Projective smooth representations in natural characteristic

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 $\mathbf{G}(\mathfrak{F})$ for a connected reductive group $\mathbf{G}$ defined over a non-archimedean local field $\mathfrak{F}$. This was proved in \cite{SS24} for finite extensions $\mathfrak{F}/{\mathbb{Q}}_p$. The argument we present in this note has the benefit of being completely elementary and, perhaps more importantly, adaptable to $\mathfrak{F}={\mathbb{F}}_q((t))$. Finally, we elucidate the fairness condition via a criterion in the Chabauty space of $G$.

12 pages