On the pro-$p$ Iwahori Hecke Ext-algebra of $\mathrm{SL}_2(\mathbb{Q}_p)$

Let $G={\rm SL}_2(\mathfrak F) $ where $\mathfrak F$ is a finite extension of $\mathbb Q_p$. We suppose that the pro-$p$ Iwahori subgroup $I$ of $G$ is a Poincar\'e group of dimension $d$. Let $k$ be a field containing the residue field of $\mathfrak F$. In this article, we study the graded Ext-algebra $E^*={\operatorname{Ext}}_{{\operatorname{Mod}}(G)}^*(k[G/I], k[G/I])$. Its degree zero piece $E^0$ is the usual pro-$p$ Iwahori-Hecke algebra $H$. We study $E^d$ as an $H$-bimodule and deduce that for an irreducible admissible smooth representation of $G$, we have $H^d(I,V)=0$ unless $V$ is the trivial representation. When $\mathfrak F=\mathbb Q_p$ with $p\geq 5$, we have $d=3$. In that case we describe $E^*$ as an $H$-bimodule and give the structure as an algebra of the centralizer in $E^*$ of the center of $H$. We deduce results on the values of the functor $H^*(I, {}_-)$ which attaches to a (finite length) smooth $k$-representation $V$ of $G$ its cohomology with respect to $I$. We prove that $H^*(I,V)$ is always finite dimensional. Furthermore, if $V$ is irreducible, then $V$ is supersingular if and only if $H^*(I,V)$ is a supersingular $H$-module.