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The spectrum of a Noetherian ring is a Noetherian topological space

Statement

Assume the Axiom of Choice.

Let $R$ be a Noetherian commutative ring. Then $Spec (R)$ is a Noetherian topological space.

Facts & Assumptions

Given: A Noetherian commutative ring $R$ and the Axiom of Choice.

[L1]

A topological space is Noetherian exactly when every descending chain of closed subsets stabilizes (Noetherian topological spaces via ACC on opens or DCC on closed subsets).

[L2]

Every Zariski-closed subset has a unique radical defining ideal (Every Zariski-closed subset has a unique radical defining ideal).

[L3]

In a Noetherian ring, every ascending chain of ideals stabilizes (A commutative ring is Noetherian exactly when every ideal is finitely generated, exactly when its ideals satisfy the ascending chain condition, and exactly when every nonempty set of ideals has a maximal member).

Proof

technique · direct

1.1L2givenchoose

Let $Z_{0} \supseteq Z_{1} \supseteq Z_{2} \supseteq \dots$ be a descending chain of closed subsets of $Spec (R)$ . By [L2], each $Z_{n}$ has a unique radical defining ideal $a_{n}$ with $Z_{n} = V (a_{n})$ .

2.1L2step 1.1algebra

Since $Z_{n + 1} \subseteq Z_{n}$ , every prime ideal in $Z_{n + 1}$ also lies in $Z_{n}$ . Therefore $a_{n} = ⋂_{p \in Z_{n}} p \subseteq ⋂_{p \in Z_{n + 1}} p = a_{n + 1} .$ Thus $a_{0} \subseteq a_{1} \subseteq a_{2} \subseteq \dots$ is an ascending chain of ideals.

3.1L3step 2.1

By [L3], the ideal chain from step 2.1 stabilizes: there is $N$ such that $a_{n} = a_{N}$ for all $n \geq N$ . Then $Z_{n} = V (a_{n}) = V (a_{N}) = Z_{N}$ for all $n \geq N$ . Hence the closed chain stabilizes.

4.1L1step 3.1∎

By [L1], stabilization of every descending closed chain means that $Spec (R)$ is Noetherian.

Depends on

Used by

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, A Primer of Commutative Algebra, v4.03, Proposition 14.4(a) (standard reference, not scraped)
The Stacks Project, Section 5.9 and Section 10.17 (standard reference, not scraped)