Projective smooth representations in natural characteristic

TL;DR

This paper studies the existence of nonzero projective smooth modules over group algebras in characteristic p, proving their non-existence for certain groups and providing an elementary, adaptable proof.

Contribution

It establishes the non-existence of projective modules for fair groups in characteristic p using an elementary and adaptable approach.

Findings

Non-existence of projective modules for fair groups in characteristic p

Elementary proof method that extends to fields like _q((t))

Criterion for fairness via Chabauty space of G

Abstract

We investigate under which circumstances there exists nonzero {\it{projective}} smooth $\field[G]$-modules, where $\field$ is a field of characteristic $p$ and $G$ is a locally pro-$p$ group. We prove the non-existence of (non-trivial) projective objects for so-called {\it{fair}} groups -- a family including $\bf{G}(\frak{F})$ for a connected reductive group $\bf{G}$ defined over a non-archimedean local field $\frak{F}$. This was proved in \cite{SS24} for finite extensions $\frak{F}/\Bbb{Q}_p$. The argument we present in this note has the benefit of being completely elementary and, perhaps more importantly, adaptable to $\frak{F}=\Bbb{F}_q(\!(t)\!)$. Finally, we elucidate the fairness condition via a criterion in the Chabauty space of $G$.