CorollaryStatement: 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.

Maximal chains in an irreducible variety

Statement

In an irreducible classical variety $X$ , a maximal proper nonempty irreducible closed subset $Z$ has codimension one. Every maximal chain of nonempty irreducible closed subsets has length $\dim X$ .

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.

[F1]

Let $X$ be irreducible affine and $0 \neq f \in k [X]$ be a nonunit. Then $V_{X} (f)$ is nonempty and every irreducible component has dimension $\dim X - 1$ , hence codimension one. 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. (A nontrivial principal section has pure codimension one).

[F2]

If $U$ is a nonempty open of an irreducible classical variety $X$ , then $\dim U = \dim X$ . Every proper closed subvariety $Z ⊊ X$ has $\dim Z < \dim X$ . 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. (Nonempty opens preserve irreducible dimension).

[F4]

A classical variety $X$ has $\dim X \leq 0$ if and only if its underlying set is finite. The empty set is included. A nonempty irreducible variety of dimension zero is one point. 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. (Zero-dimensional varieties are finite sets).

[F5]

For a nonempty irreducible closed subvariety $Z$ of an irreducible classical variety $X$ , define ${codim}_{X} Z = \dim X - \dim Z$ . These are finite integers. In a reducible ambient variety a difference of global dimensions must not be substituted for the height of a local prime; the containing component matters. 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. (Codimension of an irreducible closed subvariety).

Proof

1.1

Choose an affine open $U$ meeting $Z$ . The proper closed subset $Z \cap U$ is defined by an ideal containing a nonzero function $f$ ; since it has a point, $f$ is a nonunit. Choose a component $D$ of $V_{U} (f)$ containing $Z \cap U$ . Its closure in $X$ is irreducible, contains $Z$ and is proper because its intersection with $U$ lies in the proper zero set. Maximality forces this closure to equal $Z$ . The principal theorem and open invariance give $\dim Z = \dim D = \dim U - 1 = \dim X - 1$ .

F1F2

2.1

Dimension is finite, and every proper irreducible closed inclusion strictly decreases it. Thus any chain is finite. A maximal chain must end at $X$ and start at a point (otherwise insert a point); successive members are maximal proper irreducible closed subsets of the next member. The first step applied to each inclusion decreases dimension by exactly one. Since the starting point has dimension zero, the number of inclusions is $\dim X$ . If $X$ is a point there is only its length-zero maximal chain.

F2F4F5step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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 Corollaries 3.43–3.44, pp.76–77 (standard reference, not scraped)