Euler class groups and motivic stable cohomotopy

Article Type

Research Article

Publication Title

Journal of the European Mathematical Society

Abstract

We study maps from a smooth scheme to a motivic sphere in the Morel-Voevodsky A1-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, if X is a smooth affine k-variety of dimension d, then the Euler class group of codimension d cycles coincides with the codimension d 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 k-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.

First Page

2775

Last Page

2822

DOI

10.4171/JEMS/1156

Publication Date

1-1-2022

Comments

Open Access, Gold

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