The Stacks project

43.11 Flat pullback and rational equivalence

Suppose that $f : X \to Y$ is a flat morphism of varieties. Set $r = \dim (X) - \dim (Y)$. Let $\alpha \sim _{rat} 0$ be a $k$-cycle on $Y$ rationally equivalent to $0$. Then the pullback of $\alpha $ is rationally equivalent to zero: $f^* \alpha \sim _{rat} 0$. See Chapter I of [F] or Chow Homology, Lemma 42.20.2.

Therefore we obtain a commutative diagram

\[ \xymatrix{ Z_{k + r}(X) \ar[r] & \mathop{\mathrm{CH}}\nolimits _{k + r}(X) \\ Z_ k(Y) \ar[r] \ar[u]^{f^*} & \mathop{\mathrm{CH}}\nolimits _ k(Y) \ar[u]_{f^*} } \]

of groups of $k$-cycles.

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