Definition 81.8.1. In Situation 81.2.1 let $X, Y/B$ be good. Let $f : X \to Y$ be a morphism over $B$. Assume $f$ is proper.

Let $Z \subset X$ be an integral closed subspace with $\dim _\delta (Z) = k$. Let $Z' \subset Y$ be the image of $Z$ as in Lemma 81.7.1. We define

\[ f_*[Z] = \left\{ \begin{matrix} 0 & \text{if} & \dim _\delta (Z')< k, \\ \deg (Z/Z') [Z'] & \text{if} & \dim _\delta (Z') = k. \end{matrix} \right. \]The degree of $Z$ over $Z'$ is defined and finite if $\dim _\delta (Z') = \dim _\delta (Z)$ by Lemma 81.7.4 and Spaces over Fields, Definition 71.5.2.

Let $\alpha = \sum n_ Z [Z]$ be a $k$-cycle on $X$. The

*pushforward*of $\alpha $ as the sum\[ f_* \alpha = \sum n_ Z f_*[Z] \]where each $f_*[Z]$ is defined as above. The sum is locally finite by Lemma 81.7.5 above.

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