Block theory and Brauer's first main theorem for profinite groups

Abstract

We develop the local-global theory of blocks for profinite groups. Given a field k of characteristic p and a profinite group G, one may express the completed group algebra k[[G]] as a product ∏i∈IBi of closed indecomposable algebras, called the blocks of G. To each block B of G we associate a pro-p subgroup of G, called the defect group of B, unique up to conjugacy in G. We give several characterizations of the defect group in analogy with defect groups of blocks of finite groups. Our main theorem is a version of Brauer's first main theorem: a correspondence between the blocks of G with defect group D and the blocks of the normalizer NG(D) with defect group D.