Finite generation properties of the pro-p Iwahori-Hecke Ext-algebra

The pro-p Iwahori-Hecke Ext-algebra E^ 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 Vign\'eras and others. In the present article, we first show that the Ext-algebra E^ associated with the group SL₂(F), PGL₂(F) or GL₂(F), where F is an unramified extension of Qₚ with p ≠ 2,3, is finitely generated as a (non-commutative) algebra. We then specialize to the case of the group SL₂(Qₚ), with p ≠ 2,3, and we show that in this case the natural multiplication map from the tensor algebra TE⁰ E¹ to E^ 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^ is finitely presented as an algebra. We actually compute an explicit presentation.