[Paper Review] On the congruence kernel of isotropic groups over rings

Anastasia Stavrova|arXiv (Cornell University)|Apr 30, 2013

Advanced Algebra and Geometry20 references3 citations

TL;DR

This paper establishes the centrality of the congruence kernel in the profinite completion of the elementary subgroup $E(R)$ for isotropic simply connected reductive groups over connected noetherian rings $R$, under mild conditions on the ring and group type. The proof relies on constructing Steinberg groups for isotropic groups and analyzing their behavior under profinite and congruence completions via local-global principles and dense subgroup arguments.

ABSTRACT

Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ge 2. The elementary subgroup E(R) of G(R) is the subgroup generated by the R-points U_P^+(R) and U_P^-(R) of the unipotent radicals of two opposite parabolic subgroups P^+ and P^- of G. Assume that 2 is invertible in R if G is of type B_n,C_n,F_4,G_2 and 3 is invertible in R if G is of type G_2. We prove that the congruence kernel of E(R), defined as the kernel of the natural homomorphism between the profinite completion of E(R) and the congruence completion of E(R) with respect to congruence subgroups of finite index, is central. In the course of the proof, we construct Steinberg groups associated to isotropic reductive groups and show that they are central extensions of E(R) if R is a local ring.

Motivation & Objective

• To extend the centrality of the congruence kernel from split groups to isotropic reductive groups over general noetherian rings.
• To establish that the congruence kernel of the elementary subgroup $E(R)$ is central in its profinite completion $\widehat{E(R)}$ under mild ring-theoretic conditions.
• To generalize results of Rapinchuk–Rapinchuk and Kassabov–Nikolov to isotropic groups of rank $\geq 2$ over arbitrary connected noetherian rings.
• To construct Steinberg groups for isotropic reductive groups and show they are central extensions of $E(R)$ when $R$ is local.
• To prove that the congruence completion $\overline{E(R)}$ coincides with $\overline{G(R)}$ under the given assumptions, enabling global analysis via local data.

Proposed method

• Constructs Steinberg groups associated to isotropic reductive groups over local rings, showing they are central extensions of $E(R)$.
• Uses the local structure of $G$ at maximal ideals $m$ of $R$ to analyze completions $\widehat{E(R)}$ and $\overline{E(R)}$ via $\hat{R}_m$-completions.
• Applies the local-to-global principle by showing that the image of $E(R)$ is dense in $\widehat{E(R)}$ through the subgroup $\Delta$ generated by local completions $\hat{\Gamma}_m$.
• Employs the fact that $\widehat{U_{(\delta)}(R)} \cong U_{(\delta)}(\hat{R})$ and uses Lemma 5.4 to embed local unipotent subgroups into the profinite completion.
• Leverages the surjectivity of the map from a compact subset $S \subseteq \widehat{E(R)}$ onto $E(\hat{R})$ to show that $C \cap \Delta_m$ is dense in $C$, the congruence kernel.
• Uses the fact that $\hat{\Gamma}_m$ centralizes $\hat{\Gamma}_m'$ and $C \cap \Delta_m$, and since $\Delta$ is dense in $\widehat{E(R)}$, the kernel $C$ is central in $\widehat{E(R)}$.

Experimental results

Research questions

• RQ1 Is the congruence kernel of the elementary subgroup $E(R)$ central in its profinite completion when $G$ is an isotropic simply connected reductive group over a connected noetherian ring $R$?
• RQ2 Can the centrality result from split groups be extended to isotropic groups of rank $\geq 2$ over general rings?
• RQ3 What is the role of Steinberg groups in the structure of $E(R)$ for isotropic groups over local rings?
• RQ4 How do the profinite and congruence completions of $E(R)$ relate when $G$ has isotropic rank $\geq 2$?
• RQ5 Under what conditions does $\overline{E(R)} = \overline{G(R)}$ hold, enabling reduction to local analysis?

Key findings

• The congruence kernel $C = \ker(\widehat{E(R)} \to \overline{E(R)})$ is central in $\widehat{E(R)}$ for any connected noetherian ring $R$ and isotropic simply connected reductive group $G$ of rank $\geq 2$.
• The congruence completion $\overline{E(R)}$ coincides with $\overline{G(R)}$ under the given assumptions, allowing global analysis via local data.
• For any local ring $R$, the Steinberg group associated to $G$ is a central extension of $E(R)$, providing a key structural tool.
• The subgroup $\Delta \subseteq \widehat{E(R)}$ generated by local completions $\hat{\Gamma}_m$ is dense in $\widehat{E(R)}$, enabling global centrality arguments.
• The image of $E(R)$ is dense in $\widehat{E(R)}$ because $\sum_{m} \hat{R}_m$ is dense in $\hat{R}$, and $\widehat{U_{(\delta)}(R)} \cong U_{(\delta)}(\hat{R})$.
• The kernel $C$ is centralized by all $\hat{\Gamma}_m$, and since these generate a dense subgroup, $C$ is central in $\widehat{E(R)}$.