Bifunctors and adjoint pairs

Abstract

We use a definition of tensor products of functors to generalize some theorems of homological algebra. We show that adjoint pairs of functors between additive functor categories correspond to bifunctors and that composition of such adjoint pairs corresponds to the tensor product of the bifunctors. We also generalize some homological characterizations of finitely generated projective modules to characterizations of small projectives in a functor category. We apply our results to adjoint pairs arising from satellites and from a functor on the domain categories. 0. Introduction. Let R and S be rings, and let ModR and Mod, denote the categories of right R-modules and right S-modules respectively. Given an S-R bimodule E, we obtain a functor HomR (E, ): ModR -> Mod, with an adjoint Os E: Mods -> ModR, which we shall consider as an adjoint pair (HomR (E, ), (s E): ModR -> Mods. Furthermore, given any adjoint pair (u1, u2): ModR -> Mods (i.e., a functor u1: ModR -> Mods having u2: Mods -> ModR as an adjoint) it is well known that there is a bimodule E which represents this adjoint pair, that is, one has equivalences ulHomR (E, ) and u2 ?s E (see [3] or [6]). In fact, let E= u2(S), then using the fact that u2 is right continuous (i.e., u2 is right exact and preserves direct sums) one obtains N 0s E= u2(N) for every S-module N by examining the effect of u2 on an S-free presentation of N. Furthermore, if T is a ring and (v1, v2): Mods -> ModT is an adjoint pair represented by a T-S bimodule E', then (v1 o u1, u2 o v2): ModR ModT is an adjoint pair represented by the T-R bimodule E' Os E. Although it is well known that functor categories are generalizations of categories of modules, many of the techniques of homological algebra could not be extended because of a lack of a proper definition of a tensor product. However, in [4], a suitable definition has been given, and in this paper we demonstrate how this definition can be exploited. In particular, we show that the above theorems for modules can be easily extended to functor categories. Before giving a specific summary of the contents of this paper, we introduce some notation and conventions. Received by the editors March 29, 1970. AMS 1969 subject classifications. Primary 1810, 1820.