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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 $$u\in GW(k)^{\times}.$$

##### MSC:
 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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