Pure Direct Projective Objects in Grothendieck Categories

Abstract

We study generalizations of the concept of direct-projectivity (respectively pure-direct-projectivity) from module categories to abelian categories (respectively Grothendieck categories). We examine for which categories or under what conditions direct-projective objects are projective. Also we examine for which categories or under what conditions pure-direct-projective objects are projective, quasi-projective, pure-projective, flat or direct-projective. We investigate classes all of whose objects are direct-projective (respectively pure-direct-projective). We also give applications of some results to module categories and comodule categories.