Lemma 62.6.13. Let $S = \mathop{\mathrm{lim}}\nolimits _{i \in I} S_ i$ be the limit of a directed inverse system of Noetherian schemes with affine transition morphisms. Let $0 \in I$ and let $X_0 \to S_0$ be a finite type morphism of schemes. For $i \geq 0$ set $X_ i = S_ i \times _{S_0} X_0$ and set $X = S \times _{S_0} X_0$. If $S$ is Noetherian too, then
\[ z(X/S, r) = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} z(X_ i/S_ i, r) \]
where the transition maps are given by base change of relative $r$-cycles.
Proof.
Suppose that $i \geq 0$ and $\alpha _ i, \beta _ i \in z(X_ i/S_ i, r)$ map to the same element of $z(X/S, r)$. Then $S \to S_ i$ maps into the closed subset $E \subset S_ i$ of Lemma 62.6.12. Hence for some $j \geq i$ the morphism $S_ j \to S_ i$ maps into $E$, see Limits, Lemma 32.4.10. It follows that the base change of $\alpha _ i$ and $\beta _ i$ to $S_ j$ agree. Thus the map is injective.
Let $\alpha \in z(X/S, r)$. Applying Lemma 62.6.9 a completely decomposed proper morphism $g : S' \to S$ such that $g^*\alpha $ is in the image of (62.6.8.1). Set $X' = S' \times _ S X$. We write $g^*\alpha = \sum n_ a [Z_ a/X'/S']_ r$ for some $Z_ a \subset X'$ closed subscheme flat and of relative dimension $\leq r$ over $S'$.
Now we bring the machinery of Limits, Section 32.10 ff to bear. We can find an $i \geq 0$ such that there exist
a completely decomposed proper morphism $g_ i : S'_ i \to S_ i$ whose base change to $S$ is $g : S' \to S$,
setting $X'_ i = S'_ i \times _{S_ i} X_ i$ closed subschemes $Z_{ai} \subset X'_ i$ flat and of relative dimension $\leq r$ over $S'_ i$ whose base change to $S'$ is $Z_ a$.
To do this one uses Limits, Lemmas 32.10.1, 32.8.5, 32.8.7, 32.13.1, and 32.18.1 and More on Morphisms, Lemma 37.78.5. Consider $\alpha '_ i = \sum n_ a [Z_{ai}/X'_ i/S'_ i]_ r \in z(X'_ i/S'_ i, r)$. The image of $\alpha '_ i$ in $z(X'/S', r)$ agrees with the base change $g^*\alpha $ by construction.
Set $S''_ i = S'_ i \times _{S_ i} S'_ i$ and $X''_ i = S''_ i \times _{S_ i} X_ i$ and set $S'' = S' \times _ S S'$ and $X'' = S'' \times _ S X$. We denote $\text{pr}_1, \text{pr}_2 : S'' \to S'$ and $\text{pr}_1, \text{pr}_2 : S''_ i \to S'_ i$ the projections. The two base changes $\text{pr}_1^*\alpha '_ i$ and $\text{pr}_1^*\alpha '_ i$ map to the same element of $z(X''/S'', r)$ because $\text{pr}_1^*g^*\alpha = \text{pr}_1^*g^*\alpha $. Hence after increasing $i$ we may assume that $\text{pr}_1^*\alpha '_ i = \text{pr}_1^*\alpha '_ i$ by the first paragraph of the proof. By Lemma 62.5.9 we obtain a unique family $\alpha _ i$ of $r$-cycles on fibres of $X_ i/S_ i$ with $g_ i^*\alpha _ i = \alpha '_ i$ (this uses that $S'_ i \to S_ i$ is completely decomposed). By Lemma 62.6.3 we see that $\alpha _ i \in z(X_ i/S_ i, r)$. The uniqueness in Lemma 62.5.9 implies that the image of $\alpha _ i$ in $z(X/S, r)$ is $\alpha $ and the proof is complete.
$\square$
Comments (0)