Modular Representation Theory - Reduction (mod p)

Reduction (mod p)

In the theory initially developed by Brauer, the link between ordinary representation theory and modular representation theory is best exemplified by considering the group algebra of the group G over a complete discrete valuation ring R with residue field K of positive characteristic p and field of fractions F of characteristic 0. The structure of R is closely related both to the structure of the group algebra K and to the structure of the semisimple group algebra F, and there is much interplay between the module theory of the three algebras.

Each R-module naturally gives rise to an F-module, and, by a process often known informally as reduction (mod p), to a K-module. On the other hand, since R is a principal ideal domain, each finite-dimensional F-module arises by extension of scalars from an R-module. In general, however, not all K-modules arise as reductions (mod p) of R-modules. Those that do are liftable.