Definition 42.12.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $f$ is proper.

Let $Z \subset X$ be an integral closed subscheme with $\dim _\delta (Z) = k$. We define

\[ f_*[Z] = \left\{ \begin{matrix} 0 & \text{if} & \dim _\delta (f(Z))< k, \\ \deg (Z/f(Z)) [f(Z)] & \text{if} & \dim _\delta (f(Z)) = k. \end{matrix} \right. \]Here we think of $f(Z) \subset Y$ as an integral closed subscheme. The degree of $Z$ over $f(Z)$ is finite if $\dim _\delta (f(Z)) = \dim _\delta (Z)$ by Lemma 42.11.1.

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 42.11.2 above.

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