The Stacks project

Lemma 62.13.6. Let $S$ be a scheme. Let

\[ \xymatrix{ X' \ar[r]_ f \ar[d] & X \ar[d] \\ Y' \ar[r]^ g & Y } \]

be a cartesian diagram of schemes locally of finite type over $S$ with $g$ proper. Let $r, e \geq 0$. Let $\alpha $ be a family of $r$-cycles on the fibres of $X/Y$. Let $\beta '$ be a family of $e$-cycles on the fibres of $Y'/S$. Then we have $f_*(g^*(\alpha ) \circ \beta ') = \alpha \circ g_*\beta '$.

Proof. Unwinding the definitions, this follows from Lemma 62.11.6. $\square$

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