We introduce the concept of maximal orthogonal subcategories over artin algebras and orders, and develop higher Auslander-Reiten theory on them. Auslander-Reiten theory, especially the concept of almost split sequences and their existence theorem, is fundamental to study categories which appear in representation theory, for example, modules over artin algebras [ARS][GR][Ri], their functorially ...

For finitely generated module M over a local ring R, the conventional notions of complete intersection dimension CI-dimRM and CohenMacaulay dimension CM-dimRM do not extend to cover the case of infinitely generated modules. In this paper we introduce similar invariants for not necessarily finitely generated modules, (namely, complete intersection flat and Cohen-Macaulay flat dimensions) which f...

in this paper, we introduce a subclass of chordal graphs which contains $d$-trees and show that their complement are vertex decomposable and so is shellable and sequentially cohen-macaulay.

We introduce the notion of residual intersections modules and prove their existence. show that projective dimension one have Cohen-Macaulay intersections, namely they satisfy relevant Artin-Nagata property. then establish a formula for core orientable satisfying certain homological conditions, extending previous results Corso, Polini, Ulrich on modules. Finally, we provide examples classes our ...

We show that in a triangulated category, the existence of a cluster tilting object often implies that the homomorphism groups are bounded in size. This holds for the stable module category of a selfinjective algebra, and as a corollary we recover a theorem of Erdmann and Holm. We then apply our result to Calabi-Yau triangulated categories, in particular stable categories of maximal Cohen-Macaul...