The Category of Finitely Presented Smooth Mod $p$ Representations of $G{L}_2(F)$

Abstract

Let $F$ be a finite extension of ${\mathbb{Q}}_p$. We prove that the category of finitely presented smooth $Z$-finite representations of $G{L}_2(F)$ over a finite extension of ${\mathbb{F}}_p$ is an abelian subcategory of the category of all smooth representations. The proof uses amalgamated products of completed group rings.