Definition 82.8.1. In Situation 82.2.1 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 82.7.1. 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 82.7.4 and Spaces over Fields, Definition 72.5.2.
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 82.7.5 above.
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