Abstract
Hc G, O;ITI~ has two basic functors A(RG) 2 di’(RH) and A(RH) 2 &(RG), given by restriction and induction, which play an essential role in representation theory. An important and elementary class of RG-representations is permutation modules which are direct sums of modules Indz(R) obtained by induction from the trivial RH-module R for various Hs G. In another extreme, one has RG-modules which arise by induction from RH-projective modules, leading to the concept of relative projectivity and Green’s theory of vertices and sources [CR], [GR]. The value of these subcategories of modules in representation theory and related areas is well known. In a different direction (influenced by algebraic geometry and topology), one considers not only module categories, but various categories of chain complexes of modules and their cohomologies. This culminates in the more recent approaches to representation theory through the theory of derived categories. See [SC], [CPS] and their many references.
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