Lemma 15.31.2. A quasi-regular ideal is finitely generated.

**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 10.22.2.
$\square$

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