Projective smooth representations in natural characteristic

We investigate under which circumstances there exists nonzero {\it{projective}} smooth $\text{field}[G]$-modules, where $\text{field}$ 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 $\mathfrak{F}\mathrm{/}{\mathbb{Q}}_p$. The argument we present in this note has the benefit of being completely elementary and, perhaps more importantly, adaptable to $\mathfrak{F}={\mathbb{F}}_q(\text{\hspace{0.17em}⁣}(t)\text{\hspace{0.17em}⁣})$. Finally, we elucidate the fairness condition via a criterion in the Chabauty space of $G$.