arXiv subjects
Publications and source records attributed to John W. MacQuarrie.
We introduce block pro-fusion systems for blocks of profinite groups, prove a profinite version of Puig's structure theorem for nilpotent blocks, and use it to show that there is only one Morita equivalence class of blocks having the infinite dihedral pro-$2$ group as their defect group.
We consider a class of extensions of associative algebras, which we refer to as ``strongly proj-bounded extensions''. We prove that the finiteness of the left global dimension and the support of the Hochschild homology is preserved by strongly proj-bounded extensions, generalizing results of Cibils, Lanzillota, Marcos and Solotar. Moreover, we show that the finiteness of the big left finitistic dimension is preserved by strongly proj-bounded extensions. In order to construct examples, we describe a new class of extensions of algebras of finite relative global dimension, which may be of independent interest. The results apply both for abstract (meaning no topology) and pseudocompact algebras.
We demonstrate that the blocks of a profinite group whose defect groups are cyclic have a Brauer tree algebra structure analogous to the case of finite groups. We show further that the Brauer tree of a block with defect group $\mathbb{Z}_{p}$ is of star type.
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 $D$.