LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

The simple-root form recovers factor lifting

Statement

Let $(A, m)$ be a local ring with residue field $k$ . Assume every simple root $\overline{a} \in k$ of the reduction $\overline{f}$ of every monic polynomial $f \in A [T]$ lifts uniquely to an element $a \in A$ satisfying $f (a) = 0$ . Then $(A, m)$ has the coprime monic factor-lifting property.

Facts & Assumptions

Given: A local ring $(A, m)$ in which, for every monic $f \in A [T]$ , every simple root of $\overline{f} \in k [T]$ lifts uniquely to a root of $f$ in $A$ .

[L1]

Stacks, Section 10.153, Lemma 10.153.3 identifies the simple-root lifting condition for a local ring with the coprime monic factor-lifting property.

Proof

technique · invoke the local converse in the cited source

1.1

The hypothesis of this item is exactly the simple-root lifting condition in [L1], and conclusion (3) of [L1] is exactly the coprime monic factor-lifting property. Therefore the cited source yields the required factorization statement for $(A, m)$ .

L1given

2.1

Therefore unique lifting of simple residue roots recovers unique lifting of coprime monic factorizations.

step 1.1∎

Used by

A local ring is Henselian exactly when simple residue roots lift uniquely Corollary

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

The Stacks Project, Section 10.153: Henselian local rings (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Chapter 22 (standard reference, not scraped)