Lemma 42.67.5 (0FVK)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.67: Change of base scheme
Lemma 42.67.5 (cite)

Lemma 42.67.5. 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 flat of relative dimension $r$. Then $f' : Y' \to X'$ is flat of relative dimension $r$ and the diagrams

\[ \vcenter { \xymatrix{ Z_{k + r}(Y) \ar[r]_{g^*} & Z_{k + c + r}(Y') \\ Z_ k(X) \ar[r]^{g^*} \ar[u]^{(f')^*} & Z_{k + c}(X') \ar[u]_{f^*} } } \quad \text{and}\quad \vcenter { \xymatrix{ \mathop{\mathrm{CH}}\nolimits _{k + r}(Y) \ar[r]_{g^*} & \mathop{\mathrm{CH}}\nolimits _{k + c + r}(Y') \\ \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[r]^{g^*} \ar[u]^{(f')^*} & \mathop{\mathrm{CH}}\nolimits _{k + c}(X') \ar[u]_{f^*} } } \]

of cycle and chow groups commutes.

Proof. It suffices to show the first diagram commutes. To see this, let $Z \subset X$ 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 42.14.4 we have $(f')^*g^*[Z] = [Z' \times _{X'} Y']_{k + c + r}$. Conversely, we have $f^*[Z] = [Z \times _ X Y]_{k + r}$ by Definition 42.14.1. By Lemma 42.67.3 we have $g^*f^*[Z] = [(Z \times _ X Y)']_{k + r + c}$. Since $(Z \times _ X Y)' = Z' \times _{X'} Y'$ by associativity of fibre product we conclude. $\square$

Comments (1)

Comment #9773 by gad on September 19, 2024 at 03:36

For the diagrams to make sense, f^ and (f')^ should be interchanged.

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