ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passaudited 2026-09-01

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 residue-field basis lifts to a basis of a finite flat module over a local ring

Example

Let $(R, m)$ be a Noetherian local ring and let $M$ be a finite flat $R$ -module. If ${\overset{ˉ}{x}}_{1}, \dots, {\overset{ˉ}{x}}_{r}$ is a basis of the residue vector space $M / m M$ , then any lifts $x_{1}, \dots, x_{r} \in M$ form an $R$ -basis of $M$ .

Facts & Assumptions

Given: A Noetherian local ring $(R, m)$ , a finite flat $R$ -module $M$ , a basis ${\overset{ˉ}{x}}_{1}, \dots, {\overset{ˉ}{x}}_{r}$ of $M / m M$ , and lifts $x_{1}, \dots, x_{r} \in M$ .

[L1]

A finite flat module over a Noetherian local ring is free (A finite flat module over a local ring is free).

Verification

technique · direct

1.1

By [L1], the module $M$ is free of rank $r$ , because the residue vector-space dimension equals the rank of a free module.

L1given

1.2

The chosen lifts generate $M$ by Nakayama, and a generating set of size equal to the rank of a free module is automatically a basis. Therefore $x_{1}, \dots, x_{r}$ is an $R$ -basis of $M$ .

algebra

2.1

So residue-field bases lift to actual bases in the finite flat local case.

algebra∎

Depends on

A finite flat module over a local ring is free

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Stacks Project, Section 10.78: Finite projective modules (standard reference, not scraped)