LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-24 (gpt-6-sol)

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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.

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A point is locally cut out by dim X functions

Statement

If $X$ is irreducible of dimension $n$ and $x \in X$ is a closed point, there are an affine neighborhood $U$ of $x$ and $n$ regular functions on $U$ whose common zero set is exactly ${x}$ .

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement, including the Axiom of Choice (The Axiom of Choice).

[F1]

If $X$ is an irreducible classical variety and $x$ is a closed point, then $\dim O_{X, x} = \dim X = {codim}_{X} {x}$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Closed-point local dimension equals ambient irreducible dimension).

[F2]

Under the Axiom of Choice, if $R$ is a Noetherian commutative ring and $p \in Spec (R)$ has finite height $n$ , then in $R_{p}$ the maximal ideal $p R_{p}$ is minimal over an ideal generated by $n$ elements lifted from $p$ (Converse to Krull's height theorem in localised form).

[F3]

Every classical variety is Noetherian and has finitely many irreducible components. Every open or closed subvariety has a finite affine cover. Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Classical varieties have finite irreducible decompositions).

Proof

1.1F1F2

Choose an affine chart $V$ at $x$ , put $A = k [V]$ and $m = m_{x}$ . The local dimension result gives $ht m = n$ . The converse height theorem supplies $a_{1}, \dots, a_{n} \in m$ such that $m A_{m}$ is minimal over their localized ideal. Thus $m$ itself is minimal over $(a_{1}, \dots, a_{n})$ : any smaller prime containing this ideal would stay smaller on localization at $m$ .

2.1F3step 1.1∎

The zero set in $V$ therefore has ${x}$ as a component. Its finitely many other components avoid $x$ . Remove them, and choose a principal affine neighborhood $U$ of $x$ inside the resulting open of $V$ . The restrictions of the $a_{i}$ have common zero set exactly ${x}$ in $U$ . When $n = 0$ , the empty list of equations cuts out $V$ locally at its isolated component ${x}$ , so this construction gives $U = {x}$ .

Depends on

Used by

Every fibre component has the expected lower bound Theorem

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

Milne Proposition 3.47 specialized to a point (standard reference, not scraped)