Lemma 42.67.7 (0FVM)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.67: Change of base scheme
Lemma 42.67.7 (cite)

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Lemma 42.67.7. In Situation 42.67.1 let $X \to S$ be locally of finite type and let $X' \to S'$ be the base change by $S' \to S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module with base change $\mathcal{L}'$ on $X'$. Then the diagram

\[ \xymatrix{ \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[r]_{g^*} \ar[d]_{c_1(\mathcal{L}) \cap -} & \mathop{\mathrm{CH}}\nolimits _{k + c}(X') \ar[d]^{c_1(\mathcal{L}') \cap -} \\ \mathop{\mathrm{CH}}\nolimits _{k - 1}(X) \ar[r]^{g^*} & \mathop{\mathrm{CH}}\nolimits _{k + c - 1}(X') } \]

of chow groups commutes.

Proof. Let $p : L \to X$ be the line bundle associated to $\mathcal{L}$ with zero section $o : X \to L$. For $\alpha \in CH_ k(X)$ we know that $\beta = c_1(\mathcal{L}) \cap \alpha $ is the unique element of $\mathop{\mathrm{CH}}\nolimits _{k - 1}(X)$ such that $o_*\alpha = - p^*\beta $, see Lemmas 42.32.2 and 42.32.4. The same characterization holds after pullback. Hence the lemma follows from Lemmas 42.67.5 and 42.67.6. $\square$

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