The Stacks project

Lemma 45.8.4. Let $K/k$ be an algebraic extension of fields. Let $X$ be a finite type scheme over $k$. The kernel of the map $\mathop{\mathrm{CH}}\nolimits _ i(X) \to \mathop{\mathrm{CH}}\nolimits _ i(X_ K)$ constructed in Lemma 45.8.2 is torsion.

Proof. It clearly suffices to show that the kernel of flat pullback $\mathop{\mathrm{CH}}\nolimits _ i(X) \to \mathop{\mathrm{CH}}\nolimits _ i(X_{k'})$ by $\pi : X_{k'} \to X$ is torsion for any finite extension $k'/k$. This is clear because $\pi _* \pi ^* \alpha = [k' : k] \alpha $ by Chow Homology, Lemma 42.15.2. $\square$

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