An equivalence induced by Ext and Tor applied to the finitistic weak dimension of coherent rings

Abstract

Let R be a ring, see below for other notation. The functor categories (mod-R, Ab) and ((R-mod)op, Ab) have received considerable attention since the 1960s. The first of these has achieved prominence in the model theory of modules and most particularly in the investigation of the representation theory of Artinian algebras. Both [11, Chapter 12] and [8] contain accounts of the use (mod-R, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-R, Ab) to the study of Artinian algebras. The category ((R-mod)op, Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between R-mod and mod-R. Warfield's paper, [13], describes this for semiperfect rings, and this duality is an essential component in the construction of almost split sequences over Artinian algebras, see [4]. In comparison, the general case has been neglected. This paper seeks to remedy this situation, giving a concrete description of the resulting equivalence between (mod-R, Ab) and ((R-mod)op, Ab) for an arbitrary ring R.