Coherent rings of finite weak global dimension

Abstract

Abstract. The category of left modules over right coherent rings of finite weak global dimension has several nice features. For example, every left module over such a ring has a flat cover (Belshoff, Enochs, Xu) and, if the weak global dimension is at most two, every left module has a flat envelope (Asensio, Martínez). We will exploit these features of this category to study its objects.In particular, we will consider orthogonal complements (relative to the extension functor) of several classes of modules in this category. In the case of a commutative ring we describe an idempotent radical on its category of modules which, when the weak global dimension does not exceed 2, can be used to analyze the structure of the flat envelopes and of the ring itself.

The phantom menace in representation theory

“…The artinian line was first used towards sufficient conditions for the existence of almost split sequences in subcategories by Auslander and Smalø [5] in 1981; a decade later, this line obtained an important boost through its applications to homology and tilting theory, first observed by Auslander-Reiten [3]. Meanwhile, the latter approach was pursued separately by Asensio Mayor, Belshoff, Enochs, Guil-Asensio, Martínez Hernández, Rada, Saorín, del Valle, Xu, and others (see [2,7,20,39,41,42,52,53]). Finally, we mention that in [38], Levy independently used minimal approximations from a category of particulary accessible objects, in order to study modules over pullback rings.…”