<p>Denote the virtual cohomological dimension of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_374_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}_n(\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> by <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_374_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="107" /> </InlineMediaObject> <EquationSource Format="TEX">$t=n(n-1)/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Let <i>St</i> denote the Steinberg module of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_374_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}_n(\mathbb {Q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> tensored with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_374_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_374_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$Sh_\bullet \rightarrow St$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>h</mi> <mo>∙</mo> </msub> <mo stretchy="false">→</mo> <mi>S</mi> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation> denote the sharbly resolution of the Steinberg module. By Borel–Serre duality, the one-dimensional <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_374_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation>-vector space <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_374_Article_IEq9.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$H^0(\textrm{SL}_n(\mathbb {Z}), \mathbb {Q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mtext>SL</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is isomorphic to <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_374_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="107" /> </InlineMediaObject> <EquationSource Format="TEX">$H_t(\textrm{SL}_n(\mathbb {Z}),St)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mtext>SL</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>S</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We find an explicit generator of <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_374_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="107" /> </InlineMediaObject> <EquationSource Format="TEX">$H_t(\textrm{SL}_n(\mathbb {Z}),St)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mtext>SL</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>S</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of sharbly cycles and cosharbly cocycles. These methods may extend to other degrees of cohomology of <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_374_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}_n(\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Explicit sharbly cycles at the virtual cohomological dimension for $\textrm{SL}_n(\mathbb {Z})$

期刊论文
发表日期：

2025年7月9日

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Avner Ash,
Paul E. Gunnells,
Mark McConnell

摘要

Denote the virtual cohomological dimension of $\textrm{SL}_n(\mathbb {Z})$ ${SL}_{n} (Z)$ by $t=n(n-1)/2$ $t = n (n - 1) / 2$ . Let St denote the Steinberg module of $\textrm{SL}_n(\mathbb {Q})$ ${SL}_{n} (Q)$ tensored with $\mathbb {Q}$ $Q$ . Let $Sh_\bullet \rightarrow St$ $S h_{∙} \to S t$ denote the sharbly resolution of the Steinberg module. By Borel–Serre duality, the one-dimensional $\mathbb {Q}$ $Q$ -vector space $H^0(\textrm{SL}_n(\mathbb {Z}), \mathbb {Q})$ $H^{0} ({SL}_{n} (Z), Q)$ is isomorphic to $H_t(\textrm{SL}_n(\mathbb {Z}),St)$ $H_{t} ({SL}_{n} (Z), S t)$ . We find an explicit generator of $H_t(\textrm{SL}_n(\mathbb {Z}),St)$ $H_{t} ({SL}_{n} (Z), S t)$ in terms of sharbly cycles and cosharbly cocycles. These methods may extend to other degrees of cohomology of $\textrm{SL}_n(\mathbb {Z})$ ${SL}_{n} (Z)$ .

Explicit sharbly cycles at the virtual cohomological dimension for \(\textrm{SL}_n(\mathbb {Z})\)

摘要