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.
$\square$

## Comments (0)