The Stacks project

42.11 Preparation for proper pushforward

Lemma 42.11.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $X$, $Y$ integral and $\dim _\delta (X) = \dim _\delta (Y)$. Then either $f(X)$ is contained in a proper closed subscheme of $Y$, or $f$ is dominant and the extension of function fields $R(X)/R(Y)$ is finite.

Proof. The closure $\overline{f(X)} \subset Y$ is irreducible as $X$ is irreducible (Topology, Lemmas 5.8.2 and 5.8.3). If $\overline{f(X)} \not= Y$, then we are done. If $\overline{f(X)} = Y$, then $f$ is dominant and by Morphisms, Lemma 29.8.5 we see that the generic point $\eta _ Y$ of $Y$ is in the image of $f$. Of course this implies that $f(\eta _ X) = \eta _ Y$, where $\eta _ X \in X$ is the generic point of $X$. Since $\delta (\eta _ X) = \delta (\eta _ Y)$ we see that $R(Y) = \kappa (\eta _ Y) \subset \kappa (\eta _ X) = R(X)$ is an extension of transcendence degree $0$. Hence $R(Y) \subset R(X)$ is a finite extension by Morphisms, Lemma 29.51.7 (which applies by Morphisms, Lemma 29.15.8). $\square$

Lemma 42.11.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $f$ is quasi-compact, and $\{ Z_ i\} _{i \in I}$ is a locally finite collection of closed subsets of $X$. Then $\{ \overline{f(Z_ i)}\} _{i \in I}$ is a locally finite collection of closed subsets of $Y$.

Proof. Let $V \subset Y$ be a quasi-compact open subset. Since $f$ is quasi-compact the open $f^{-1}(V)$ is quasi-compact. Hence the set $\{ i \in I \mid Z_ 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

\[ \{ i \in I \mid f(Z_ i) \cap V \not= \emptyset \} = \{ i \in I \mid \overline{f(Z_ i)} \cap V \not= \emptyset \} \]

the lemma is proved. $\square$


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