Endomorphism rings and category equivalences

Abstract

The use of category equivalences for the study of endomorphism rings stems from the Morita theorem. In a sense, this theorem can be viewed as stating that if P is a finitely generated projective generator of R-mod and S = End( RP), then properties of P correspond to properties of S through the equivalence between the categories R-mod and S-mod given by the functor Hom,(P, -). Generalizations of this theorem were given in [4, 51, In [S], P is only assumed to be finitely generated and projective, and Hom,(P, -) gives in this case an equivalence between S-mod and a quotient category of R-mod, while in [S] it is shown that if P is a finitely generated quasiprojective self-generator, then the equivalence induced by the same functor is now defined between the category a[P] of all the R-modules subgenerated by P and S-mod. Later on, other category equivalences were constructed, in an analogous way to those already mentioned, by replacing S-mod by a certain quotient category of S-mod. Thus, in [14] Morita contexts are used to obtain a category equivalence between quotient categories of both R-mod and S-mod for an arbitrary R-module M. On the other hand, if M is a C-quasiprojective module, then it is shown in [8 3 that the functor Hom,(M, -) induces an equivalence between quotient categories of o[M] and S-mod, and the latter quotient category coincides with S-mod when M is finitely generated. All the above constructions can be considered as particular cases of the following: if V is a locally finitely generated Grothendieck category and M is an object of V with S = End,(M), then the class of the M-distinguished objects of g (in the sense of [lo]) is the torsionfree class of a torsion