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Maximal chains in an affine domain all have the same length

Statement

Assume the Axiom of Choice. Let $k$ be a field and let $A$ be a finite-type $k$ -domain. Every saturated prime chain from $(0)$ to a maximal ideal of $A$ has length $\dim A$ .

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice), a field $k$ , a finite-type $k$ -domain $A$ , and a saturated prime chain $(0) = p_{0} ⊊ p_{1} ⊊ \dots ⊊ p_{r} = m .$

[L1]

In a domain, the minimal prime $(0)$ has height zero (Minimal primes are exactly the primes of height zero).

[L2]

Under Choice, every maximal ideal of an affine domain has height equal to the full dimension (Maximal ideals of an affine domain have full height).

[L3]

Under Choice, for primes $p \subseteq q$ in an affine domain, $ht (q / p) + {trdeg}_{k} Frac (A / q) = {trdeg}_{k} Frac (A / p)$ (Transcendence degrees along affine prime quotients add correctly).

[L4]

Under Choice, for every prime $p$ of an affine domain, $ht (p) + {trdeg}_{k} Frac (A / p) = {trdeg}_{k} Frac (A)$ (The dimension formula for affine domains).

Proof

technique · direct

1.1L1given

By [L1], $ht (p_{0}) = 0$ .

2.1L3L4step 1.1given

For each $0 \leq i < r$ , the quotient $A / p_{i}$ is a domain and the chain is saturated, so there is no prime strictly between $0$ and $p_{i + 1} / p_{i}$ in $A / p_{i}$ . Hence $ht (p_{i + 1} / p_{i}) = 1$ . Applying [L3] to $p_{i} \subseteq p_{i + 1}$ and comparing [L4] at $p_{i}$ and $p_{i + 1}$ gives $ht (p_{i + 1}) = ht (p_{i}) + 1$ .

3.1L2step 2.1∎

Starting from step 1.1 and iterating step 2.1, we obtain $ht (m) = r$ . By [L2], $ht (m) = \dim A$ . Therefore the saturated chain has length $\dim A$ , and every such chain has the same length.

Depends on

Used by

Why the equal-chain statement stops at affine domains Remark

Dependency tree · two levels

17 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, §§18, 21 (standard reference, not scraped)
Melvin Hochster, Dimension theory and systems of parameters (standard reference, not scraped)