The Stacks project

Proof. Since we are proving an equality of cycles on $X$, we may work locally on $Z$, see Lemma 62.11.2. Thus we may assume $Z$ is affine. In particular $\gamma $ is a finite linear combination of prime cycles. Since $- \cap -$ is linear in the second variable (Lemma 62.11.1), it suffices to prove the equality when $\gamma = [W]$ for some integral closed subscheme $W \subset Z$ of $\delta $-dimension $e$.

Let $z \in W$ be the generic point. Write $\beta _ z = \sum m_ j[V_ j]$ in $Z_ s(Y_ z)$. Then $\beta \cap \gamma $ is equal to $\sum m_ j[\overline{V}_ j]$ where $\overline{V}_ j \subset Y$ is an integral closed subscheme mapped by $Y \to Z$ into $W$ with generic fibre $V_ j$. Let $y_ j \in V_ j$ be the generic point. We may and do view also as the generic point of $\overline{V}_ j$ (mapping to $z$ in $W$). Write $\alpha _{y_ j} = \sum n_{jk} [W_{jk}]$ in $Z_ r(X_{y_ j})$. Then $\alpha \cap (\beta \cap \gamma )$ is equal to

where $\overline{W}_{jk} \subset X$ is an integral closed subscheme mapped by $X \to Y$ into $\overline{V}_ j$ with generic fibre $W_{jk}$.

On the other hand, let us consider

By the construction of $- \cap -$ this is equal to the cycle

on $X_ z$. Thus by definition we obtain

where $\widetilde{W}_{jk} \subset X$ is an integral closed subscheme which is mapped by $X \to Z$ into $W$ with generic fibre $(\overline{W}_{jk})_ z$. Clearly, we must have $\widetilde{W}_{jk} = \overline{W}_{jk}$ and the proof is complete. $\square$