Projective bundle theorem in MW-motivic cohomology

Abstract

We present a version of projective bundle theorem in MW-motives (resp. Chow-Witt rings), which says that ${\widetilde{CH}}^{\ast }(\mathbb{P}(E))$ is determined by ${\widetilde{CH}}^{\ast }(X),{\widetilde{CH}}^{\ast }(X,det(E{)}^{\vee }),C{H}^{\ast }(X)$ and $S{q}^2$ for smooth quasi-projective schemes $X$ and vector bundles $E$ over $X$ with $e(E^{\vee })=0\in H^n(X,W(det(E)))$, provided that ${}_{2}C{H}^{\ast }(X)=0$.

As an application, we compute the MW-motives of blow-ups with smooth centers. Moreover, we discuss the invariance of Chow-Witt cycles of projective bundles under automorphisms of vector bundles.