---- > [!definition] Definition. ([[locally Noetherian scheme]]) > We say a [[scheme]] $X$ is **locally Noetherian** if it can be [[cover|covered]] by [[affine scheme|affine]] open subsets $\text{Spec }A_{i}$ with $A_{i}$ [[Noetherian ring|Noetherian]]. > We say $X$ is **Noetherian** if it can be [[cover|covered]] by *a finite number of* [[affine scheme|affine]] open subsets $\text{Spec }A_{i}$ with $A_{i}$ [[Noetherian ring|Noetherian]]. ^definition > [!justification] > A Noetherian scheme is [[Noetherian topological space|Noetherian]] as a [[topological space]], [[Noetherian topological space#^7a38e7|since it equals a finite union]] of Noetherian [[Zariski topology on a ring spectrum|topological spaces]] $\text{Spec }A_{i}$. ^justification > [!equivalence] > A [[scheme]] $X$ is locally Noetherian if and only if for every open affine subset $U=\text{Spec }A$, $A$ is a [[Noetherian ring|Noetherian]] [[ring]]. > > > [!proposition] Corollary. > > In particular, an [[affine scheme]] $X=\text{Spec }A$ is a Noetherian scheme if and only if $A$ is a [[Noetherian ring]]. > > > > [!basicproperties] > - The [[(pre)sheaf stalk|stalks]] of a locally Noetherian [[scheme]] are [[Noetherian ring|Noetherian]] [[ring|rings]]. ^properties > [!proof] Proof of Equivalence. > > $\leftarrow.$ This is immediate. > > $\to.$ The following results will be crucial: > - [[the Noetherian gluing lemma]] (cf. the hint in the example sheet) > - [[Nike's lemma]] > - [[localization preserves Noetherianity]] > > Suppose $X$ is locally Noetherian; cover it with $\{ \text{Spec }A_{i} \}$ where $A_{i}$ is [[Noetherian ring|Noetherian]]. Let $U \subset X$ be any [[affine scheme|open affine]], $U=\text{Spec }A$. WTS $A$ is Noetherian. Write $U=\bigcup_{i}\underbrace{\text{Spec }A \cap \text{Spec }A_{i}}_{:=U_{i}}=\bigcup_{i}U_{i}.$ > Using [[Nike's lemma]], each $U_{i}$ can be [[cover|covered]] by [[basis for a topology|basic]] open sets $D(f_{i})$ with $f_{i} \in A$, so $U$ can too be covered as such: $U=\bigcup_{i}D(f_{i}) \ \ \ \ f_{i} \in A.$ > Since $U=\text{Spec }A$ is [[compact|quasicompact]] ([[the spectrum of a ring is quasi-compact]]) we may take such a cover to be finite: $U=D(f_{1}) \cup \dots \cup D(f_{r}).$By [[localizations at finitely many elements cover Spec iff they generate the unit ideal]], it follows that $\langle f_{1},\dots,f_{r} \rangle=\langle 1 \rangle=A$. > > Recall that (also using [[Nike's lemma]]) $D(f_{i})$ corresponds also to $\text{Spec }(A_{i})_{g}$, the spectrum of $A_{i}$ [[localization|localized]] at some $g \in A_{i}$, $D(f_{i})=D(g)$. Since $A_{i}$ is Noetherian by assumption, so is $(A_{i})_{g}$. Hence $\text{Spec }A_{f_{i}} \cong \text{Spec }(A_{i})_{g}$, [[the category of affine schemes is dual to that of rings|inducing]] a [[ring isomorphism]] $(A_{i})_{g} \xrightarrow{\sim} A_{f_{i}}$ witnessing that $A_{f_{i}}$ is [[Noetherian ring|Noetherian]]. [[The Noetherian gluing lemma]] now guarantees $A$ is [[Noetherian ring|Noetherian]]. ---- #### ---- #### 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 > ```