Lemma 62.12.1. Let $(S, \delta )$ be as in Section 62.11. Let $f : X' \to X$ be a proper morphism of schemes locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Chow Homology, Definition 42.29.1. Form the diagram
\[ \xymatrix{ D' \ar[d]_ g \ar[r]_{i'} & X' \ar[d]^ f \\ D \ar[r]^ i & X } \]
as in Chow Homology, Remark 42.29.7. If $\mathcal{L}|_ D \cong \mathcal{O}_ D$, then $i^*f_*\alpha ' = g_*(i')^*\alpha '$ in $Z_ k(D)$ for any $\alpha ' \in Z_{k + 1}(X')$.
Proof.
The statement makes sense as all operations are defined on the level of cycles, see Chow Homology, Remark 42.29.6 for the gysin maps. Suppose $\alpha = [W']$ for some integral closed subscheme $W' \subset X'$. Let $W = f(W') \subset X$. In case $W' \not\subset D'$, then $W \not\subset D$ and we see that
\[ [W' \cap D']_ k = \text{div}_{\mathcal{L}'|_{W'}}({s'|_{W'}}) \quad \text{and}\quad [W \cap D]_ k = \text{div}_{\mathcal{L}|_ W}(s|_ W) \]
and hence $f_*$ of the first cycle equals the second cycle by Chow Homology, Lemma 42.26.3. Hence the equality holds as cycles. In case $W' \subset D'$, then $W \subset D$ and both sides are zero by construction.
$\square$
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