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

1.1

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

L1given

1.2

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.

L2L3given

2.1

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

step 1.1step 1.2∎

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)