CorollaryStatement: 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.

Complete equicharacteristic local rings have coefficient fields

Statement

Assume the Axiom of Choice.

Every complete equicharacteristic Noetherian local ring contains a coefficient field.

Facts & Assumptions

Given: A complete equicharacteristic Noetherian local ring $(A, m)$ and the Axiom of Choice.

[L1]

Stacks, Section 10.160, Theorem 10.160.8 gives a coefficient ring in every complete local ring; in the equicharacteristic case that coefficient ring is a field.

[L2]

A coefficient field is exactly a subfield mapping isomorphically to the residue field (Equicharacteristic local rings and coefficient fields).

Proof

technique · apply the Cohen structure theorem directly

1.1

By [L1], the complete local ring $A$ contains a coefficient ring $Λ \subseteq A$ . Because $A$ is equicharacteristic, the cited source says that $Λ$ is a field mapping isomorphically to $A / m$ .

L1givenchoose

2.1

By [L2], any such subfield is a coefficient field in the library's terminology. Therefore $Λ$ is a coefficient field of $A$ .

L2step 1.1

3.1

Therefore every complete equicharacteristic Noetherian local ring has a coefficient field.

step 2.1∎

Depends on

Equicharacteristic local rings and coefficient fields

Used by

Dependency tree · two levels

3 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

Melvin Hochster, The structure theory of complete local rings (standard reference, not scraped)
The Stacks Project, Section 10.160: The Cohen structure theorem (standard reference, not scraped)