The Stacks project

Remark 62.12.4. Let $(S, \delta )$ be as in Section 62.11. Let $X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $r \geq 0$. Let $c$ be a rule that to every morphism $g : Y' \to Y$ locally of finite type and every $e \in \mathbf{Z}$ associates an operation

\[ c \cap - : Z_ e(Y') \to Z_{r + e}(X') \]

compatible with proper pushforward, flat pullback, and gysin maps as in Lemma 62.12.2. Then we claim there is a relative $r$-cycle $\alpha $ on $X/Y$ such that $c \cap = g^*\alpha \cap -$ for every $g$ as above. If we ever need this, we will carefully state and prove this here.

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View Remark 62.12.4 as pdf