Euler class groups and motivic stable cohomotopy

Abstract

We study maps from a smooth scheme to a motivic sphere in the Morel-Voevodsky ${\mathbb A}^1$-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 \`a 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.