Projective smooth representations in natural characteristic

We investigate under which circumstances there exists nonzero projective smooth F [ G ] F[G] -modules, where F F is a field of characteristic p p and G G is a locally pro- p p group. We prove the non-existence of (non-trivial) projective objects for so-called fair groups – a family including G ( F ) \mathbf {G}(\frak {F}) for a connected reductive group G \mathbf {G} defined over a non-archimedean local field F \frak {F} . This was proved in [J. Lond. Math. Soc. (2) 111 (2025), p. 29] for finite extensions F / Q p \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 F = F q ( ( t ) ) \frak {F}=\Bbb {F}_q(\!(t)\!) . Finally, we elucidate the fairness condition via a criterion in the Chabauty space of G G .