# On the exactness of ordinary parts over a local field of characteristic $p$

## TL;DR

This paper proves the exactness of the ordinary part functor for admissible smooth representations of a reductive group over a local field of characteristic p, under specific ring conditions, and explores related extension properties.

## Contribution

It establishes the exactness of the ordinary part functor in a new setting where the residue characteristic matches the characteristic of the field, with implications for extension groups.

## Key findings

- Exactness of the ordinary part functor under specified conditions
- Results on Yoneda extensions between admissible smooth representations
- Conditions when p is nilpotent in the coefficient ring

## Abstract

Let $G$ be a connected reductive group over a non-archimedean local field $F$ of residue characteristic $p$, $P$ be a parabolic subgroup of $G$, and $R$ be a commutative ring. When $R$ is artinian, $p$ is nilpotent in $R$, and $\mathrm{char}(F)=p$, we prove that the ordinary part functor $\mathrm{Ord}_P$ is exact on the category of admissible smooth $R$-representations of $G$. We derive some results on Yoneda extensions between admissible smooth $R$-representations of $G$.