----- > [!proposition] Proposition. ([[localizations at finitely many elements cover Spec iff they generate the unit ideal]]) > Let $A$ be a ([[commutative ring|commutative]]) [[ring]] and $f_{1},\dots,f_{r} \in A$. Let $A_{f_{i}}$ denote the [[localization]] of $A$ at the element $f_{i}$. > Then $A_{f_{1}},\dots,A_{f_{r}}$ [[cover|cover]] $\text{Spec }A$:[^1] $\text{Spec }A=\text{Spec }A_{f_{1}} \cup \dots \cup \text{Spec } A_{f_{r}}$ > if and only if $f_{1},\dots,f_{r}$ [[ideal generated by a subset|generate]] the [[ideal|unit ideal]] $\langle 1 \rangle=A$: $\langle 1 \rangle =\langle f_{1},\dots,f_{r} \rangle .$ ^proposition [^1]: More precisely, [[Zariski topology on a ring spectrum|the Zariski basic open sets]] $D(f_{1}),\dots,D(f_{r}) \subset \text{Spec }A$, where $D(f_{i})$ is [[homeomorphism|homeomorphic]] to $\text{Spec }A_{f_{i}}$, cover $\text{Spec }A$. > [!proof]+ Proof. ([[localizations at finitely many elements cover Spec iff they generate the unit ideal]]) > > $\to$. Suppose $D(f_{1}),\dots,D(f_{r})$ cover $\text{Spec }A$. Then every $\mathfrak{p} \in \text{Spec }A$ fails to contain at least one of the $f_{i}$, and hence $\mathfrak{p} \not \supset \langle f_{1},\dots,f_{r} \rangle$ for all [[prime ideal|prime ideals]] $\mathfrak{p} \in \text{Spec }A$. Since [[existence of maximal ideals in commutative rings|any proper ideal has a maximal ideal containing it]], and all [[maximal ideal|maximal]] [[ideal|ideals]] are in $\text{Spec }A$, it follows that $\langle f_{1},\dots,f_{r} \rangle$ must not be proper. > > $\leftarrow.$ Suppose $\langle f_{1},\dots,f_{r} \rangle=\langle 1 \rangle$. Let $\mathfrak{p} \in \text{Spec }A$. Then since $\mathfrak{p}$ is proper, it must not contain at least one of the $f_{i}$, and thus belongs to $D(f_{i})$. > ----- #### ---- #### 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 > ```