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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