Elementary torsion theories and locally finitely presented categories

Abstract

Locally finitely presented Grothendieck abelian categories are shown to be just the “elementary” localizations of Module categories. All categories V are assumed to be Grothendieck abelian (e.g. see [l, 21 or [5]). 55’ is locally finifeely presented if 9? has a generating family Ce = {Gi: i E I} with each Gi a finitely presented object of %. I have shown elsewhere [4] that in such categories one may define a canonical first-order (many-sorted) logic which behaves well one may prove Los’ Theorem, produce K-Saturated extensions of structures etc. Given a torsion theory (= a hereditary torsion theory in the terminology of [5]) (T, S) on Q and C in %‘, Q$ will be the filter of T-dense subobjects of C (where C’s C is T-dense if C/C’ is a member of the torsion class T). %? will denote the quotient category of V with respect to (9,s) and ( -)swill denote the T-localization functor from %’ to %? Tr will be the functor taking objects C of % to their T-torsion subobjects. We say that C is T-finitely generated (T-fig.) if C contains a finitely generated T-dense subobject. Recall that a torsion theory (9,9) is of finite type if for every Gi E 59 (as above), %$I has a cofinal subset of finitely generated members equivalently (by an easy argument) iff this condition holds on %s for any finitely generated c E %. We shall say that a torsion theory (9,T) is elementary if: (i) (9, 9) is of finite type; (ii) for each Gi, Gi E ‘3 (as above) and r: Gi --, Gi with im r E %z, ker r is T-finitely generated (that is (finitely generated) members of %z are **T-finitely presented” in some sense). It may be shown that this notion does not depend on the choice of ‘3 (directly or as a consequence of the following result). The term “elementary” is justified by the following theorem: