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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 aPoincar\'e group of dimension $d$. Let $k$ be a field containing the residuefield of $\mathfrak F$.  In this article, we study the graded Ext-algebra$E^*={\operatorname{Ext}}_{{\operatorname{Mod}}(G)}^*(k[G/I], k[G/I])$. Itsdegree 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 irreducibleadmissible smooth representation of $G$, we have $H^d(I,V)=0$ unless $V$ is thetrivial representation.  When $\mathfrak F=\mathbb Q_p$ with $p\geq 5$, we have $d=3$. In that case wedescribe $E^*$ as an $H$-bimodule and give the structure as an algebra of thecentralizer in $E^*$ of the center of $H$. We deduce results on the values ofthe 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.

On the pro-$p$ Iwahori Hecke Ext-algebra of ${\rm SL}_2(\mathbb Q_p)$