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

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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.
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Complete local rings are Henselian

Statement

If $(A, m)$ is a local ring that is complete and separated for its maximal-ideal topology, then $A$ is Henselian.

Facts & Assumptions

Given: A local ring $(A, m)$ complete and separated for the $m$ -adic topology.

[L1]

Every complete separated adic pair is Henselian (Complete separated adic pairs are Henselian).

Proof

technique · apply the pair statement to the maximal ideal

1.1

By [L1], the pair $(A, m)$ is Henselian because $A$ is complete and separated for the $m$ -adic topology.

L1given

2.1

By definition, a local ring is Henselian exactly when its maximal-ideal pair is Henselian. Therefore $A$ is Henselian.

step 1.1given∎

Depends on

Complete separated adic pairs are Henselian

Used by

Dependency tree · two levels

9 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)