LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04

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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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Completeness resolves the purely inseparable prime-field case

Statement

Let $(A, m)$ be a complete equicharacteristic local ring of characteristic $p > 0$ , with residue field $k$ . If every element of $k$ is purely inseparable over the prime field $F_{p}$ , then the canonical copy of $F_{p}$ inside $A$ is contained in a coefficient field of $A$ .

Facts & Assumptions

Given: A complete equicharacteristic local ring $(A, m)$ of characteristic $p > 0$ whose residue field $k$ is purely inseparable over $F_{p}$ .

[L1]

The prime field already lifts in the equicharacteristic case (The prime field lifts in the equicharacteristic case).

[L2]

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

[L3]

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

Proof

technique · apply the complete-local source theorem to the prime-field case

1.1

By [L1], the prime field $F_{p}$ has its canonical copy inside $A$ .

L1given

2.1

By [L3], the cited Cohen structure theorem yields a coefficient ring $C \subseteq A$ . Because $A$ is equicharacteristic, that coefficient ring is a field, hence a coefficient field in the sense of [L2]. Every subfield of characteristic $p$ contains the prime field, so the canonical copy of $F_{p}$ from step 1.1 lies in $C$ .

L2L3step 1.1choose

3.1

Therefore, in the purely inseparable case over the prime field, completeness supplies a coefficient field containing the canonical prime-field lift.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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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)