LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Vanishing ideals and zero loci form a Galois connection

Statement

Let $k$ be an algebraically closed field, let $X \subseteq k^{n}$ , and let $J \subseteq k [x_{1}, \dots, x_{n}]$ be an ideal. Then $X \subseteq V (J) ⟺ J \subseteq I (X) .$ Consequently, $X \subseteq V (I (X)) and J \subseteq I (V (J)) .$

Facts & Assumptions

Given: An algebraically closed field $k$ , a subset $X \subseteq k^{n}$ , and an ideal $J \subseteq k [x_{1}, \dots, x_{n}]$ .

[L1]

$V (J)$ is the set of points where every polynomial in $J$ vanishes (An affine algebraic set in affine space).

[L2]

$I (X)$ is the set of polynomials vanishing at every point of $X$ (The vanishing ideal of a subset of affine space).

Proof

technique · direct

1.1L1L2given

If $X \subseteq V (J)$ and $f \in J$ , then every point of $X$ lies in $V (J)$ , so [L1] says $f$ vanishes at every point of $X$ . Therefore $f \in I (X)$ by [L2], and hence $J \subseteq I (X)$ .

1.2L1L2given

Conversely, if $J \subseteq I (X)$ and $x \in X$ , then every polynomial in $J$ lies in $I (X)$ , so [L2] says every polynomial in $J$ vanishes at $x$ . By [L1], this means $x \in V (J)$ . Hence $X \subseteq V (J)$ .

2.1step 1.1step 1.2∎

Applying steps 1.1 and 1.2 with $J = I (X)$ yields $X \subseteq V (I (X))$ , and applying them with $X = V (J)$ yields $J \subseteq I (V (J))$ .

Vanishing ideals and zero loci form a Galois connection

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 1 level

Sources