Flat and coherent functors

Abstract

The main objective of this paper is to characterize flat group valued functors. We obtain the following theorem, announced in [;7: Let X be a small additive category with dual X0 and S an object in [x0, AB], the category of all additive functors from X to the category Ab of Abelian groups. Then S is flat, i.e., the functor S @r : [X, AB] + AB is exact if and only if the fiber X/S of the Yoneda embedding X --+ [x0, AB] over S is filtered from above, or if and only if S is a filtered direct limit of representable functors. There are several other equivalent statements, and it is, mutatis mutandis, enough to aSsume X preadditive. A similar theorem has been obtained by B. Stenstrom in [II]. He proves that a functor is flat if and only if it is a filtered direct limit of projective (instead of representable) functors. For Abelian X the result was obtained by J. Fisher [5J; in this particular case “flat” means “left exact,” and a short proof is possible. Our result is a generalization of the well-known characterization of flat modules by means of generators and relations, and has applications in the study of the exactness of the direct limit functor [7], and in the singular homology theory of sheaves [S]. Using the above characterization of flat functora we show in analogy to the results of S. U. Chase [4] on coherent rings that the category [X0, AB] is locally coherent, i.e., has a family of coherent generators, if and only if