Purity and Almost Split Morphisms in Abstract Homotopy Categories: A Unified Approach via Brown Representability

Abstract

Our aim in this paper is to develop a theory of purity and to prove in a unified conceptual way the existence of almost split morphisms, almost split sequences and almost split triangles in abstract homotopy categories, a rather omnipresent class of categories of interest in representation theory. Our main tool for doing this is the classical Brown representability theorem. Mathematics Subject Classifications (2000): 16Gxx, 18Gxx, 18Exx, 55U35. Abstract homotopy categories were introduced by E. Brown in the mid-sixties as the proper framework for the study of the homotopy theory of CW-complexes. In this setting, he proved in (18) his celebrated representability theorem, a variant of which has recently found important applications in the stable module category of a modular group algebra and more generally in compactly generated triangulated categories, mainly through the work of Rickard (42) and Neeman (40). Our main purpose in this paper is to develop a theory of purity and a theory of existence of almost split morphisms in an abstract homotopy category, using the Brown representability theorem as a main catalyzing tool. This aim is justified by the fact that abstract homotopy categories are om- nipresent in representation theory. They include module categories, locally finitely presented categories with products, compactly generated triangulated categories, categories of projective modules, and stable categories modulo projectives over left coherent and right perfect rings, categories of injective modules and stable categories modulo injectives over right Morita rings, (stable) categories of Cohen- Macaulay modules over Gorenstein rings, and many others. Abstract homotopy