On pure global dimension of locally finitely presented Grothendieck categories

Abstract

The author gives a general categorical setting of classicalsubjects from ring theory, such as pure global dimension anddetermination of indecomposable modules over Artinian rings. Heintroduces the pure global dimension of a locally finitelypresented Grothendieck category by the vanishing of the Pextfunctors defined in analogy with the module case. One of the mainresults gives a through and useful characterization of locallyfinitely presented Grothendieck categories \germ A for whichthe pure global dimension p.gl.\dim\germ A=0; for instance,p.gl.\dim\germ A=0 if and only if \germ A is pure perfect ifand only if \germ A is locally Noetherian and any object is adirect sum of finitely generated objects. A special section isdevoted to the study of stable and factorizable systems, and acharacterization of Mittag Leffler objects [cf. M. Raynaud and L.Gruson, Invent. Math. 13 (1971), 1 89; MR 46#7219] is obtained.The author gives a number of applications of the theory; forexample, the following generalization of a known result: A ringR has vanishing left and right pure global dimension if andonly if R is left and right Artinian and there is a finiteupper bound for the lengths of indecomposable finitely presentedleft and right R modules. Moreover, it is proved that the pureglobal dimension of the category of commutative and cocommutativeHopf algebras over a countable field of prime characteristic isone