The Stacks project

Lemma 10.24.3. Let $R$ be a ring. If $\mathop{\mathrm{Spec}}(R) = U \amalg V$ with both $U$ and $V$ open then $R \cong R_1 \times R_2$ with $U \cong \mathop{\mathrm{Spec}}(R_1)$ and $V \cong \mathop{\mathrm{Spec}}(R_2)$ via the maps in Lemma 10.21.2. Moreover, both $R_1$ and $R_2$ are localizations as well as quotients of the ring $R$.

Proof. By Lemma 10.21.3 we have $U = D(e)$ and $V = D(1-e)$ for some idempotent $e$. By Lemma 10.24.2 we see that $R \cong R_ e \times R_{1 - e}$ (since clearly $R_{e(1-e)} = 0$ so the glueing condition is trivial; of course it is trivial to prove the product decomposition directly in this case). The lemma follows. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 10.24: Glueing functions

Post a comment

Your email address will not be published.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 toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00EM. Beware of the difference between the letter 'O' and the digit '0'.