Lemma 114.4.5. Let $S$ be a multiplicative set of $A$. Then the map

$f: \mathop{\mathrm{Spec}}(S^{-1}A)\longrightarrow \mathop{\mathrm{Spec}}(A)$

induced by the canonical ring map $A \to S^{-1}A$ is a homeomorphism onto its image and $\mathop{\mathrm{Im}}(f) = \{ \mathfrak p \in \mathop{\mathrm{Spec}}(A) : \mathfrak p\cap S = \emptyset \}$.

Proof. This is a duplicate of Algebra, Lemma 10.17.5. $\square$

Comment #589 by Wei Xu on

$\Spec(R)$ should be $\Spec(A)$, one is in the statement, one is in the proof.

Comment #590 by Wei Xu on

I mean, $\operatorname{Spec}(R)$ should be $\operatorname{Spec}(A)$, one is in the statement, one is in the proof.

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