LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Simple-root lifting also lifts idempotents

Statement

Let $(A, m)$ be a local ring whose simple residue roots lift uniquely. Then every idempotent of the residue field $A / m$ lifts uniquely to an idempotent of $A$ .

Facts & Assumptions

Given: A local ring $(A, m)$ in which every simple residue root of a monic polynomial lifts uniquely.

[L1]

In a Henselian local ring, factor lifting implies unique lifting of simple residue roots (Factor lifting implies simple-root lifting).

Proof

technique · apply the simple-root criterion to $T^2-T$

1.1

The residue ring $A / m$ is a field, so its only idempotents are $0$ and $1$ . For $\overline{e} \in {0, 1}$ , the polynomial $p (T) = T^{2} - T$ satisfies $p (\overline{e}) = 0$ and $p^{'} (\overline{e}) = 2 \overline{e} - 1 \in {- 1, 1},$ hence $\overline{e}$ is a simple residue root.

givenalgebra

2.1

By the assumed simple-root lifting property, there is a unique lift $e \in A$ of $\overline{e}$ with $p (e) = 0$ . The equation $p (e) = 0$ is exactly $e^{2} = e$ , so $e$ is idempotent.

step 1.1given

3.1

Therefore the simple-root form lifts residue idempotents uniquely. The role of [L1] is only to identify this as the same mechanism already proved for Henselian local rings.

L1step 2.1∎

Depends on

Factor lifting implies simple-root lifting

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

The Stacks Project, Section 10.153: Henselian local rings (standard reference, not scraped)