The Stacks project

Lemma 42.29.8. Let $(S, \delta )$ be as in Situation 42.7.1. 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 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 Remark 42.29.7. For any $(k + 1)$-cycle $\alpha '$ on $X'$ we have $i^*f_*\alpha ' = g_*(i')^*\alpha '$ in $\mathop{\mathrm{CH}}\nolimits _ k(D)$ (this makes sense as $f_*$ is defined on the level of cycles).

Proof. 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 Lemma 42.26.3. Hence the equality holds as cycles. In case $W' \subset D'$, then $W \subset D$ and $f_*(c_1(\mathcal{L}|_{W'}) \cap [W'])$ is equal to $c_1(\mathcal{L}|_ W) \cap [W]$ in $\mathop{\mathrm{CH}}\nolimits _ k(W)$ by the second assertion of Lemma 42.26.3. By Remark 42.19.6 the result follows for general $\alpha '$. $\square$

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