The Stacks project

Proof. The formation of $\alpha \circ \beta$ is compatible with base change by Lemma 62.13.3. Thus we may assume $S$ is the spectrum of a discrete valuation ring with generic point $\eta$ and closed point $0$ and we have to show that $sp_{X/S}((\alpha \circ \beta )_\eta ) = (\alpha \circ \beta )_0$. Since we are trying to prove an equality of cycles, we may work locally on $Y$ and $X$ (this uses Lemmas 62.13.2 and 62.4.4 to see that the constructions commute with restriction). Thus we may assume $X$ and $Y$ are affine. By Lemma 62.6.9 we can find a completely decomposed proper morphism $g : Y' \to Y$ such that $g^*\alpha$ is in the image of (62.6.8.1).

Since the family of morphisms $g_\eta : Y'_\eta \to Y_\eta$ is completely decomposed, we can find $\beta '_\eta \in Z_ e(Y'_\eta )$ such that $\beta _\eta = \sum g_{\eta , *}\beta '_\eta$, see Chow Homology, Lemma 42.22.4. Set $\beta '_0 = sp_{Y'/S}(\beta '_\eta )$ so that $\beta ' = (\beta '_\eta , \beta '_0)$ is a relative $e$-cycle on $Y'/S$. Then $g_*\beta '$ and $\beta$ are relative $e$-cycles on $Y/S$ (Lemma 62.6.2) which have the same value at $\eta$ and hence are equal (Lemma 62.6.6). By linearity (Lemma 62.13.1) it suffices to show that $\alpha \circ g_*\beta '$ is a relative $(r + e)$-cycle.

Set $X' = X \times _ Y Y'$ and denote $f : X' \to X$ the projection. By Lemma 62.13.6 we see that $\alpha \circ g_*\beta ' = f_*(g^*\alpha \circ \beta ')$. By Lemma 62.6.2 it suffices to show that $g^*\alpha \circ \beta '$ is a relative $(r + e)$-cycle. Using Lemma 62.6.10 and bilinearity this reduces us to the case discussed in the next paragraph.

Assume $\alpha = [Z/X/Y]_ r$ and $\beta = [W/Y/S]$ where $Z \subset X$ is a closed subscheme flat and of relative dimension $\leq r$ over $Y$ and $W \subset Y$ is a closed subscheme flat and of relative dimension $\leq e$ over $S$. By Lemma 62.13.5 we see that

and $Z \times _ X W \subset X$ is a closed subscheme flat over $S$ of relative dimension $\leq r + e$. This is a relative $(r + e)$-cycle by Lemma 62.6.8. $\square$