Lemma 10.29.5. Let $R$ be a ring. Let $f$ be an element of $R$. Let $S = R_ f$. Then the image of a constructible subset of $\mathop{\mathrm{Spec}}(S)$ is constructible in $\mathop{\mathrm{Spec}}(R)$.

Proof. We repeatedly use Lemma 10.29.1 without mention. Let $U, V$ be quasi-compact open in $\mathop{\mathrm{Spec}}(S)$. We will show that the image of $U \cap V^ c$ is constructible. Under the identification $\mathop{\mathrm{Spec}}(S) = D(f)$ of Lemma 10.17.6 the sets $U, V$ correspond to quasi-compact opens $U', V'$ of $\mathop{\mathrm{Spec}}(R)$. Hence it suffices to show that $U' \cap (V')^ c$ is constructible in $\mathop{\mathrm{Spec}}(R)$ which is clear. $\square$

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