62.16 Relative cycles in the non-Noetherian case
We urge the reader to skip this section.
Let $f : X \to S$ be a morphism of schemes of finite presentation. Let $r \geq 0$. Denote $Hilb(X/S, r)$ the set of closed subschemes $Z \subset X$ such that $Z \to S$ is flat, of finite presentation, and of relative dimension $\leq r$. We consider the group homomorphism
62.16.0.1
\begin{equation} \label{relative-cycles-equation-cycle-classes-general} \begin{matrix} \text{free abelian group}
\\ \text{on }Hilb(X/S, r)
\end{matrix} \longrightarrow \begin{matrix} \text{families of }r\text{-cycles}
\\ \text{on fibres of }X/S
\end{matrix} \end{equation}
sending $\sum n_ i[Z_ i]$ to $\sum n_ i[Z_ i/X/S]_ r$.
Lemma 62.16.1. Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a morphism of finite presentation. Let $r \geq 0$ and let $\alpha $ be a family of $r$-cycles on fibres of $X/S$. The following are equivalent
there exists a cartesian diagram
\[ \xymatrix{ X \ar[r] \ar[d] & X_0 \ar[d] \\ S \ar[r] & S_0 } \]
where $X_0 \to S_0$ is a finite type morphism of Noetherian schemes and $\alpha _0 \in z(X_0/S_0, r)$ such that $\alpha $ is the base change of $\alpha _0$ by $S \to S_0$
there exists a completely decomposed proper morphism $g : S' \to S$ of finite presentation such that $g^*\alpha $ is in the image of (62.16.0.1).
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
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, 32.8.15, 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.
$\square$
Discussion. If $f : X \to S$, $r$, and $\alpha $ are as in Lemma 62.16.1, then it makes sense to say that $\alpha $ is a relative $r$-cycle on $X/S$ if the equivalent conditions (1) and (2) of Lemma 62.16.1 hold. This definition has many good properties; for example it doesn't conflict with the earlier definition in case $S$ is Noetherian and most of the results of Section 62.6 generalize to this setting.
We may still generalize further as follows. Assume $S$ is arbitrary and $f : X \to S$ is locally of finite presentation. Let $r \geq 0$ and let $\alpha $ be a family of $r$-cycles $\alpha $ on fibres of $X/S$. Then $\alpha $ is an relative $r$-cycle on $X/S$ if for $U \subset X$ and $V \subset S$ affine open with $f(U) \subset V$ the restriction $\alpha |_ U$ is a relative $r$-cycle on $U/V$ as defined in the previous paragraph. Again many of the earlier results generalize to this setting.
If we ever need these generalizations we will carefully state and prove them here.
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