Modular representation theory: Difference between revisions

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[[group ring|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, such as the [[p-adic number#p-adic integers|''p''-adic integers]]. The structure of ''R''[''G''] is closely related both to

the structure of the group algebra ''K''[''G''] and to the structure of the semisimple group algebra ''F''[''G''], and there is much interplay

between the module theory of the three algebras.

Each ''R''[''G'']-module naturally gives rise to an ''F''[''G'']-module, and, by a process often known informally as '''reduction (mod ''p'')''',

to a ''K''[''G'']-module. On the other hand, since ''R'' is a [[principal ideal ___domain]], each finite-dimensional ''F''[''G'']-module

and, by a process often known informally as '''reduction (mod ''p'')''',

toarises aby extension of scalars from an ''KR''[''G'']-module.{{citation Onneeded}} theIn othergeneral, handhowever, sincenot all ''RK''[''G'']-modules isarise as reductions (mod ''p'') aof

[[principal ideal ___domain]], each finite-dimensional ''F''[''G'']-module

arises by extension of scalars from an ''R''[''G'']-module. In general,

however, not all ''K''[''G'']-modules arise as reductions (mod ''p'') of

''R''[''G'']-modules. Those that do are '''liftable'''.