Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07‡ rests on unproved material (inherited)
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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

A polynomial variable increases depth by one

Statement

Let (R,m) be Noetherian local and 0≠M finite. Put S=R[X](m,X) and N=M[X](m,X). Then depth⁡S(N)=depth⁡R(M)+1,dim⁡S(N)=dim⁡R(M)+1.

Facts & Assumptions

Given: X belongs to the maximal ideal of S.

Proof

technique · direct
1.1

Multiplication by X on M[X] is injective coefficientwise and its cokernel is M. These properties survive localization, so X is N-regular and N/XN≅M.

given
2.1

As an S-module, N/XN is M with X acting by zero; regular sequences from (m,X) on it are exactly regular sequences from m on M. Hence the regular-quotient depth formula gives depth⁡S(N)=depth⁡R(M)+1.

step 1.1algebra
3.1

Put A=R/Ann⁡R(M). Then Supp⁡S(N)=Spec⁡(A[X])(mA,X). The polynomial-dimension theorem gives dim⁡A[X]=dim⁡A+1, and its lower chain is attained below (mA,X) because A is local. Therefore the displayed localization has dimension dim⁡A+1=dim⁡R(M)+1. This is the asserted dimension equality and does not require every support prime to be extended from R or to contain X.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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