82.9 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 étale locally on the source and the target we get a morphism of schemes which has relative dimension r. The precise definition is equivalent, but in fact slightly different, see Morphisms of Spaces, Definition 67.33.2.
Lemma 82.9.1. In Situation 82.2.1 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 67.30.6 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 66.13.3. 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 67.34.1 (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 82.9.2. In Situation 82.2.1 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
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