Relative derived functors (Q921106)

The Gabriel-Popescu Theorem implies that every Grothendieck category \(\underline D\) may be identified as a quotient category of a module category \(\underline{R\text{-mod}}\), i.e., up to equivalence \(\underline D=\underline{(R,\sigma)}\)-mod, the quotient category of \(\underline{R\text{-mod}}\) with respect to some idempotent kernel functor \(\sigma\) or, equivalently, by the Serre subcategory \(\underline T_{\sigma}\) associated to \(\sigma\). In particular, \(\underline D\) is a Giraud subcategory of \(\underline{R\text{-mod}}\). In this paper the authors start from an arbitrary Grothendieck category \(\underline C\) and a Giraud subcategory \(\underline D\) and study the relationship between functors in \(\underline C\) and \(\underline D\), with respect to extension and restriction properties. This allows to relate derived functors in \(\underline C\) and in \(\underline D\). With \(\underline C=\underline{R\text{-mod}}\), this provides a general method to characterize derived functors for arbitrary Grothendieck categories \(\underline D\) by linking them to appropriate derived functors in \(\underline{R\text{-mod}}\). In particular, this yields some new results about generalized local cohomology, i.e. with respect to arbitrary idempotent kernel functors instead of just idempotent kernel functors of the type \(\sigma_ I\) for some ideal \(I\) of \(R\) (the classical theory).