The Stacks project

Remark 42.33.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. For $i = 1, 2$ let $Z_ i \to X$ be a morphism of schemes locally of finite type. Let $c_ i \in A^{p_ i}(Z_ i \to X)$, $i = 1, 2$ be bivariant classes. For any $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ we can ask whether

\[ c_1 \cap c_2 \cap \alpha = c_2 \cap c_1 \cap \alpha \]

in $\mathop{\mathrm{CH}}\nolimits _{k - p_1 - p_2}(Z_1 \times _ X Z_2)$. If this is true and if it holds after any base change by $X' \to X$ locally of finite type, then we say $c_1$ and $c_2$ commute. Of course this is the same thing as saying that

\[ res(c_1) \circ c_2 = res(c_2) \circ c_1 \]

in $A^{p_1 + p_2}(Z_1 \times _ X Z_2 \to X)$. Here $res(c_1) \in A^{p_1}(Z_1 \times _ X Z_2 \to Z_2)$ is the restriction of $c_1$ as in Remark 42.33.5; similarly for $res(c_2)$.

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