Definition 42.14.1. 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 flat of relative dimension $r$.

Let $Z \subset Y$ be an integral closed subscheme of $\delta $-dimension $k$. We define $f^*[Z]$ to be the $(k+r)$-cycle on $X$ to the scheme theoretic inverse image

\[ f^*[Z] = [f^{-1}(Z)]_{k+r}. \]This makes sense since $\dim _\delta (f^{-1}(Z)) = k + r$ by Lemma 42.13.1.

Let $\alpha = \sum n_ i [Z_ i]$ be a $k$-cycle on $Y$. The

*flat pullback of $\alpha $ by $f$*is the sum\[ f^* \alpha = \sum n_ i f^*[Z_ i] \]where each $f^*[Z_ i]$ is defined as above. The sum is locally finite by Lemma 42.13.2.

We denote $f^* : Z_ k(Y) \to Z_{k + r}(X)$ the map of abelian groups so obtained.

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