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On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ of Chevalley groups of simply-laced type

Sergei Sinchuk

TL;DR

The paper proves an $A^1$-invariance result for unstable $ ext{K}_2$ of simply-laced Chevalley groups $ ext{G}_{ ext{Λ}}( ext{Φ},-)$ with $ ext{Φ}$ containing $ ext{A}_4$ but excluding $ ext{E}_8$, establishing $ ext{K}_2^G(R[t]) o ext{K}_2^G(R)$ for regular $R$ over a field and interpreting $ ext{K}_2^G(R)$ as the $ ext{A}^1$-fundamental group $oldsymbol{ ext{π}}_1^{ ext{A}^1}(G)(R)$ in Morel–Voevodsky motivic homotopy theory. The work hinges on a refined Horrocks-type theorem for $ ext{K}_2$ that adapts Suslin–Tulenbaev strategies to ADE and beyond, using Allcock’s affine Curtis–Tits amalgamations, weight automorphisms, and a relative Curtis–Tits framework to control relative Steinberg presentations. A second contribution provides a Horrocks-dedekind stability result: for a Dedekind domain $A$, $ ext{K}_2( ext{A}_4,A) o ext{K}_2( ext{Φ},A[X_1,\dots,X_n])$ is surjective, and hence equality under suitable conditions (e.g., if $ ext{K}_2(A)$ is generated by Dennis–Stein symbols). Together, these results extend homotopy-invariance phenomena for unstable $K$-theory to a broad class of root systems (including $ ext{D}_ ext{ℓ}$, $ ext{E}_6$, $ ext{E}_7$) and connect unstable $ ext{K}_2$ to motivic fundamental groups, with implications for Bass–Quillen-type questions in the geometric and arithmetic settings.

Abstract

In this paper we study the $\mathbb{A}^1$-invariance of the unstable functor $\mathrm{K}_2(Φ, R)$ in the case when $Φ$ is an irreducible root system of type $\mathsf{ADE}$ containing $\mathsf{A}_4$ and not of type $\mathsf{E}_8$. We show that in the geometric case, i. e. when $R$ is a regular ring containing a field $k$ one has $\mathrm{K}_2(Φ, R[t]) = \mathrm{K}_2(Φ, R)$, which allows one to interpret the unstable $\mathrm{K}_2$ groups as $\mathbb{A}^1$-fundamental groups of Chevalley--Demazure group schemes in the $\mathbb{A}^1$-homotopy category. We also prove a variant of "early stability" theorem which allows one to find a generating set of $\mathrm{K}_2(Φ, A[X_1, \ldots X_n])$ in the case when $A$ is a Dedekind domain.

On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ of Chevalley groups of simply-laced type

TL;DR

The paper proves an -invariance result for unstable of simply-laced Chevalley groups with containing but excluding , establishing for regular over a field and interpreting as the -fundamental group in Morel–Voevodsky motivic homotopy theory. The work hinges on a refined Horrocks-type theorem for that adapts Suslin–Tulenbaev strategies to ADE and beyond, using Allcock’s affine Curtis–Tits amalgamations, weight automorphisms, and a relative Curtis–Tits framework to control relative Steinberg presentations. A second contribution provides a Horrocks-dedekind stability result: for a Dedekind domain , is surjective, and hence equality under suitable conditions (e.g., if is generated by Dennis–Stein symbols). Together, these results extend homotopy-invariance phenomena for unstable -theory to a broad class of root systems (including , , ) and connect unstable to motivic fundamental groups, with implications for Bass–Quillen-type questions in the geometric and arithmetic settings.

Abstract

In this paper we study the -invariance of the unstable functor in the case when is an irreducible root system of type containing and not of type . We show that in the geometric case, i. e. when is a regular ring containing a field one has , which allows one to interpret the unstable groups as -fundamental groups of Chevalley--Demazure group schemes in the -homotopy category. We also prove a variant of "early stability" theorem which allows one to find a generating set of in the case when is a Dedekind domain.

Paper Structure

This paper contains 17 sections, 49 theorems, 101 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Steinberg groups, $\mathop{\mathrm{K}}\nolimits_2$-groups and symbols
  5. Relative Steinberg groups
  6. Weight automorphisms
  7. Another presentation of the relative linear Steinberg group
  8. Curtis--Tits type presentations
  9. Curtis--Tits type presentation of affine Steinberg groups
  10. Relationship between $\mathop{\mathrm{St}}\nolimits(\Phi, A[X, X^{-1}])$ and $\mathop{\mathrm{St}}\nolimits(\Phi, A[X])$.
  11. Relative Curtis--Tits decompositions
  12. Horrocks theorem for $\mathop{\mathrm{K}}\nolimits_2$
  13. Formalism of triples
  14. First reductions
  15. Construction of an $\varpi_k$-pair
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $k$ be an arbitrary field and $R$ be a regular ring containing $k$. Let $\Phi$ be an irreducible simply-laced root system containing $\mathsf{A}_4$ but not of type $\mathsf{E}_8$. Then for any lattice $\Lambda$ as above and $G = \mathrm{G}_\Lambda(\Phi, -)$ one has $\mathop{\mathrm{K}}\nolimits_

Figures (1)

  • Figure 1: Root markings on extended Dynkin diagrams

Theorems & Definitions (108)

  • Theorem 1.1: The $\mathop{\mathrm{K}}\nolimits_2$-analogue of Lindel--Popescu theorem
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 98 more