Lemma 62.13.1. The construction above is bilinear, i.e., we have $(\alpha _1 + \alpha _2) \circ \beta \alpha _1 \circ \beta + \alpha _1 \circ \beta $ and $\alpha \circ (\beta _1 + \beta _2) = \alpha \circ \beta _1 + \alpha \circ \beta _2$.
62.13 Composition of families of cycles on fibres
Let $X \to Y \to S$ be morphisms of schemes, both locally of finite type. Let $r, e \geq 0$. Let $\alpha $ be a family of $r$-cycles on fibres of $X/Y$ and let $\beta $ be a family of $e$-cycles on fibres of $Y/S$. Then we obtain a family of of $(r + e)$-cycles $\alpha \circ \beta $ on the fibres of $X/S$ by setting
\[ (\alpha \circ \beta )_ s = (Y_ s \to Y)^*\alpha \cap \beta _ s \]
More precisely, the expression $(Y_ s \to Y)^*\alpha $ denotes the base change of $\alpha $ by $Y_ s \to Y$ to a family of $r$-cycles on the fibres of $X_ s/Y_ s$ and the operation $- \cap -$ was defined and studied in Section 62.111.
Proof. Omitted. Hint: on fibres the construction is bilinear by Lemma 62.11.1. $\square$
Lemma 62.13.2. If $U \subset X$ and $V \subset Y$ are open and $f(U) \subset V$, then $(\alpha \circ \beta )|_ U$ is equal to $\alpha |_ U \circ \beta |_ V$.
Proof. Omitted. Hint: on fibres use Lemma 62.11.2. $\square$
Lemma 62.13.3. The formation of $\alpha \circ \beta $ is compatible with base change.
Proof. Let $g : S' \to S$ be a morphism of schemes. Denote $X' \to Y'$ the base change of $X \to Y$ by $g$. Denote $\alpha '$ the base change of $\alpha $ with respect to $Y' \to Y$. Denote $\beta '$ the base change of $\beta $ with respect to $S' \to S$. The assertion means that $\alpha ' \circ \beta '$ is the base change of $\alpha \circ \beta $ by $g : S' \to S$.
Let $s' \in S'$ be a point with image $s \in S$. Then
\[ (\alpha ' \circ \beta ')_{s'} = (Y'_{s'} \to Y')^*\alpha ' \cap \beta '_{s'} \]
We observe that
\[ (Y'_{s'} \to Y')^*\alpha ' = (Y'_{s'} \to Y')^*(Y' \to Y)^*\alpha = (Y'_{s'} \to Y_ s)^*(Y_ s \to Y)^*\alpha \]
and that $\beta '_{s'}$ is the base change of $\beta _ s$ by $s' = \mathop{\mathrm{Spec}}(\kappa (s')) \to \mathop{\mathrm{Spec}}(\kappa (s)) = s$. Hence the result follows from Lemma 62.11.3 applied to $(Y_ s \to Y)^*\alpha $, $\beta _ s$, $X_ s \to Y_ s \to s$, and base change by $s' \to s$. $\square$
Lemma 62.13.4. Let $f : X \to Y$ and $Y \to S$ be morphisms of schemes, both locally of finite type. Let $r, e \geq 0$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type, with $\dim (\text{Supp}(\mathcal{F}_ y)) \leq r$ for all $y \in Y$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module of finite type, with $\dim (\text{Supp}(\mathcal{G}_ s)) \leq e$ for all $s \in S$. If $\alpha = [\mathcal{F}/X/Y]_ r$ and $\beta = [\mathcal{G}/Y/S]_ e$ (Example 62.5.2) and $\mathcal{F}$ is flat over $Y$, then $\alpha \circ \beta = [\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}/X/S]_{r + e}$.
Proof. First we observe that $\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}$ is a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $s \in S$. Observe that
\[ (\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G})_ s = \mathcal{F}_ s \otimes _{\mathcal{O}_{X_ s}} f_ s^*\mathcal{G}_ s \]
by right exactness of tensor products. Moreover $\mathcal{F}_ s$ is flat over $Y_ s$ as a base change of a flat module. Thus the equality $(\alpha \circ \beta )_ s = [(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G})_ s]_{r + e}$ follows from Lemma 62.11.4. $\square$
Lemma 62.13.5. Let $f : X \to Y$ and $Y \to S$ be morphisms of schemes, both locally of finite type. Let $r, e \geq 0$. Let $Z \subset X$ be a closed subscheme of relative dimension $\leq r$ over $Y$. Let $W \subset Y$ be a closed subscheme of relative dimension $\leq e$ over $S$. If $\alpha = [Z/X/Y]_ r$ and $\beta = [W/Y/S]_ e$ (Example 62.5.4) and $Z$ is flat over $Y$, then $\alpha \circ \beta = [Z \times _ Y W/X/S]_{r + e}$.
Proof. This is a special case of Lemma 62.13.4 if we take $\mathcal{F} = \mathcal{O}_ Z$ and $\mathcal{F} = \mathcal{O}_ W$. $\square$
Lemma 62.13.6. Let $S$ be a scheme. Let 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 $f_*(g^*(\alpha ) \circ \beta ') = \alpha \circ g_*\beta '$.
\[ \xymatrix{ X' \ar[r]_ f \ar[d] & X \ar[d] \\ Y' \ar[r]^ g & Y } \]
Proof. Unwinding the definitions, this follows from Lemma 62.11.6. $\square$
Lemma 62.13.7. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X \to Y \to Z$ be morphisms of schemes locally of finite type over $S$. Let $r, s, e \geq 0$. Then where $\alpha $ is a family of $r$-cycles on fibres of $X/Y$, $\beta $ is a family of $s$-cycles on fibres of $Y/Z$, and $\gamma \in Z_ e(Z)$.
\[ (\alpha \circ \beta ) \cap \gamma = \alpha \cap (\beta \cap \gamma ) \quad \text{in}\quad Z_{r + s + e}(X) \]
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
\[ \sum m_ j n_{jk} [\overline{W}_{jk}] \]
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
\[ (\alpha \circ \beta )_ z = (Y_ z \to Y)^*\alpha \cap \beta _ z = (Y_ z \to Y)^*\alpha \cap (\sum m_ j [V_ j]) \]
By the construction of $- \cap -$ this is equal to the cycle
\[ \sum m_ j n_{jk} [(\overline{W}_{jk})_ z] \]
on $X_ z$. Thus by definition we obtain
\[ (\alpha \circ \beta ) \cap [W] = \sum m_ j n_{jk} [\widetilde{W}_{jk}] \]
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. $\square$
[1] To be sure, we use $s = \mathop{\mathrm{Spec}}(\kappa (s))$ as the base scheme with $\delta (s) = 0$.
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