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Comparing Euler classes. (English) Zbl 1372.14013
Let \(k\) be a perfect field of characteristic different than 2, \(X\) a \(d\)-dimensional smooth \(k\) scheme and \(\xi :{\mathcal E}\rightarrow X\) a rank \(r\) vector bundle on \(X.\) There are two ways to define an Euler class \(e(\xi)\) of \(\xi\). One is via “characteristic class” approach and is as follows. Let \(s_{0}: X\rightarrow \mathcal E\) be the zero section of \(\xi\) then one can form a pullback and the Gysin pushforward homomorphism in Chow-Witt groups. The Euler class is then defined as \[ e_{cw}(\xi)=({\xi}^*)^{-1}(s_0)_*<1>\, \in \widetilde{\mathrm{CH}}^{r}(X,{\det}({\xi})^{\vee}). \] The second definition was given in [F. Morel, \(\mathbb A^1\)-algebraic topology over a field. Berlin: Springer (2012; Zbl 1263.14003)] and relies on a construction of an Euler class as the primary obstruction to existence of a non-vavanishing section of \(\xi .\) The first non-trivial stage of a Moore-Postnikov factorization in \({\mathbb A}^1\)-homotopy theory of the map \(\mathrm{Gr}_{r-1}\rightarrow \mathrm{Gr}_{r},\) where \(\mathrm{Gr}_r\) is the infinite Grassmanian with the universal rank \(r\) vector bundle \({\gamma}_r\) yields a canonical equivariant cohomology class \({o}_r \in H^r_{\text{Nis}}(\mathrm{Gr}_r, {{\mathbf K}}_r^{MW}(\det\,{\gamma}_r^{\vee}))\) [loc. cit., Appendix B]. Then \[ e_{ob}(\xi)={\xi}^*(o_r). \] But one can construct an isomorphism \[ H^r_{\text{Nis}}(\mathrm{Gr}_r, {\mathbf{K}}_r^{MW}(\det\,{\gamma}_r^{\vee})) \,\rightarrow \,\widetilde{\mathrm{CH}}^{r}(X,{\det}({\xi})^{\vee}) \] and this makes possible to compare both definitions. The main theorem of the paper is that if \({\xi}\) is oriented then both classes differ by a unit \(u\in GW(k)^{\times}.\)

14F42 Motivic cohomology; motivic homotopy theory
14C25 Algebraic cycles
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
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