Lemma 42.67.6. In Situation 42.67.1 let Y \to X \to S be locally of finite type and let Y' \to X' \to S' be the base change by S' \to S. Assume f : Y \to X is proper. Then f' : Y' \to X' is proper and the diagram
\vcenter { \xymatrix{ Z_ k(Y) \ar[r]_{g^*} \ar[d]_{f_*} & Z_{k + c}(Y') \ar[d]^{f'_*} \\ Z_ k(X) \ar[r]^{g^*} & Z_{k + c}(X') } } \quad \text{and}\quad \vcenter { \xymatrix{ \mathop{\mathrm{CH}}\nolimits _ k(Y) \ar[r]_{g^*} \ar[d]_{f_*} & \mathop{\mathrm{CH}}\nolimits _{k + c}(Y') \ar[d]^{f'_*} \\ \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[r]^{g^*} & \mathop{\mathrm{CH}}\nolimits _{k + c}(X') } }
of cycle and chow groups commutes.
Proof.
It suffices to show the first diagram commutes. To see this, let Z \subset Y be an integral closed subscheme of \delta -dimension k and denote Z' \subset X' its base change. By construction we have g^*[Z] = [Z']_{k + c}. By Lemma 42.12.4 we have (f')_*g^*[Z] = [f'_*\mathcal{O}_{Z'}]_{k + c}. By the same lemma we have f_*[Z] = [f_*\mathcal{O}_ Z]_ k. By Lemma 42.67.3 we have g^*f_*[Z] = [(X' \to X)^*f_*\mathcal{O}_ Z]_{k + r}. Thus it suffices to show that
(X' \to X)^*f_*\mathcal{O}_ Z \cong f'_*\mathcal{O}_{Z'}
as coherent modules on X'. As X' \to X is flat and as \mathcal{O}_{Z'} = (Y' \to Y)^*\mathcal{O}_ Z, this follows from flat base change, see Cohomology of Schemes, Lemma 30.5.2.
\square
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