On the category of finitely presented mod $p$ representations of $GL_2(F)$, $F/\mathbb{Q}_p$ finite

Jack Shotton

Published 2017-10-16Version 1

Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $\mathbb{F}$ be a finite field of characteristic $p$. A smooth representation of $GL_2(F)$ over $\mathbb{F}$ is finitely presented if it can be written as the cokernel of a map between representations induced from compact-mod-centre open subgroups of $GL_2(F)$. We prove that the category of finitely presented smooth representations is an abelian subcategory of all smooth representations. This amounts to showing that the kernel of a map between finitely presented smooth representations is finitely presented. The proof uses amalgamated products of completed group rings.