Pro-p Iwahori-Hecke algebras are Gorenstein

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. For k an arbitrary field, we study the homological properties of the Iwahori-Hecke k-algebra H' and of the pro-p Iwahori-Hecke k-algebra H of G. We prove that both of these algebras are Gorenstein rings with self-injective dimension bounded above by the rank of G. If G is semisimple, we also show that this upper bound is sharp, that both H and H' are Auslander-Gorenstein and that there is a duality functor on the finitely generated modules of H (respectively H'). We obtain the analogous Gorenstein and Auslander-Gorenstein properties for the graded rings associated to H and H'. When k has characteristic p, we prove that in most cases H and H' have infinite global dimension. In particular, we deduce that the category of smooth k-representations of G=PGL(2,Q_p) generated by their invariant vectors under the pro-p-Iwahori subgroup has infinite global dimension (at least if k is algebraically closed).