abelian category having enough projective objects and enough injective objects. We prove that if A admits an additive generating object, then the extension dimension and the weak resolution dimension of A are identical, and they are at most the representation dimension of A minus two. By using it, for a right Morita ring Λ, we establish the relation between the extension dimension of the category mod Λ of finitely generated right Λ-modules and the representation dimension as well as the right global dimension of Λ. In particular, we give an upper bound for the extension dimension of mod Λ in terms of the projective dimension of certain class of simple right Λ-modules and the radical layer length of Λ. In addition, we investigate the behavior of the extension dimension under some ring extensions and recollements.

The derived dimensions of $(m,n)$-Igusa-Todorov algebras

“…For a subclass U of mod Λ, we use add U to denote the subcategory of mod Λ consisting of direct summands of finite direct sums of objects in U. Let us recall some notions and basic facts(for example, see [4,25]). Let U 1 , U 2 , • • • , U n be subcategories of mod Λ.…”

“…Roughly speaking, it is an invariant that measures how quickly the category can be built from one object. Many authors have studied the upper bound of tri.dim T , see [3,5,7,9,12,14,17,18,22,25,23] and so on. There are a lot of triangulated categories having infinite dimension, for instance, Oppermann and Št'ovíček proved in [14] that all proper thick subcategories of the bounded derived category of finitely generated modules over a Noetherian algebra containing perfect complexes have infinite dimension.…”

“…Similar to the dimension of triangulated categories, the (extension) dimension of an abelian category was introduced by Beligiannis in [4], also see [8]. The size of the extension dimension reflects how far an artin algebra is from a finite representation type, some relate result can be see [25,22,21] and so on. We denote ext.dim mod Λ by the extension dimension of mod Λ, and LL(Λ) by the Loewy length of Λ, gl.dim Λ by the global dimension of Λ, rep.dim Λ by the representation dimension of Λ (see [2]), w.resol.dim mod Λ by the weak resolution dimension of Λ(see [13]).…”

“…In [19], Wei introduced the notion of n-Igusa-Todorov algebra. The relation between extension dimension and 0-Igusa-Todorov algebra is that artin algebra Λ is 0-Igusa-Todorov algebra if and only if ext.dim mod Λ 1(see [25,Proposition 3.15]). We will introduce the notion of (m, n)-Igusa-Todorov algebras, where m, n are two nonnegative integers, which is a generalization of n-Igusa-Todorov algebra.…”

“…Let Λ be an artin algebra and V a set of some simple modules in mod Λ. We have (4) ( [22]) tri.dim D b (mod Λ) 2(pd V + ℓℓ tV (Λ)) + 1;…”