## 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

of groups of $k$-cycles.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)