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      Euler class groups and motivic stable cohomotopy

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          Abstract

          We study motivic cohomotopy sets in algebraic geometry using the Morel-Voevodsky \({{\mathbb A}^1}\)-homotopy category: these sets are defined in terms of maps from a smooth scheme to a motivic sphere. Following Borsuk, we show that in the presence of suitable dimension hypotheses on the source, our motivic cohomotopy sets can be equipped with abelian group structures. We then explore links between these motivic cohomotopy groups, Euler class groups \`a la Nori--Bhatwadekar--Sridharan and Chow-Witt groups. Using these links, we show that, at least for \(k\) an infinite field having characteristic unequal to \(2\), the Euler class group of codimension \(d\) cycles on a smooth affine \(k\)-variety of dimension \(d\) coincides with the codimension \(d\) Chow-Witt 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.

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            Vector bundles over affine surfaces

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              The Euler class group of a polynomial algebra

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                Author and article information

                Journal
                1601.05723

                Geometry & Topology
                Geometry & Topology

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