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

Abstract

Let F be a locally compact nonarchimedean field with residue characteristic p and G the group of F-rational points of a connected split reductive group over 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 ModI(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 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.