Projective bundle theorem in MW-motivic cohomology. (English) Zbl 1480.14017
Let \(k\) be an infinite perfect field of characteristic \(\neq 2\). Let \(S\) be a separated smooth \(k\)-scheme, \(X\) a quasi-projective \(k\)-scheme equipped with a smooth \(S\)-scheme structure. Let \(E\) be a vector bundle of rank \(n\) over \(X\). The author computes the MW-motives (as defined by B. Calmès, F. Déglise and J. Fasel) of projective spaces and establishes a version of projective bundle theorem in MW-motives (resp. Chow-Witt rings). It is shown in particular that if \(e(E^{\vee})=0\in H^n(X,\,W(\det(E^{\vee})))\) and \({}_2CH(X)=0\), then \(\widetilde{CH}^*(\mathbb{P}(E))\) can be determined in terms of \(\widetilde{CH}^*(X)\), \(\widetilde{CH}^*(X,\,\det(E^{\vee}))\), \(CH^*(X)\) and the operator \(Sq^2\). As an application, the MW-motives of blow-ups with smooth centers are computed. The paper also discusses the invariance of Chow-Witt cycles of projective bundles under automorphisms of vector bundles.
Reviewer: Yong Hu (Guangdong)
MSC:
| 14F42 | Motivic cohomology; motivic homotopy theory |
| 11E81 | Algebraic theory of quadratic forms; Witt groups and rings |
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