Lemma 10.29.4 (00F8)—The Stacks project

Lemma 10.29.4. Let $R$ be a ring and let $T \subset \mathop{\mathrm{Spec}}(R)$ be constructible. Then there exists a ring map $R \to S$ of finite presentation such that $T$ is the image of $\mathop{\mathrm{Spec}}(S)$ in $\mathop{\mathrm{Spec}}(R)$.

Proof. The spectrum of a finite product of rings is the disjoint union of the spectra, see Lemma 10.21.2. Hence if $T = T_1 \cup T_2$ and the result holds for $T_1$ and $T_2$, then the result holds for $T$. By Lemma 10.29.3 we may assume that $T = D(f) \cap V(g_1, \ldots , g_ m)$. In this case $T$ is the image of the map $\mathop{\mathrm{Spec}}((R/(g_1, \ldots , g_ m))_ f) \to \mathop{\mathrm{Spec}}(R)$, see Lemmas 10.17.6 and 10.17.7. $\square$

Comments (3)

Comment #4937 by awllower on February 22, 2020 at 03:08

I think the last ring map should be $R_f\rightarrow R_f/(g_1,\ldots,g_m)$ ?

Comment #4938 by Laurent Moret-Bailly on February 22, 2020 at 03:42

The proof contains the result that the constructible sets of $\mathrm{Spec}(R)$ are the finite unions of sets of the form $D(f)\cap V(g_1,\dots,g_m)$ . This fact is useful by itself, so why not make it a separate statement (or incorporate it in 00F6)? It would make 00F8 immediate, and probably simplify other proofs, too.

Comment #5203 by Johan on June 10, 2020 at 18:41

Thanks and fixed here.

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