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Tag 00E4

Lemma 10.16.6. Let $R$ be a ring. Let $f \in R$. The map $R \to R_f$ induces via the functoriality of $\mathop{\rm Spec}$ a homeomorphism $$\mathop{\rm Spec}(R_f) \longrightarrow D(f) \subset \mathop{\rm Spec}(R).$$ The inverse is given by $\mathfrak p \mapsto \mathfrak p \cdot R_f$.

Proof. This is a special case of Lemma 10.16.5. $\square$

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 3026–3035 (see updates for more information).

\begin{lemma}
\label{lemma-standard-open}
Let $R$ be a ring. Let $f \in R$.
The map $R \to R_f$ induces via the functoriality of
$\Spec$ a homeomorphism
$$\Spec(R_f) \longrightarrow D(f) \subset \Spec(R).$$
The inverse is given by $\mathfrak p \mapsto \mathfrak p \cdot R_f$.
\end{lemma}

\begin{proof}
This is a special case of Lemma \ref{lemma-spec-localization}.
\end{proof}

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