Finiteness properties of profinite blocks and blocks with defect group $\mathbb Z_p^n$

Abstract

We prove that any block of a profinite group with a topologically finitely generated defect group $D$ is isomorphic to a block of a virtually pro-$p$ group, and if $D$ is finitely presented then so is the basic algebra of the block over $k$. Using these general results we then prove that any block with defect group $D=\mathbb Z_p^n$ is Morita equivalent to $\mathcal O_α[[\mathbb Z_p^n\rtimes E]]$ for a finite $p'$-group $E$ acting faithfully and a $2$-cocycle $α\in H^2(E,k^\times)$. In particular, Donovan's conjecture holds for such blocks of profinite groups.