The Stacks project

Lemma 29.22.2. Let $f : X \to Y$ be a morphism of schemes. Assume

  1. $f$ is quasi-compact and locally of finite presentation, and

  2. $Y$ is quasi-compact and quasi-separated.

Then the image of every constructible subset of $X$ is constructible in $Y$.

Proof. By Properties, Lemma 28.2.5 it suffices to prove this lemma in case $Y$ is affine. In this case $X$ is quasi-compact. Hence we can write $X = U_1 \cup \ldots \cup U_ n$ with each $U_ i$ affine open in $X$. If $E \subset X$ is constructible, then each $E \cap U_ i$ is constructible too, see Topology, Lemma 5.15.4. Hence, since $f(E) = \bigcup f(E \cap U_ i)$ and since finite unions of constructible sets are constructible, this reduces us to the case where $X$ is affine. In this case the result is Algebra, Theorem 10.29.10. $\square$

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