LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge 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.

Dimension can be computed on an open cover

Statement

For every open cover $T = ⋃_{i \in I} U_{i}$ of a Noetherian space, $\dim T = \sup_{i} \dim U_{i}$ , with empty supremum $- \infty$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

For a Noetherian topological space $T$ , define $\dim T$ as the supremum of the lengths $s$ of strict chains $Z_{0} ⊊ \dots ⊊ Z_{s}$ of nonempty irreducible closed subsets of $T$ . Thus a one-member chain has length zero. Set $\dim \emptyset = - \infty$ , and allow $\dim T = + \infty$ . The supremum of an empty family of dimensions is $- \infty$ . (Chain dimension and the empty-space convention).

Proof

1.1

If $T$ is empty all terms have dimension $- \infty$ . Otherwise, for a chain of irreducible closed subsets $C_{0} ⊊ \dots ⊊ C_{s}$ in an open $U$ , their closures in $T$ are irreducible and closed, and $\overline{C_{j}} \cap U = C_{j}$ . Hence their closures remain strictly nested and $\dim U \leq \dim T$ .

2.1

For a chain $Z_{0} ⊊ \dots ⊊ Z_{s}$ in $T$ , choose $x \in Z_{0}$ and a covering open $U_{i}$ containing $x$ . Each $Z_{j} \cap U_{i}$ is nonempty, irreducible and closed in $U_{i}$ . A nonempty open of an irreducible space is dense: two disjoint nonempty opens would give a cover by two proper closed subsets. Thus $\overline{Z_{j} \cap U_{i}} = Z_{j}$ . The intersections form a strict chain of the same length. Taking suprema proves the claim.

F1step 1.1∎

Depends on

Chain dimension and the empty-space convention

Used by

Affine and projective n-space have dimension n Corollary
Dimension equals transcendence degree Theorem
Dimensions add under products Theorem
Fibres have pure expected dimension over a dense open Theorem

Dependency tree · two levels

3 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 §3l chain interpretation; §5j p.115 (standard reference, not scraped)