A canonical torsion theory for pro-p Iwahori-Hecke modules

Let $\mathfrak F$ be a locally compact nonarchimedean field with residue characteristic $p$ and $G$ the group of $\mathfrak{F}$-rational points of a connected split reductive group over $\mathfrak{F}$. We define a torsion pair in the category Mod$(H)$ of modules over the pro-$p$-Iwahori Hecke $k$-algebra $H$ of $G$, where $k$ is an arbitrary field. We prove that, under a certain hypothesis, the torsionfree class embeds fully faithfully into the category Mod${}^I(G)$ of smooth $k$-representations of $G$ generated by their pro-$p$-Iwahori fixed vectors. If the characteristic of $k$ is different from $p$ then this hypothesis is always satisfied and the torsionfree class is the whole category Mod$(H)$. If $k$ contains the residue field of $\mathfrak F$ then we study the case $G = \mathbf{SL_2}(\mathfrak F)$. We show that our hypothesis is satisfied, and we describe explicitly the torsionfree and the torsion classes. If $\mathfrak F\neq \mathbb Q_p$ and $p\neq 2$, then an $H$-module is in the torsion class if and only if it is a union of supersingular finite length submodules; it lies in the torsionfree class if and only if it does not contain any nonzero supersingular finite length module. If $\mathfrak{F} = \mathbb{Q}_p$, the torsionfree class is the whole category Mod$(H)$, and we give a new proof of the fact that Mod$(H)$ is equivalent to Mod${}^I(G)$. These results are based on the computation of the $H$-module structure of certain natural cohomology spaces for the pro-$p$-Iwahori subgroup $I$ of $G$.