> [!proposition] Proposition. ([[the spectrum of a ring is quasi-compact]]) > The [[prime ideal|spectrum]] $\text{Spec }A$ of a ([[commutative ring|commutative]]) [[ring]] $A$ is (quasi-)[[compact]] as a [[topological space]] endowed with the [[Zariski topology on a ring spectrum|Zariski topology]]. ^proposition > [!proof]- Proof. ([[the spectrum of a ring is quasi-compact]]) > [[compact#^c1aa5b|It suffices]] to consider a [[cover]] of $\text{Spec }A$ by principal opens $D(f_{i})$, where $f_{i} \in A$ and $i \in I$: $\text{Spec }A=\bigcup_{i\in I}D(f_{i})$, and produce a finite subcover. > > By [[De Morgan's Laws|De Morgan's first law]], $\bigcap_{i \in I}^{}V(f_{i})=\emptyset$. This means that no [[prime ideal]] contains every $f_{i}$, or equivalently that no prime ideal contains $\sum_{i \in I} \langle f_{i} \rangle$, meaning that $\sum_{i \in I}\langle f_{i} \rangle=\langle 1 \rangle$. This means $1$ can be written as a *finite* sum: $1=\sum_{j=1}^{r} c_{j}f_{j}$ for some $c_{j} \in A$. Hence $\langle 1 \rangle=\langle f_{1},\dots,f_{r} \rangle$. The result now follows: suppose $\mathfrak{p} \in \text{Spec }A$. Since $\mathfrak{p}$ is proper, it must fail to contain at least one of the $f_{j}$, and thus $\mathfrak{p} \in D(f_{j})$. Hence $\text{Spec }A=\bigcup_{j=1}^{r}D(f_{j})$ admits a finite open cover by basic opens. > > > ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```