Proof. Let $I \subset R$ be a quasi-regular ideal. Since $V(I)$ is quasi-compact, there exist $g_1, \ldots , g_ m \in R$ such that $V(I) \subset D(g_1) \cup \ldots \cup D(g_ m)$ and such that $I_{g_ j}$ is generated by a quasi-regular sequence $g_{j1}, \ldots , g_{jr_ j} \in R_{g_ j}$. Write $g_{ji} = g'_{ji}/g_ j^{e_{ij}}$ for some $g'_{ij} \in I$. Write $1 + x = \sum g_ j h_ j$ for some $x \in I$ which is possible as $V(I) \subset D(g_1) \cup \ldots \cup D(g_ m)$. Note that $\mathop{\mathrm{Spec}}(R) = D(g_1) \cup \ldots \cup D(g_ m) \bigcup D(x)$ Then $I$ is generated by the elements $g'_{ij}$ and $x$ as these generate on each of the pieces of the cover, see Algebra, Lemma 10.23.2. $\square$

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.

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 07CW. Beware of the difference between the letter 'O' and the digit '0'.