The Stacks project

Lemma 10.33.2. Let $R$ be a ring. Let $S \subset R$ be a multiplicative subset. Assume the image of the map $\mathop{\mathrm{Spec}}(S^{-1}R) \to \mathop{\mathrm{Spec}}(R)$ is closed. If $R$ is Noetherian, or $\mathop{\mathrm{Spec}}(R)$ is a Noetherian topological space, or $S$ is finitely generated as a monoid, then $R \cong S^{-1}R \times R'$ for some ring $R'$.

Proof. By Lemma 10.33.1 we have $S^{-1}R \cong R/I$ for some ideal $I \subset R$. By Lemma 10.24.3 it suffices to show that $V(I)$ is open. If $R$ is Noetherian then $\mathop{\mathrm{Spec}}(R)$ is a Noetherian topological space, see Lemma 10.31.5. If $\mathop{\mathrm{Spec}}(R)$ is a Noetherian topological space, then the complement $\mathop{\mathrm{Spec}}(R) \setminus V(I)$ is quasi-compact, see Topology, Lemma 5.12.13. Hence there exist finitely many $f_1, \ldots , f_ n \in I$ such that $V(I) = V(f_1, \ldots , f_ n)$. Since each $f_ i$ maps to zero in $S^{-1}R$ there exists a $g \in S$ such that $gf_ i = 0$ for $i = 1, \ldots , n$. Hence $D(g) = V(I)$ as desired. In case $S$ is finitely generated as a monoid, say $S$ is generated by $g_1, \ldots , g_ m$, then $S^{-1}R \cong R_{g_1 \ldots g_ m}$ and we conclude that $V(I) = D(g_1 \ldots g_ m)$. $\square$

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