LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-24 (gpt-6-sol)

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.

A regular parameter quotient preserves the depth--dimension gap

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $(R, m)$ be Noetherian local, $0 \neq M$ finite, and let $x \in m$ be $M$ -regular and part of a system of parameters for $M$ . Then $\dim_{R} (M / x M) - {depth}_{R} (M / x M) = \dim_{R} (M) - {depth}_{R} (M) .$

Facts & Assumptions

Given: The Axiom of Choice; regularity makes $M / x M$ nonzero; complete $x$ to a system of parameters $x, x_{2}, \dots, x_{d}$ for $M$ , where $d = \dim_{R} (M)$ .

Proof

technique · direct

1.1given

By lem-depth-quotient-by-regular-element, $depth (M / x M) = depth (M) - 1$ .

1.2givenalgebra

Put $A = R / {Ann}_{R} (M)$ . The finite-length terminal quotient shows that the images of $x, x_{2}, \dots, x_{d}$ form a system of parameters of the local ring $A$ , as in thm-dimension-and-parameters-for-modules. Moreover ${Supp}_{R} (M / x M) = {Supp}_{R} (M) \cap V (x) ≅ Spec (A / (x)) .$ Applying lem-parameter-dimension-drop-is-exact to $A$ gives $\dim_{R} (M / x M) = d - 1$ .

2.1step 1.1step 1.2algebra∎

Subtracting the equalities in steps 1.1 and 1.2 cancels the common decrement and gives the displayed identity.

Depends on

Used by

Cohen--Macaulayness and a regular parameter quotient Corollary

Dependency tree · two levels

18 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

Depth and Cohen--Macaulay modules source treatment (standard reference, not scraped)