We study the <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_805_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {A}}^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">A</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-invariance of the unstable functor <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_805_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{K}_2(\Phi , R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>K</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in the case when <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_805_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> is an irreducible root system of type <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_805_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\textsf{ADE}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ADE</mi> </math></EquationSource> </InlineEquation> containing <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_805_Article_IEq11.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">${\textsf{A}}_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="sans-serif">A</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> and not of type <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_805_Article_IEq12.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\textsf{E}}_8$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="sans-serif">E</mi> <mn>8</mn> </msub> </math></EquationSource> </InlineEquation>. We show that in the geometric case, i. e. when R is a regular ring containing a field k one has <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_805_Article_IEq13.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="173" /> </InlineMediaObject> <EquationSource Format="TEX">${{\,\textrm{K}\,}}_2(\Phi , R[t]) = {{\,\textrm{K}\,}}_2(\Phi , R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mspace width="0.166667em" /> <mtext>K</mtext> <mspace width="0.166667em" /> </mrow> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo>,</mo> <mi>R</mi> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mrow> <mspace width="0.166667em" /> <mtext>K</mtext> <mspace width="0.166667em" /> </mrow> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which allows one to interpret the unstable <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_805_Article_IEq14.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">${{\,\textrm{K}\,}}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mspace width="0.166667em" /> <mtext>K</mtext> <mspace width="0.166667em" /> </mrow> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> groups as <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_805_Article_IEq15.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {A}}^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">A</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-fundamental groups of Chevalley–Demazure group schemes in the <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_805_Article_IEq16.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {A}}^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">A</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-homotopy category. We also prove a variant of “early stability” theorem which allows one to find a generating set of <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_805_Article_IEq17.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </InlineMediaObject> <EquationSource Format="TEX">${{\,\textrm{K}\,}}_2(\Phi , A[X_1, \ldots X_n])$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mspace width="0.166667em" /> <mtext>K</mtext> <mspace width="0.166667em" /> </mrow> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo>,</mo> <mi>A</mi> <mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in the case when A is a Dedekind domain.

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

期刊论文
发表日期：

2025年2月4日

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Sergei Sinchuk

摘要

We study the ${\mathbb {A}}^1$ $A^{1}$ -invariance of the unstable functor $\textrm{K}_2(\Phi , R)$ $K_{2} (Φ, R)$ in the case when $\Phi $ $Φ$ is an irreducible root system of type $\textsf{ADE}$ $ADE$ containing ${\textsf{A}}_4$ $A_{4}$ and not of type ${\textsf{E}}_8$ $E_{8}$ . We show that in the geometric case, i. e. when R is a regular ring containing a field k one has ${{\,\textrm{K}\,}}_2(\Phi , R[t]) = {{\,\textrm{K}\,}}_2(\Phi , R)$ $K_{2} (Φ, R [t]) = K_{2} (Φ, R)$ , which allows one to interpret the unstable ${{\,\textrm{K}\,}}_2$ $K_{2}$ groups as ${\mathbb {A}}^1$ $A^{1}$ -fundamental groups of Chevalley–Demazure group schemes in the ${\mathbb {A}}^1$ $A^{1}$ -homotopy category. We also prove a variant of “early stability” theorem which allows one to find a generating set of ${{\,\textrm{K}\,}}_2(\Phi , A[X_1, \ldots X_n])$ $K_{2} (Φ, A [X_{1}, \dots X_{n}])$ in the case when A is a Dedekind domain.

On the \({\mathbb {A}}^1\)-invariance of \({{\,\textrm{K}\,}}_2\) of Chevalley groups of simply-laced type

摘要