The Stacks project

Proof. Let a diagram and $\alpha _0 \in z(X_0/S_0, r)$ as in (1) be given. By Lemma 62.6.9 there exists a proper surjective morphism $g_0 : S'_0 \to S_0$ such that $g_0^*\alpha _0$ is in the image of (62.16.0.1). Namely, since $S'_0$ is Noetherian, every closed subscheme of $S'_0 \times _{S_0} X_0$ is of finite presentation over $S'_0$. Setting $S' = S \times _{S_0} S'_0$ and using base change by $S' \to S'_0$ we see that (2) holds.

Conversely, assume that (2) holds. Choose a surjective proper morphism $g : S' \to S$ of finite presentation such that $g^*\alpha $ is in the image of (62.16.0.1). Set $X' = S' \times _ S X$. Write $g^*\alpha = \sum n_ a [Z_ a/X'/S']_ r$ for some $Z_ a \subset X'$ closed subscheme flat, of finite presentation, and of relative dimension $\leq r$ over $S'$.

Write $S = \mathop{\mathrm{lim}}\nolimits S_ i$ as a directed limit with affine transition morphisms with $S_ i$ of finite type over $\mathbf{Z}$, see Limits, Proposition 32.5.4. We can find an $i$ large enough such that there exist

1. a completely decomposed proper morphism $g_ i : S'_ i \to S_ i$ whose base change to $S$ is $g : S' \to S$,

2. 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, 32.8.15, 32.13.1, and 32.18.1 and 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 base change of $\alpha '_ i$ to a family of $r$-cycles on fibres of $X'/S'$ 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 relative $r$-cycles $\text{pr}_1^*\alpha '_ i$ and $\text{pr}_1^*\alpha '_ i$ on $X''_ i/S''_ i$ base change to the same family of $r$-cycles on fibres of $X''/S''$ because $\text{pr}_1^*g^*\alpha = \text{pr}_1^*g^*\alpha $. Hence the morphism $S'' \to S''_ i$ maps into $E = \{ s \in S''_ i : (\text{pr}_1^*\alpha '_ i)_ s = (\text{pr}_1^*\alpha '_ i)_ s\} $. By Lemma 62.6.12 this is a closed subset. Since $S'' = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} S''_{i'}$ we see from Limits, Lemma 32.4.10 that for some $i' \geq i$ the morphism $S''_{i'} \to S''_ i$ maps into $E$. Therefore, after replacing $i$ by $i'$, we may assume that $\text{pr}_1^*\alpha '_ i = \text{pr}_1^*\alpha '_ i$. 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 base change of $\alpha _ i$ is $\alpha $ and we see (1) holds. $\square$