Why is the localization of a commutative Noetherian ring still Noetherian?

Question

This is an unproven proposition I've come across in multiple places.

Suppose $A$ is a commutative Noetherian ring, and $S$ a multiplicative subset of $A$. Then $S^{-1}A$ is Noetherian.

Why is this? I thought about taking some chain of submodules $$ S^{-1}M_1\subset S^{-1}M_2\subset\cdots $$ and pulling back to a chain $$ M_1\subset M_2\subset\cdots $$ of submodules of $A$ which must eventually stablize. Is there more to it than this? I kind of wary of assuming all submodules of $S^{-1}A$ have form $S^{-1}M$ for $M\leq A$.

Answer 1

This is a standard property of localizations:

Theorem. Let $R$ be a commutative ring, and let $S\neq\emptyset$ be a multiplicative subset. Let $M$ be an $R$-module. Let $\varphi\colon R\to S^{-1}R$ be the canonical map ($\varphi(r) = \frac{rs}{s}$), and likewise, by abuse of notation, let $\varphi\colon M\to S^{-1}M$ be the natural map $(\varphi(m) = \frac{sm}{s}$ with $s\in S$).

1. For every submodule $N$ of $M$, $S^{-1}N = \{\frac{a}{s}\mid a\in N\}$ is a submodule of $S^{-1}M$.

2. If $L$ is a submodule of $S^{-1}M$, then $\varphi^{-1}(L) = \{m\in M\mid \varphi(m)\in L\}$ is a submodule of $M$.

3. If $N$ is a submodule of $M$, then $N\subseteq \varphi^{-1}(S^{-1}N)$. Moreover, if $N=\varphi^{-1}(L)$ for some submodule $L$ of $S^{-1}M$, then $L=S^{-1}N$.

Proof. $S^{-1}N$ is nonempty, as it contains $\frac{0}{s}$; it is closed under differences, since $\frac{a}{s}-\frac{b}{t} = \frac{ta-sb}{st}\in S^{-1}N$ if $a,b\in N$. And it is closed under scalar multiplication, since $a\in N$ implies $ra\in N$ for all $r\in R$, so $\frac{r}{t}(\frac{a}{s}) = \frac{ra}{ts}\in S^{-1}N$ if $a\in N$.

Now let $L$ be a submodule of $S^{-1}M$; since $\varphi$ is a module homomorphism, the pullback of a submodule is a submodule, so 2 is immediate.

Again, let $N$ be a submodule of $M$. Then for every $a\in N$ we have $\varphi(a) = \frac{sa}{s}\in S^{-1}N$, since $sa\in N$, hence $a\in \varphi^{-1}(S^{-1}N)$. Now assume that $N=\varphi^{-1}(L)$. If $a\in N$ and $s\in S$, then $\frac{a}{s} = \frac{ssa}{sss} = \frac{s}{ss}\frac{sa}{s} =\frac{s}{ss}\varphi(a)\in L$ (since $\varphi(a)\in L$), hence $S^{-1}N\subseteq L$. Conversely, let $\frac{m}{t}\in L$. Then $\frac{tt}{t}\frac{m}{t} = \frac{(tt)m}{tt}=\varphi(m)\in L$, hence $m\in \varphi^{-1}(L) = N$; thus, $\frac{m}{t}\in S^{-1}N$, proving that $L\subseteq S^{-1}N$. Therefore, $L=S^{-1}N$, as claimed. $\Box$

Corollary. Every submodule of $S^{-1}M$ is of the form $S^{-1}N$ for some submodule $N$ of $M$.

Answer 2

Let $f\colon A \to S^{-1}A$ be the canonical homomorphism. It is true that every ideal $\mathfrak b$ of $S^{-1}A$ is of the form $S^{-1}\mathfrak a = f(\mathfrak a)(S^{-1}A)$ for some ideal $\mathfrak a$ of $A$. We can even take $\mathfrak a = f^{-1}(\mathfrak b)$, and this should help you prove that $S^{-1}A$ is Noetherian using increasing chains.

You can find related facts in and around Proposition 6.4 of Milne.

Answer 3

To verify that a commutative ring $R$ is Noetherian, it suffices to check that the prime ideals of $R$ are finitely-generated. There is a bijective correspondence between prime ideals of $S^{-1}R$ and prime ideals of $R$ not intersecting $S$ which is given by the map $I \mapsto I \cap R$ with an inverse $J \mapsto \{\frac{a}{s} | s \in S\}$. A prime ideal in $S^{-1} R$ is thus finitely-generated with generators $\{g_i/1\}$, where $g_i$ are the generators of the corresponding prime ideal in $R$.