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<p>Exterior power operations provide an additional structure on <i>K</i>-groups of schemes which lies at the heart of Grothendieck’s Riemann-Roch theory. Over the past decades, various authors have constructed such operations on higher <i>K</i>-theory. In this paper, we prove that these constructions actually yield the same operations, ultimately matching up the explicit combinatorial description by Harris, the first author and Taelman on the one hand and the recent, conceptually clear-cut construction by Barwick, Glasman, Mathew and Nikolaus on the other hand. This also leads to the proof of a conjecture by the first author about composition of these operations in the equivariant context, completing the proof that higher equivariant <i>K</i>-groups satisfy all axioms of a <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3681_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-ring.</p>

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Comparison of exterior power operations on higher K-theory of schemes

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年2月21日

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  • Bernhard Köck,
  • Ferdinando Zanchetta

摘要

Exterior power operations provide an additional structure on K-groups of schemes which lies at the heart of Grothendieck’s Riemann-Roch theory. Over the past decades, various authors have constructed such operations on higher K-theory. In this paper, we prove that these constructions actually yield the same operations, ultimately matching up the explicit combinatorial description by Harris, the first author and Taelman on the one hand and the recent, conceptually clear-cut construction by Barwick, Glasman, Mathew and Nikolaus on the other hand. This also leads to the proof of a conjecture by the first author about composition of these operations in the equivariant context, completing the proof that higher equivariant K-groups satisfy all axioms of a \(\lambda \) λ -ring.