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Let $\mathfrak F$ be a locally compact nonarchimedean field with residuecharacteristic $p$ and $G$ the group of $\mathfrak{F}$-rational points of aconnected split reductive group over $\mathfrak{F}$. We define a torsion pairin 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 certainhypothesis, the torsionfree class embeds fully faithfully into the categoryMod${}^I(G)$ of smooth $k$-representations of $G$ generated by theirpro-$p$-Iwahori fixed vectors.  If the characteristic of $k$ is different from $p$ then this hypothesis isalways 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 wedescribe explicitly the torsionfree and the torsion classes. If $\mathfrakF\neq \mathbb Q_p$ and $p\neq 2$, then an $H$-module is in the torsion class ifand only if it is a union of supersingular finite length submodules; it lies inthe torsionfree class if and only if it does not contain any nonzerosupersingular finite length module. If $\mathfrak{F} = \mathbb{Q}_p$, thetorsionfree class is the whole category Mod$(H)$, and we give a new proof ofthe fact that Mod$(H)$ is equivalent to Mod${}^I(G)$. These results are basedon the computation of the $H$-module structure of certain natural cohomologyspaces for the pro-$p$-Iwahori subgroup $I$ of $G$.

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