## 43.10 Proper pushforward and rational equivalence

Suppose that $f : X \to Y$ is a proper morphism of varieties. Let $\alpha \sim _{rat} 0$ be a $k$-cycle on $X$ rationally equivalent to $0$. Then the pushforward of $\alpha$ is rationally equivalent to zero: $f_* \alpha \sim _{rat} 0$. See Chapter I of [F] or Chow Homology, Lemma 42.20.3.

Therefore we obtain a commutative diagram

$\xymatrix{ Z_ k(X) \ar[r] \ar[d]_{f_*} & \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[d]^{f_*} \\ Z_ k(Y) \ar[r] & \mathop{\mathrm{CH}}\nolimits _ k(Y) }$

of groups of $k$-cycles.

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