| Résume||Let $G$ be a split connected reductive group over a finite extension of $Q_p$, and let $H$ be the (pro-$p$) Iwahori-Hecke algebra of $G$ with coefficients in an arbitrary field $k$. In the classical case, where $k$ has characteristic zero, $H$ is known, by Bernstein, to be a regular ring.
This means that any $H$-module has a finite projective resolution. This is no longer the case if $k$ has characteristic $p$. In joint work with R. Ollivier we prove that $H$ always is a Gorenstein ring, i.e., has finite injective dimension as a module over itself.|