----- > [!proposition] Proposition. ([[the Noetherian gluing lemma]]) > Given a [[ring]] $A$ and a finite collection [[relatively prime integers|of]] elements $f_{1},\dots,f_{r} \in A$ which generate the [[ideal|unit ideal]] $\langle 1 \rangle=A$, suppose $A_{f_{i}}$ is Noetherian for each $i$. Then $A$ is Noetherian. ^proposition > [!proof]- Proof. ([[the Noetherian gluing lemma]]) > > > [!proposition] Small Lemma. > > In general: if there are ideals $I_{1},\dots,I_{r} \subset R$ such that $I_{1}+\dots+I_{r}=\langle 1 \rangle$, then for any $N \geq 1$ it holds that $I_{1}^{N}+\dots+I_{r}^{N}=\langle 1 \rangle$. > ^proposition > > > [!proof] Proof of Small Lemma. > > When the $I_{i}$ are [[principal ideal|principal]], this is usually proven via monomial theorem. Here is a technique that sidesteps that. We have $\begin{align} > \emptyset &= V(\langle 1 \rangle ) \\ > &= V\left( \sum_{i=1}^{r} I_{i} \right) \\ > &= \bigcap_{i=1}^{r} V(I_{i}) \\ > &= \bigcap_{i=1}^{r} V(I_{i}^{N}) \\ > &= V\left( \sum_{i=1}^{r} I_{i}^{N} \right) > \end{align}$ > since in general $V(I)=\emptyset \iff I=\langle 1 \rangle$, it follows that $\sum_{i=1}^{r}I_{i}^{N}=\langle 1 \rangle$ as hoped. > ^proof > > > > > [!proposition] Bigger Lemma. > > Let $\mathfrak{a} \subset A$ be an [[ideal]], and let $\varphi_{i}:A \to A_{f_{i}}$ denote the relevant [[localization|localization map]], $i \in [r]$. Then $\mathfrak{a}= \bigcap_{i=1}^{r} \mathfrak{a}^{ec \text{ wrt }i} =\bigcap_{i=1}^{r} \varphi_{i} ^{-1}\big( \varphi_{i}(\mathfrak{a}) \cdot A_{f_{i}} \big).$ > ^proposition > > > > > [!proof] Proof of Lemma. > > Possibly relevant: [[extension of an ideal]], [[contraction of an ideal]], [[extension and contraction under localization]]. > > > > $\subset.$ This is clear, because $\mathfrak{a} \subset \mathfrak{a}^{ec \text{ wrt }i}$ for each $i$. So if $a \in \mathfrak{a}$ then $a$ is in each term of the intersection, and hence in the intersection itself. > > > > $\supset.$ Given $b \in A$ contained in the intersection, write $\varphi_{i}(b)=\frac{b}{1}=\frac{a_{i}}{f_{i}^{n_{i}}}$ in $A_{f_{i}}$ for each $i$, where $a_{i} \in \mathfrak{a}$ and $n_{i} \geq 1$. Increasing the $n_{i}$ if necessary, we can make then all equal to a fixed $n$. This means that in $A$ we have $f_{i}^{m_{i}}(f_{i}^{n}b - a_{i})=0$ for some $m_{i}$. As before, we can take all the $m_{i}$ to equal some fixed $m$. Thus $f_{i}^{m+n}b \in \mathfrak{a}$ for each $i$. The *small lemma* above says that that since $f_{1},\dots,f_{r}$ generate the [[ideal|unit ideal]], the same is true for their $N$th powers for any $N \in \mathbb{N}$. Put $N=n+m$. Then we have $1=\sum_{i=1}^{r}c_{i}f_{i}^{N}$ for some $c_{i} \in A$. Multiplying both side by $b$: $b=\sum_{i=1}^{r} c_{i}f_{i}^{N} b=\sum_{i=1}^{r}c_{i}f_{i}^{m+n}b$ > > where since we have established $f_{i}^{m+n}b \in \mathfrak{a}$, it follows that $b \in A$. > > > > > > Now we show that $A$ is [[Noetherian ring|Noetherian]]. Then $\mathfrak{a}_{1} \subset \mathfrak{a}_{2} \subset \dots$ be an [[ascending chain condition|ascending chain]] of [[ideal|ideals]] in $\mathfrak{a}$. Then for each $i$, [[extension of an ideal|extension]] under the [[localization|localization map]] preserves the chain: $\varphi_{i}(\mathfrak{a}_{1})\cdot A_{f_{i}} \subset \varphi_{i}(\mathfrak{a}_{2}) \cdot A_{f_{i}}\subset \dots$ > is an ascending chain of ideals in $A_{f_{i}}$, which must [[ascending chain - maximality characterization of Noetherian modules|which must]] stabilize since $A_{f_{i}}$ is [[Noetherian ring|Noetherian]]. There are only finitely many $f_{i}$; assume that each chain has stabilized once we hit $M$ inclusions. The *bigger lemma* implies $\mathfrak{a}_{M}=\mathfrak{a}_{M+1}=\dots$, finishing the proof. > ----- #### ---- #### 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 > ```