The Stacks project

Lemma 27.8.9. Let $S$ be a graded ring. The scheme $\text{Proj}(S)$ is quasi-compact if and only if there exist finitely many homogeneous elements $f_1, \ldots , f_ n \in S_{+}$ such that $S_{+} \subset \sqrt{(f_1, \ldots , f_ n)}$. In this case $\text{Proj}(S) = D_+(f_1) \cup \ldots \cup D_+(f_ n)$.

Proof. Given such a collection of elements the standard affine opens $D_{+}(f_ i)$ cover $\text{Proj}(S)$ by Algebra, Lemma 10.57.3. Conversely, if $\text{Proj}(S)$ is quasi-compact, then we may cover it by finitely many standard opens $D_{+}(f_ i)$, $i = 1, \ldots , n$ and we see that $S_{+} \subset \sqrt{(f_1, \ldots , f_ n)}$ by the lemma referenced above. $\square$


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