TheoremStatement: 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.

Equivalent elementary forms of Hensel's property

Statement

For a local ring $(A, m)$ , the following are equivalent:

$A$ is Henselian.
Every simple root of every monic polynomial over $A / m$ lifts uniquely to $A$ .
For every finite $A$ -algebra $B$ , the map from idempotents of $B$ to idempotents of $B / m B$ is a bijection.

Facts & Assumptions

Given: A local ring $(A, m)$ .

[L1]

Henselianity is equivalent to unique simple-root lifting (A local ring is Henselian exactly when simple residue roots lift uniquely).

[L2]

In Stacks, Section 15.11, Lemma 15.11.6, a pair $(A, I)$ is Henselian if and only if for every finite $A$ -algebra $B$ the map $B \to B / I B$ induces a bijection on idempotents.

[L3]

A local ring $A$ is Henselian exactly when its maximal-ideal pair $(A, m)$ is Henselian (Henselian pairs and Henselian local rings).

Proof

technique · combine the local criterion with the pair-level finite-algebra characterization from the cited source

1.1L1given

By [L1], conditions (1) and (2) are equivalent.

1.2L2L3given

By [L3], condition (1) says exactly that the pair $(A, m)$ is Henselian. Then [L2] identifies this with the finite-algebra idempotent bijection in condition (3). Hence conditions (1) and (3) are equivalent.

2.1step 1.1step 1.2∎

Since (1) is equivalent to both (2) and (3), all three conditions are equivalent.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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 15.11: Henselian pairs (standard reference, not scraped)
The Stacks Project, Section 10.153: Henselian local rings (standard reference, not scraped)