Finite generation properties of the pro-$p$ Iwahori-Hecke $\mathrm{Ext}$-algebra

The pro-$p$ Iwahori-Hecke $\mathrm{Ext}$-algebra $E^{\ast }$ is a graded algebra that has been introduced and studied by Ollivier-Schneider, with the long-term goal of investigating the category of smooth mod-$p$ representations of $p$-adic reductive groups and its derived category. Its $0$th graded piece is the pro-$p$ Iwahori-Hecke algebra studied by Vignéras and others. In the present article, we first show that the $\mathrm{Ext}$-algebra $E^{\ast }$ associated with the group ${\mathrm{SL}}_2(\mathfrak{F})$, ${\mathrm{PGL}}_2(\mathfrak{F})$ or ${\mathrm{GL}}_2(\mathfrak{F})$, where $\mathfrak{F}$ is an unramified extension of ${\mathbb{Q}}_p$ with $p\mathrm{\ne }2,3$, is finitely generated as a (non-commutative) algebra. We then specialize to the case of the group ${\mathrm{SL}}_2({\mathbb{Q}}_p)$, with $p\mathrm{\ne }2,3$, and we show that in this case the natural multiplication map from the tensor algebra $T_{E^0}^{\ast }{E}^1$ to $E^{\ast }$ is surjective and that its kernel is finitely generated as a two-sided ideal. Using this fact as main input, we then show that $E^{\ast }$ is finitely presented as an algebra. We actually compute an explicit presentation.