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Locally finitely presented categories and functor rings

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Garcı´a, J.L., Go´mez Sa´nchez, P.L. and Martı´nez Herna´ndez, J. Osaka J. Math. 42 (2005), 173–187 LOCALLY FINITELY PRESENTED CATEGORIES AND FUNCTOR RINGS J.L. GARC´IA, P.L. G ´OMEZ S ´ANCHEZ and J. MART´INEZ HERN ´ANDEZ (Received June 25, 2003) Abstract By using the correspondence between locally finitely presented additive cate- gories and rings with enough idempotents, we study several properties of such rings in terms of the associated categories, and conversely. In particular, it is shown that a ring (with enough idempotents) is right perfect and the categories of finitely pre- sented right and left -modules are dual to each other if and only if the categories of projective and of injective right -modules are equivalent. 1. Introduction An important tool in the study of purity has been the use of Gabriel’s functor rings. In 1994, Crawley-Boevey [5] gave a quite general version of this technique, by introducing the concept of a locally finitely presented additive category (briefly, an l.f.p. additive category). First, an object of an additive category A is finitely pre- sented if the functor HomA( ) preserves direct limits. Then, we shall say that the additive category A is locally finitely presented in case every directed system of ob- jects and morphisms has a direct limit, the class of finitely presented objects of A is skeletally small and every object of A is the direct limit of finitely presented objects. One may define the functor ring (see [9]) associated to the category A as the ring = Hom( ) where is a set of isomorphism classes of finitely presented objects of A. With the natural sum and multiplication, is a ring with enough idempotents = = being the identity on . The category A may be embedded as a full subcategory of the category Mod( ) of all the unitary right -modules (i.e., modules such that = ) in such a way that pure exact sequences in A are those that are carried into exact sequences of Mod( ) through the embedding; and A can be thus seen as the c

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