JournalsjemsVol. 24, No. 8pp. 2775–2822

Euler class groups and motivic stable cohomotopy (with an appendix by Mrinal Kanti Das)

  • Aravind Asok

    University of Southern California, Los Angeles, USA
  • Jean Fasel

    Université Grenoble Alpes, France
Euler class groups and motivic stable cohomotopy (with an appendix by Mrinal Kanti Das) cover
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Abstract

We study maps from a smooth scheme to a motivic sphere in the Morel–Voevodsky A1\mathbb{A}^1-homotopy category, i.e., motivic cohomotopy sets. Following Borsuk, we show that, in the presence of suitable hypotheses on the dimension of the source, motivic cohomotopy sets can be equipped with functorial abelian group structures. We then explore links between motivic cohomotopy groups, Euler class groups à la Nori–Bhatwadekar–Sridharan and Chow–Witt groups. We show that, again under suitable hypotheses on the base field k{k}, if X{X} is a smooth affine kk-variety of dimension dd, then the Euler class group of codimension dd cycles coincides with the codimension dd Chow–Witt group; the identification proceeds by comparing both groups with a suitable motivic cohomotopy group. As a byproduct, we describe the Chow group of zero cycles on a smooth affine kk-scheme as the quotient of the free abelian group on zero cycles by the subgroup generated by reduced complete intersection ideals; this answers a question of S. Bhatwadekar and R. Sridharan.

Cite this article

Aravind Asok, Jean Fasel, Euler class groups and motivic stable cohomotopy (with an appendix by Mrinal Kanti Das). J. Eur. Math. Soc. 24 (2022), no. 8, pp. 2775–2822

DOI 10.4171/JEMS/1156