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On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ modeled on linear and even orthogonal groups

Andrei Lavrenov, Sergey Sinchuk, Egor Voronetsky

TL;DR

The paper proves an unstable ${ m K}_2$-invariance (an unstable Bass–Quillen-type result) for split linear and even orthogonal groups by establishing $ ext{A}^1$-invariance of ${ m K}_2( ext{Φ},-)$ over regular rings $R$ containing a field $k$ for $ ext{Φ} subseteq ext{C}_ ext{something}$ (specifically $ ext{Φ}= ext{A}_ ext{ℓ}$ with $ ext{ℓ}\ge 4$ or $ ext{D}_ ext{ℓ}$ with $ ext{ℓ}\ge 7$ and ${ m char}(k) eq 2$). The main result identifies ${ m K}_2^ ext{G}(R)$ as the unstable $ ext{A}^1$-fundamental group of the simply-connected Chevalley–Demazure group scheme $ ext{G}_ ext{Λ}( ext{Φ},-)$ and embeds ${ m K}_2^ ext{G}(A)$ for semilocal regular $k$-algebras $A$ into the Milnor ${ m K}_2$ of the fraction field. The approach develops an axiomatic framework for functors on rings, verifies the necessary properties (including Panin’s lifting property and pro-group patching), and uses Nisnevich glueing for Steinberg groups to obtain descent results. Consequently, unstable ${ m K}_2$-groups are representable in unstable $ ext{A}^1$-homotopy theory and related to Milnor ${ m K}_2$ through central extensions, offering a robust K2-analogue of the Bass–Quillen conjecture in the geometric setting.

Abstract

Let $k$ be an arbitrary field. In this paper we show that in the linear case ($Φ=\mathsf{A}_\ell$, $\ell \geq 4$) and even orthogonal case ($Φ= \mathsf{D}_\ell$, $\ell\geq 7$, $\mathrm{char}(k)\neq 2$) the unstable functor $\mathrm{K}_2(Φ, -)$ possesses the $\mathbb{A}^1$-invariance property in the geometric case, i. e. $\mathrm{K}_2(Φ, R[t]) = \mathrm{K}_2(Φ, R)$ for a regular ring $R$ containing $k$. As a consequence, the unstable $\mathrm{K}_2$ groups can be represented in the unstable $\mathbb{A}^1$-homotopy category $\mathscr{H}_\bullet(k)$ as fundamental groups of the simply-connected Chevalley--Demazure group schemes $\mathrm{G}(Φ,-)$. Our invariance result can be considered as the $\mathrm{K}_2$-analogue of the geometric case of Bass--Quillen conjecture. We also show for a semilocal regular $k$-algebra $A$ that $\mathrm{K}_2(Φ, A)$ embeds as a subgroup into $\mathrm{K}^\mathrm{M}_2(\mathrm{Frac}\,A)$.

On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ modeled on linear and even orthogonal groups

TL;DR

The paper proves an unstable -invariance (an unstable Bass–Quillen-type result) for split linear and even orthogonal groups by establishing -invariance of over regular rings containing a field for (specifically with or with and ). The main result identifies as the unstable -fundamental group of the simply-connected Chevalley–Demazure group scheme and embeds for semilocal regular -algebras into the Milnor of the fraction field. The approach develops an axiomatic framework for functors on rings, verifies the necessary properties (including Panin’s lifting property and pro-group patching), and uses Nisnevich glueing for Steinberg groups to obtain descent results. Consequently, unstable -groups are representable in unstable -homotopy theory and related to Milnor through central extensions, offering a robust K2-analogue of the Bass–Quillen conjecture in the geometric setting.

Abstract

Let be an arbitrary field. In this paper we show that in the linear case (, ) and even orthogonal case (, , ) the unstable functor possesses the -invariance property in the geometric case, i. e. for a regular ring containing . As a consequence, the unstable groups can be represented in the unstable -homotopy category as fundamental groups of the simply-connected Chevalley--Demazure group schemes . Our invariance result can be considered as the -analogue of the geometric case of Bass--Quillen conjecture. We also show for a semilocal regular -algebra that embeds as a subgroup into .

Paper Structure

This paper contains 10 sections, 34 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. General formalism
  3. Proof of the main results
  4. Proof of Theorems \ref{['theorem:LP-for-K2']} and \ref{['theorem:Gersten']}
  5. K2(R) as $\mathbb{A}^1$-fundamental group
  6. Nisnevich glueing for K2(Ф,R)
  7. Plan of the proof
  8. Overview of pro-completion
  9. Overview of Steinberg pro-groups
  10. Proof of Theorem \ref{['glueing']}

Key Result

Theorem 1.1

Let $k$ be an arbitrary field and $R$ be a regular ring containing $k$. Let $\Phi$ be either $\mathsf{A}_\ell$ for $\ell\geq4$, or $\mathsf{D}_\ell$ for $\ell\geq 7$. In the latter case assume additionally that the characteristic of $k$ is not $2$. Then for any lattice $\Lambda$ as above and $G = \m

Theorems & Definitions (76)

  • Theorem 1.1: The ${\mathrm{K}}_2$-analogue of Lindel--Popescu theorem
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 66 more