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Maximal ideals of an affine domain have full height

Statement

Let k be a field, let A be a finite-type k-domain, and let m be a maximal ideal of A. Then

ht⁡(m)=dim⁡A.

Facts & Assumptions

Given: A field k, a finite-type k-domain A, and a maximal ideal m⊂A.

[L1]

The residue field A/m is a finite extension of k (A maximal ideal of an affine algebra has finite residue field over the base field).

[L2]

In an affine domain,

ht⁡(m)+dim⁡(A/m)=dim⁡A

(Height plus quotient dimension equals ambient dimension in an affine domain).

Proof

technique · direct
1.1

By [L1], the quotient A/m is a field. Hence dim⁡(A/m)=0.

L1given
2.1

Applying [L2] now yields ht⁡(m)=dim⁡A.

L2step 1.1
3.1

Thus every maximal ideal of an affine domain has full height.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

7 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