CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

The product of Zariski topologies is too coarse on A1 times A1

Statement refuted

For every field $k$ , the Zariski topology of $A_{k}^{1} \times A_{k}^{1}$ equals the product of the two factor Zariski topologies.

Facts & Assumptions

Given: An infinite field $k$ and the diagonal $D = V (x - y) \subseteq A_{k}^{1} \times A_{k}^{1}$ .

Counterexample

1.1

The affine-product coordinate ring is $k [x, y]$ , so $D$ is Zariski closed.

givenalgebra

2.1

Each factor has the cofinite Zariski topology. For a point $(a, b)$ off $D$ , every basic product neighbourhood $U \times V$ has $U \cap V \neq \emptyset$ . Thus it contains $(c, c) \in D$ for some $c \in U \cap V$ .

step 1.1

3.1

Hence no product-topology neighbourhood of $(a, b)$ lies in the complement of $D$ . The complement is not open, so $D$ is not closed in the product topology, although it is closed in the affine-product Zariski topology. This refutes the statement.

step 2.1∎

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

MIT 18.725 Algebraic Geometry, Lecture 7, Remark 10 (standard reference, not scraped)