The Stacks project

80.9 Preparation for flat pullback

This section is the analogue of Chow Homology, Section 42.13.

Recall that a morphism of algebraic spaces is said to have relative dimension $r$ if ├ętale locally on the source and the target we get a morphism of schemes which has relative dimension $d$. The precise definition is equivalent, but in fact slightly different, see Morphisms of Spaces, Definition 65.33.2.

Lemma 80.9.1. In Situation 80.2.1 let $X, Y/B$ be good. Let $f : X \to Y$ be a morphism over $B$. Assume $f$ is flat of relative dimension $r$. For any closed subset $T \subset |Y|$ we have

\[ \dim _\delta (|f|^{-1}(T)) = \dim _\delta (T) + r. \]

provided $|f|^{-1}(T)$ is nonempty. If $Z \subset Y$ is an integral closed subscheme and $Z' \subset f^{-1}(Z)$ is an irreducible component, then $Z'$ dominates $Z$ and $\dim _\delta (Z') = \dim _\delta (Z) + r$.

Proof. Since the $\delta $-dimension of a closed subset is the supremum of the $\delta $-dimensions of the irreducible components, it suffices to prove the final statement. We may replace $Y$ by the integral closed subscheme $Z$ and $X$ by $f^{-1}(Z) = Z \times _ Y X$. Hence we may assume $Z = Y$ is integral and $f$ is a flat morphism of relative dimension $r$. Since $Y$ is locally Noetherian the morphism $f$ which is locally of finite type, is actually locally of finite presentation. Hence Morphisms of Spaces, Lemma 65.30.6 applies and we see that $f$ is open. Let $\xi \in X$ be a generic point of an irreducible component of $X$. By the openness of $f$ we see that $f(\xi )$ is the generic point $\eta $ of $Z = Y$. Thus $Z'$ dominates $Z = Y$. Finally, we see that $\xi $ and $\eta $ are in the schematic locus of $X$ and $Y$ by Properties of Spaces, Proposition 64.13.3. Since $\xi $ is a generic point of $X$ we see that $\mathcal{O}_{X, \xi } = \mathcal{O}_{X_\eta , \xi }$ has only one prime ideal and hence has dimension $0$ (we may use usual local rings as $\xi $ and $\eta $ are in the schematic loci of $X$ and $Y$). Thus by Morphisms of Spaces, Lemma 65.34.1 (and the definition of morphisms of given relative dimension) we conclude that the transcendence degree of $\kappa (\xi )$ over $\kappa (\eta )$ is $r$. In other words, $\delta (\xi ) = \delta (\eta ) + r$ as desired. $\square$

Here is the lemma that we will use to prove that the flat pullback of a locally finite collection of closed subschemes is locally finite.

Lemma 80.9.2. In Situation 80.2.1 let $X, Y/B$ be good. Let $f : X \to Y$ be a morphism over $B$. Assume $\{ T_ i\} _{i \in I}$ is a locally finite collection of closed subsets of $|Y|$. Then $\{ |f|^{-1}(T_ i)\} _{i \in I}$ is a locally finite collection of closed subsets of $X$.

Proof. Let $U \subset |X|$ be a quasi-compact open subset. Since the image $|f|(U) \subset |Y|$ is a quasi-compact subset there exists a quasi-compact open $V \subset |Y|$ such that $|f|(U) \subset V$. Note that

\[ \{ i \in I : |f|^{-1}(T_ i) \cap U \not= \emptyset \} \subset \{ i \in I : T_ i \cap V \not= \emptyset \} . \]

Since the right hand side is finite by assumption we win. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0EP4. Beware of the difference between the letter 'O' and the digit '0'.