LemmaStatement: Literature-sourcedProof: AI-generatedprecheck 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.

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The prime field lifts in the equicharacteristic case

Statement

Let $(A, m)$ be an equicharacteristic local ring with residue field $k = A / m$ . Then the prime field of $k$ has a canonical copy inside $A$ , and the residue map identifies that copy with the prime field of $k$ .

Facts & Assumptions

Given: An equicharacteristic local ring $(A, m)$ with residue field $k$ .

[L1]

Every field has a prime subfield, isomorphic either to $Q$ or to $F_{p}$ according to its characteristic (A field's prime subfield is isomorphic to $Q$ in characteristic zero and to $F_{p}$ in characteristic $p$ ).

[L2]

Equicharacteristic means $char (A) = char (k)$ (Equicharacteristic local rings and coefficient fields).

Proof

technique · compare the two characteristics

1.1

By [L2], the ring $A$ and its residue field $k$ have the same characteristic. If that characteristic is $0$ , then the unique map $Z \to A$ kills no nonzero integer, so it extends to an embedding $Q ↪ A$ . If the characteristic is $p > 0$ , then the image of $Z \to A$ is a copy of $F_{p}$ inside $A$ .

L1L2givenalgebra

2.1

Reducing these embedded prime fields modulo $m$ gives the prime subfield of $k$ , because $A$ and $k$ have the same characteristic and prime subfields are unique by [L1].

L1step 1.1

3.1

Therefore the prime field of the residue field lifts canonically in the equicharacteristic case.

step 2.1∎

Depends on

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

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)