## Tag `00E4`

Chapter 10: Commutative Algebra > Section 10.16: The spectrum of a ring

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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