LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-06‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Classical k-points give closed points over an algebraically closed field

Statement

Let $k$ be algebraically closed and let $A$ be a reduced finite-type $k$ -algebra. The closed points of $Spec A$ are exactly the kernels of the $k$ -algebra maps $A \to k$ . They need not exhaust all points of $Spec A$ .

Facts & Assumptions

Given: An algebraically closed field $k$ and a reduced finite-type $k$ -algebra $A$ .

[F1]

Every maximal ideal of an affine $k$ -algebra is the kernel of a $k$ -algebra map to $k$ (Over an algebraically closed field, maximal ideals of an affine algebra are kernels of points).

[F2]

Closed points of an affine scheme are its maximal ideals (Closed points of an affine scheme).

Proof

technique · direct

1.1

A closed point is maximal by [F2], and [F1] identifies every such ideal with a kernel $A \to k$ .

F1F2

1.2

Every unital $k$ -algebra map $A \to k$ is surjective, so its kernel is maximal and hence closed by [F2].

F2algebra

2.1

Thus classical $k$ -points and closed points agree, but this makes no assertion that every prime is maximal.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

James S. Milne, Algebraic Geometry, 10.24 (standard reference, not scraped)