Modular Representation Theory - Projective Modules

Projective Modules

In ordinary representation theory, every indecomposable module is irreducible, and so every module is projective. However, the simple modules with characteristic dividing the group order are rarely projective. Indeed, if a simple module is projective, then it is the only simple module in its block, which is then isomorphic to the endomorphism algebra of the underlying vector space, a full matrix algebra. In that case, the block is said to have 'defect 0'. Generally, the structure of projective modules is difficult to determine.

For the group algebra of a finite group, the (isomorphism types of) projective indecomposable modules are in a one-to-one correspondence with the (isomorphism types of) simple modules: the socle of each projective indecomposable is simple (and isomorphic to the top), and this affords the bijection, as non-isomorphic projective indecomposables have non-isomorphic socles. The multiplicity of a projective indecomposable module as a summand of the group algebra (viewed as the regular module) is the dimension of its socle (for large enough fields of characteristic zero, this recovers the fact that each simple module occurs with multiplicity equal to its dimension as a direct summand of the regular module).

Each projective indecomposable module (and hence each projective module) in positive characteristic p may be lifted to a module in characteristic 0. Using the ring R as above, with residue field K, the identity element of G may be decomposed as a sum of mutually orthogonal primitive idempotents ( not necessarily central) of K. Each projective indecomposable K-module is isomorphic to e.K for a primitive idempotent e that occurs in this decomposition. The idempotent e lifts to a primitive idempotent, say E, of R, and the left module E.R has reduction (mod p) isomorphic to e.K.