This section is the analogue of Chow Homology, Section 42.11.
Lemma 81.7.1. In Situation 81.2.1 let $X,Y/B$ be good and let $f : X \to Y$ be a morphism over $B$. If $Z \subset X$ is an integral closed subspace, then there exists a unique integral closed subspace $Z' \subset Y$ such that there is a commutative diagram with $Z \to Z'$ dominant. If $f$ is proper, then $Z \to Z'$ is proper and surjective.
\[ \xymatrix{ Z \ar[r] \ar[d] & X \ar[d]^ f \\ Z' \ar[r] & Y } \]
Proof. Let $\xi \in |Z|$ be the generic point. Let $Z' \subset Y$ be the integral closed subspace whose generic point is $\xi ' = f(\xi )$, see Remark 81.2.3. Since $\xi \in |f^{-1}(Z')| = |f|^{-1}(|Z'|)$ by Properties of Spaces, Lemma 65.4.3 and since $Z$ is the reduced with $|Z| = \overline{\{ \xi \} }$ we see that $Z \subset f^{-1}(Z')$ as closed subspaces of $X$ (see Properties of Spaces, Lemma 65.12.4). Thus we obtain our morphism $Z \to Z'$. This morphism is dominant as the generic point of $Z$ maps to the generic point of $Z'$. Uniqueness of $Z'$ is clear. If $f$ is proper, then $Z \to Y$ is proper as a composition of proper morphisms (Morphisms of Spaces, Lemmas 66.40.3 and 66.40.5). Then we conclude that $Z \to Z'$ is proper by Morphisms of Spaces, Lemma 66.40.6. Surjectivity then follows as the image of a proper morphism is closed. $\square$
Remark 81.7.2. In Situation 81.2.1 let $X/B$ be good. Every $x \in |X|$ can be represented by a (unique) monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ where $k$ is a field, see Decent Spaces, Lemma 67.11.1. Then $k$ is the residue field of $x$ and is denoted $\kappa (x)$. Recall that $X$ has a dense open subscheme $U \subset X$ (Properties of Spaces, Proposition 65.13.3). If $x \in U$, then $\kappa (x)$ agrees with the residue field of $x$ on $U$ as a scheme. See Decent Spaces, Section 67.11.
Remark 81.7.3. In Situation 81.2.1 let $X/B$ be good. Assume $X$ is integral. In this case the function field $R(X)$ of $X$ is defined and is equal to the residue field of $X$ at its generic point. See Spaces over Fields, Definition 71.4.3. Combining this with Remark 81.2.3 we find that for any $x \in X$ the residue field $\kappa (x)$ is the function field of the unique integral closed subspace $Z \subset X$ whose generic point is $x$.
Lemma 81.7.4. In Situation 81.2.1 let $X, Y/B$ be good and let $f : X \to Y$ be a morphism over $B$. Assume $X$, $Y$ integral and $\dim _\delta (X) = \dim _\delta (Y)$. Then either $f$ factors through a proper closed subspace of $Y$, or $f$ is dominant and the extension of function fields $R(X) / R(Y)$ is finite.
Proof. By Lemma 81.7.1 there is a unique integral closed subspace $Z \subset Y$ such that $f$ factors through a dominant morphism $X \to Z$. Then $Z = Y$ if and only if $\dim _\delta (Z) = \dim _\delta (Y)$. On the other hand, by our construction of dimension functions (see Situation 81.2.1) we have $\dim _\delta (X) = \dim _\delta (Z) + r$ where $r$ the transcendence degree of the extension $R(X)/R(Z)$. Combining this with Spaces over Fields, Lemma 71.5.1 the lemma follows. $\square$
Lemma 81.7.5. In Situation 81.2.1 let $X, Y/B$ be good. Let $f : X \to Y$ be a morphism over $B$. Assume $f$ is quasi-compact, and $\{ T_ i\} _{i \in I}$ is a locally finite collection of closed subsets of $|X|$. Then $\{ \overline{|f|(T_ i)}\} _{i \in I}$ is a locally finite collection of closed subsets of $|Y|$.
Proof. Let $V \subset |Y|$ be a quasi-compact open subset. Then $|f|^{-1}(V) \subset |X|$ is quasi-compact by Morphisms of Spaces, Lemma 66.8.3. Hence the set $\{ i \in I : T_ i \cap |f|^{-1}(V) \not= \emptyset \} $ is finite by a simple topological argument which we omit. Since this is the same as the set
the lemma is proved. $\square$
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)