Separable functors applied to graded rings (Q1120649)

A functor F is called separable if there exists a map Hom(F(M),F(N))\(\to Hom(M,N)\) for all objects M, N, which is left inverse to F and natural in a certain sense. For a given homomorphism of rings p: \(R\to S\) the restriction functor S-Mod\(\to R\)-Mod resp. the induction functor R- Mod\(\to S\)-Mod \((M\to S\otimes_ RM)\) is separable if and only if S/R is separable (: \(S\otimes_ RS\to S\) splits) resp. p splits as an R- bimodule map. The main part is concerned with the category of graded modules over a ring R, which is strongly graded by a group G, the (with respect to this grading) forgetful functor U and its (always existing) right adjoint F. U is separable in general, F if and only if \(R/R_ e\) is separable, where \(R_ e\) denotes the unit component of R. Among other constructions and characterizations of the separability of F the authors consider such properties of a module M which are transferred to the graded module F(M), mainly in the case of finite G. F(M) is semi-simple of finite length if M is simple.