The Stacks project

62.14 Composition of relative cycles

Let $S$ be a locally Noetherian scheme. Let $X \to Y$ be a morphism of schemes locally of finite type over $S$. We are going to define a map

\[ z(X/Y, r) \otimes _\mathbf {Z} z(Y/S, e) \longrightarrow z(X/S, r + e),\quad \alpha \otimes \beta \longmapsto \alpha \circ \beta \]

using the construction in Section 62.13. We already know the construction is bilinear (Lemma 62.13.1) hence we obtain the displayed arrow once we show the following.

Lemma 62.14.1. If $\alpha $ and $\beta $ are relative cycles, then so is $\alpha \circ \beta $.

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 (

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

\[ \alpha \circ \beta = [Z \times _ X W/X/S]_{r + e} \]

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. $\square$

Lemma 62.14.2. Let $f : X \to Y$ and $g : Y \to S$ be a morphisms of schemes. Assume $S$ locally Noetherian, $g$ locally of finite type and flat of relative dimension $e \ge 0$, and $f$ locally of finite type and flat of relative dimension $r \geq 0$. Then $[X/X/Y]_ r \circ [Y/Y/S]_ e = [X/X/S]_{r + e}$ in $z(X/S, r + e)$.

Proof. Special case of Lemma 62.13.5. $\square$

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