The Stacks project

Remark 42.32.3. We will see later (Lemma 42.36.3) that if $X$ is a vector bundle of rank $r$ over $Y$ then the pullback map $\mathop{\mathrm{CH}}\nolimits _ k(Y) \to \mathop{\mathrm{CH}}\nolimits _{k + r}(X)$ is an isomorphism. This is true whenever $X \to Y$ satisfies the assumptions of Lemma 42.32.1, see [Lemma 2.2, Totaro-group]. We will sketch a proof in Remark 42.32.8 using higher chow groups.