The Stacks project

Proof. By the result of Lemma 42.67.9 we obtain a system of cycle groups $Z_ k(X_ i)$ and a system of chow groups $\mathop{\mathrm{CH}}\nolimits _ k(X_ i)$ as well as maps $\mathop{\mathrm{colim}}\nolimits Z_ k(X_ i) \to Z_ k(X')$ and $\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{CH}}\nolimits _ i(X_ i) \to \mathop{\mathrm{CH}}\nolimits _ k(X')$. We may replace $S$ by a quasi-compact open through which $X \to S$ factors, hence we may and do assume all the schemes occurring in this proof are Noetherian (and hence quasi-compact and quasi-separated).

Let us show that the map $\mathop{\mathrm{colim}}\nolimits Z_ k(X_ i) \to Z_ k(X')$ is surjective. Namely, let $Z' \subset X'$ be an integral closed subscheme of $\delta '$-dimension $k$. By Limits, Lemma 32.10.1 we can find an $i$ and a morphism $Z_ i \to X_ i$ of finite presentation whose base change is $Z'$. After increasing $i$ we may assume $Z_ i$ is a closed subscheme of $X_ i$, see Limits, Lemma 32.8.5. Then $Z' \to X_ i$ factors through $Z_ i$ and we may replace $Z_ i$ by the scheme theoretic image of $Z' \to X_ i$. In this way we see that we may assume $Z_ i$ is an integral closed subscheme of $X_ i$. By Lemma 42.67.2 we conclude that $\dim _{\delta _ i}(Z_ i) = \dim _{\delta '}(Z') = k$. Thus $Z_ k(X_ i) \to Z_ k(X')$ maps $[Z_ i]$ to $[Z']$ and we conclude surjectivity holds.

Let us show that the map $\mathop{\mathrm{colim}}\nolimits Z_ k(X_ i) \to Z_ k(X')$ is injective. Let $\alpha _ i = \sum n_ j[Z_ j] \in Z_ k(X_ i)$ be a cycle whose image in $Z_ k(X')$ is zero. We may and do assume $Z_ j \not= Z_{j'}$ if $j \not= j'$ and $n_ j \not= 0$ for all $j$. Denote $Z'_ j \subset X'$ the base change of $Z_ j$. By Lemma 42.67.2 each irreducible component of $Z'_ j$ has $\delta '$-dimension $k$. Moreover, as $Z_ j$ is irreducible and $Z'_ j \to Z_ j$ is flat (as the base change of $S' \to S$) we see that $Z'_ j \to Z_ j$ is dominant. Hence if $Z'_ j$ is nonempty, then some irreducible component, say $Z'$, of $Z'_ j$ dominates $Z_ j$. It follows that $Z'$ cannot be an irreducible component of $Z'_{j'}$ for $j' \not= j$. Hence if $Z'_ j$ is nonempty, then we see that $(S' \to S_ i)^*\alpha _ i = \sum [Z'_ j]_ r$ is nonzero (as the coefficient of $Z'$ would be nonzero). Thus we see that $Z'_ j = \emptyset$ for all $j$. However, this means that the base change of $Z_ j$ by some transition map $S_{i'} \to S_ i$ is empty by Limits, Lemma 32.4.3. Thus $\alpha _ i$ dies in the colimit as desired.

The surjectivity of $\mathop{\mathrm{colim}}\nolimits Z_ k(X_ i) \to Z_ k(X')$ implies that $\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{CH}}\nolimits _ k(X_ i) \to \mathop{\mathrm{CH}}\nolimits _ k(X')$ is surjective. To finish the proof we show that this map is injective. Let $\alpha _ i \in \mathop{\mathrm{CH}}\nolimits _ k(X_ i)$ be a cycle whose image $\alpha ' \in \mathop{\mathrm{CH}}\nolimits _ k(X')$ is zero. Then there exist integral closed subschemes $W_ l' \subset X'$, $l = 1, \ldots , r$ of $\delta "$-dimension $k + 1$ and nonzero rational functions $f'_ l$ on $W'_ l$ such that $\alpha ' = \sum _{l = 1, \ldots , r} \text{div}_{W'_ l}(f'_ l)$. Arguing as above we can find an $i$ and integral closed subschemes $W_{i, l} \subset X_ i$ of $\delta _ i$-dimension $k + 1$ whose base change is $W'_ l$. After increasing $i$ we may assume we have rational functions $f_{i, l}$ on $W_{i, l}$. Namely, we may think of $f'_ l$ as a section of the structure sheaf over a nonempty open $U'_ l \subset W'_ l$, we can descend these opens by Limits, Lemma 32.4.11 and after increasing $i$ we may descend $f'_ l$ by Limits, Lemma 32.4.7. We claim that

after possibly increasing $i$.

To prove the claim, let $Z'_{l, j} \subset W'_ l$ be a finite collection of integral closed subschemes of $\delta '$-dimension $k$ such that $f'_ l$ is an invertible regular function outside $\bigcup _ j Y'_{l, j}$. After increasing $i$ (by the arguments above) we may assume there exist integral closed subschemes $Z_{i, l, j} \subset W_ i$ of $\delta _ i$-dimension $k$ such that $f_{i, l}$ is an invertible regular function outside $\bigcup _ j Z_{i, l, j}$. Then we may write

and

To prove the claim it suffices to show that $n_{l, i} = n_{i, l, j}$. Namely, this will imply that $\beta _ i = \alpha _ i - \sum \nolimits _{l = 1, \ldots , r} \text{div}_{W_{i, l}}(f_{i, l})$ is a cycle on $X_ i$ whose pullback to $X'$ is zero as a cycle! It follows that $\beta _ i$ pulls back to zero as a cycle on $X_{i'}$ for some $i' \geq i$ by an easy argument we omit.

To prove the equality $n_{l, i} = n_{i, l, j}$ we choose a generic point $\xi ' \in Z'_{l, j}$ and we denote $\xi \in Z_{i, l, j}$ the image which is a generic point also. Then the local ring map

is flat as $W'_ l \to W_{i, l}$ is the base change of the flat morphism $S' \to S_ i$. We also have $\mathfrak m_\xi \mathcal{O}_{W'_ l, \xi '} = \mathfrak m_{\xi '}$ because $Z_{i, l, j}$ pulls back to $Z'_{l, j}$! Thus the equality of

follows from Algebra, Lemma 10.52.13 and the construction of $\text{ord}$ in Algebra, Section 10.121. $\square$