The Stacks project

Lemma 42.61.5. Exterior product is associative. More precisely, let $k$ be a field, let $X, Y, Z$ be schemes locally of finite type over $k$, let $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(X)$, $\beta \in \mathop{\mathrm{CH}}\nolimits _*(Y)$, $\gamma \in \mathop{\mathrm{CH}}\nolimits _*(Z)$. Then $(\alpha \times \beta ) \times \gamma = \alpha \times (\beta \times \gamma )$ in $\mathop{\mathrm{CH}}\nolimits _*(X \times _ k Y \times _ k Z)$.

Proof. Omitted. Hint: associativity of fibre product of schemes. $\square$

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