LemmaStatement: Literature-sourcedProof: 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.

Completion preserves regular sequences

Statement

Let $(R, m)$ be Noetherian local, let $M$ be finite, and put $\hat{M} = M \otimes_{R} \hat{R}$ . If $x_{1}, \dots, x_{r} \in m$ is $M$ -regular, then its images form a $\hat{M}$ -regular sequence.

Facts & Assumptions

Given: $R \to \hat{R}$ is faithfully flat.

Proof

technique · direct

1.1

Tensor each injective multiplication map on the successive quotients with the flat module $\hat{R}$ . Flatness preserves its injectivity and identifies the resulting quotient with the corresponding quotient of $\hat{M}$ .

given

2.1

The terminal quotient remains nonzero because faithful flatness reflects the zero module. Hence the base-changed sequence satisfies both parts of the regular-sequence definition.

step 1.1algebra∎

Depends on

Used by

Completion reflects depth Lemma

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

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