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

If k contains the residue field of F then we study the case G=SL2(F). We show that our hypothesis is satisfied, and we describe explicitly the torsionfree and the torsion classes. If F≠Qp and p≠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 F=Qp, the torsionfree class is the whole category Mod(H), and we give a new proof of the fact that Mod(H) is equivalent to ModI(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.