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 ${\mathcal{O}}_{\alpha }[[{\mathbb{Z}}_p^n⋊E]]$ for a finite $p^{\prime }$-group $E$ acting faithfully and a $2$-cocycle $\alpha \in H^2(E,k^{\times })$. In particular, Donovan's conjecture holds for such blocks of profinite groups.

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