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 krrGss as a product ś iPI Bi 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 Brauer correspondence between the blocks of G with defect group D and the blocks of the normalizer NGpDq with defect group D.

Blocks of profinite groups with cyclic defect group

“…As with krrGss, the natural projections ϕ MN , ϕ N are algebra homomorphisms. We recall some further results from [5]: Lemma 2.14 ([5, Proposition 4.7 and Corollary 5.10]). If the block B of krrGss has defect group D, then B can be expressed as the inverse limit of a surjective inverse system of blocks X N of krG{N s having defect group DN {N , as N runs through some cofinal inverse system of open normal subgroups of G.…”

“…The study of the modular representation theory of profinite groups was begun in [13,12], while the study of blocks and defect groups has been initiated recently in [5]. In this article we classify, in Theorem 4.3, the blocks of an arbitrary profinite group whose defect groups are cyclic (meaning either a finite cyclic p-group or the p-adic integers Z p ): they are Brauer tree algebras in strict analogy with the finite case.…”