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

TL;DR

This paper extends block theory and Brauer's first main theorem to profinite groups, establishing a local-global framework and a correspondence between blocks and defect groups in this broader setting.

Contribution

It develops the local-global theory of blocks for profinite groups and proves a Brauer correspondence for blocks with a given defect group.

Findings

Established a decomposition of the completed group algebra into blocks.

Defined and characterized defect groups for profinite groups.

Proved a Brauer correspondence between blocks of G and its normalizer.

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 $\prod_{i\in I}B_i$ 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 $N_G(D)$ with defect group $D$.