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<p>This paper utilizes the properties of transforms of currents under equidimensional cycles, as introduced in (dos Santos et al. in J Lond Math Soc (2) 106(3): 2511–2561, 2022), to establish the multiplicative nature of the resulting regulator map, in the derived category. The construction relies on a synthetic presentation of the fundamental triples of currents from [6], which exhibits group-like behavior under an extended Eilenberg–Zilber morphism. A key component of the analysis is a character of the permutation Hopf algebra <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2024_3645_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\( \mathfrak {S}\textsf {Sym}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">S</mi> <mi mathvariant="sans-serif">Sym</mi> </mrow> </math></EquationSource> </InlineEquation> that takes values in the function field of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2024_3645_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\( \mathbb {A}_{\mathbb {Q}}^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">A</mi> <mrow> <mi mathvariant="double-struck">Q</mi> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation>.</p>

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Multiplicative properties of the current transform regulator

  • 期刊论文

  • 发表日期：

    2025年1月28日

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  • Paulo Lima-Filho

摘要

This paper utilizes the properties of transforms of currents under equidimensional cycles, as introduced in (dos Santos et al. in J Lond Math Soc (2) 106(3): 2511–2561, 2022), to establish the multiplicative nature of the resulting regulator map, in the derived category. The construction relies on a synthetic presentation of the fundamental triples of currents from [6], which exhibits group-like behavior under an extended Eilenberg–Zilber morphism. A key component of the analysis is a character of the permutation Hopf algebra \( \mathfrak {S}\textsf {Sym}\) S Sym that takes values in the function field of \( \mathbb {A}_{\mathbb {Q}}^\infty \) A Q ∞ .