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Let $\mathfrak F$ be a locally compact nonarchimedean field of positiveresidue characteristic $p$ and $k$ a field of characteristic $p$. Let $G$ bethe group of $\mathfrak{F}$-rational points of a connected reductive group over$\mathfrak{F}$ which we suppose $\mathfrak F$-split. Given a pro-$p$ Iwahorisubgroup $I$ of $G$, we consider the space $\mathbf X$ of $k$-valued functionswith compact support on $G/I$. It is naturally an object in the category${\operatorname{Mod}}{(G)}$ of all smooth $k$-representations of $G$.  We study the graded Ext-algebra$E^*={\operatorname{Ext}}_{{\operatorname{Mod}}(G)}^*(\mathbf X, \mathbf X)$.Its degree zero piece $E^0$ is the usual pro-$p$ Iwahori-Hecke algebra $H$. Wedescribe the product in $E^*$ and provide an involutive anti-automorphism of$E^*$. When $I$ is a Poincar\'e group of dimension $d$, the${\operatorname{Ext}}$-algebra $E^*$ is supported in degrees $i\in\{0\dots d\}$and we establish a duality theorem between $E^i$ and $E^{d-i}$. Under the samehypothesis (and assuming that $\mathbf G$ is almost simple and simplyconnected), we compute $E^d$ as an $H$-module on the left and on the right. Weprove that it is a direct sum of the trivial character, and of supersingularmodules.

The modular pro-$p$ Iwahori-Hecke ${\operatorname{Ext}}$-algebra