Fully bounded grothendieck categories part II: Graded modules

Abstract

In Part I of this paper the second author studies boundedness conditions in Grothendieck categories with a Noetherian generator. However, if R is a Z-graded ring, then R is not necessarily a generator for R-gr, the Grothendieck category of graded left R-modules. As a matter of fact, we may consider G = @&R(n) as a generator for R-gr but G is neither a ring, nor is G Noetherian in R-gr even if R is a left Noetherian graded ring. Thus the graded case does not fall into the cases dealt with in Part I. Another new feature here is that in the graded case the existence of a functor R -gr + R-mod is evident, whereas in the locally Noetherian case substantial use has to be made of the Gabriel-Popescu embedding theorem, in order to obtain a somewhat similar situation. Although the functor R-gr --, R-mod is not as nice as one might hope after a first optimistic glance, we aim to study graded fully boundedness properties and relate these to phenomena in R-mod. A similar philosophy is behind the first author’s results on graded rings and modules of quotients, and it may thus be expected that the graded boundedness conditions may be brought to bear on certain properties of graded localizations, exactly as in the ungraded case. The final section of the paper characterizes graded fully left bounded rings that are fully left bounded. For basic theory on graded rings we refer to [l, 171. Some of the less known results have been summarized in Section 1.