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

TL;DR

This paper proves that the Ext-algebra associated with certain p-adic groups is finitely generated and finitely presented, providing explicit algebraic descriptions and deepening understanding of the algebraic structures underlying smooth mod-p representations.

Contribution

It establishes finite generation and presentation of the pro-p Iwahori-Hecke Ext-algebra for specific p-adic groups, with explicit algebraic presentations and computations.

Findings

The Ext-algebra is finitely generated for SL2, PGL2, and GL2 over unramified extensions.

For SL2(Qp), the algebra is finitely presented with an explicit presentation.

The multiplication map from the tensor algebra to the Ext-algebra is surjective with finitely generated kernel.

Abstract

The pro-$p$ Iwahori-Hecke $\operatorname{Ext}$-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 Vign\'eras and others. In the present article, we first show that the $\operatorname{Ext}$-algebra $E^\ast$ associated with the group $\operatorname{SL}_2(\mathfrak{F})$, $\operatorname{PGL}_2(\mathfrak{F})$ or $\operatorname{GL}_2(\mathfrak{F})$, where $\mathfrak{F}$ is an unramified extension of $\mathbb{Q}_p$ with $p \neq 2,3$, is finitely generated as a (non-commutative) algebra. We then specialize to the case of the group $\operatorname{SL}_2(\mathbb{Q}_p)$, with $p \neq 2,3$, and we show that in this case the natural multiplication map from the tensor algebra $T^\ast_{E^0} E^1$ 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.