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Over an algebraically closed field, maximal ideals of an affine algebra are kernels of points

Statement

Let $k$ be an algebraically closed field and let $A$ be an affine $k$ -algebra. Then every maximal ideal of $A$ is the kernel of a $k$ -algebra map $A \to k$ .

Facts & Assumptions

Given: An algebraically closed field $k$ and an affine $k$ -algebra $A$ .

[L1]

Every affine $k$ -algebra admits a presentation $A ≅ k [x_{1}, \dots, x_{n}] / I$ for some ideal $I$ .

[L2]

Maximal ideals of a quotient correspond to maximal ideals upstairs that contain the defining ideal ( $R / M$ is a field if and only if $M$ is a maximal ideal).

[L3]

Over an algebraically closed field, maximal ideals of the polynomial ring are evaluation ideals (Over an algebraically closed field, every maximal ideal is an evaluation ideal).

[L4]

$k$ -points of a quotient algebra are exactly its $k$ -algebra maps to $k$ (k-points of k[x_1, ..., x_n]/I are exactly k-algebra maps to k).

Proof

technique · direct

1.1

Choose a presentation $A ≅ k [x_{1}, \dots, x_{n}] / I$ as in [L1], and let $m$ be a maximal ideal of $A$ . Its inverse image in the polynomial ring is a maximal ideal $M$ containing $I$ .

L1L2givenchoose

2.1

By [L3], there exists $a \in k^{n}$ with $M = (x_{1} - a_{1}, \dots, x_{n} - a_{n})$ . Since $I \subseteq M$ , the point $a$ annihilates every element of $I$ . Therefore [L4] gives a $k$ -algebra map $A \to k$ corresponding to $a$ , and its kernel is exactly $m$ .

L3L4step 1.1

3.1

Hence every maximal ideal of $A$ is the kernel of a $k$ -point.

step 2.1∎

Depends on

Used by

A radical ideal omitting a function admits a point that kills the ideal but not the function Corollary

Dependency tree · two levels

12 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, Remark 13.4 (standard reference, not scraped)