## 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 $\mathop{\mathrm{Spec}}$ a homeomorphism $$ \mathop{\mathrm{Spec}}(R_f) \longrightarrow D(f) \subset \mathop{\mathrm{Spec}}(R). $$ 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 3030–3039 (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}
```

## Comments (0)

## Add a comment on tag `00E4`

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

All contributions are licensed under the GNU Free Documentation License.

There are no comments yet for this tag.

There are also 2 comments on Section 10.16: Commutative Algebra.