ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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.

A proper localization is flat but need not be faithfully flat

Example

Assume the Axiom of Choice for the faithfully-flat characterization used below.

The localization map

Z \to Z ⁣ [\frac{1}{2}]

is flat but not faithfully flat.

Facts & Assumptions

Given: The Axiom of Choice and the localization map $Z \to Z [1 / 2]$ .

[L1]

Every localization is flat (Every localization is flat, and localizing a flat module preserves flatness).

[L2]

Faithful flatness is equivalent to preserving proper ideals under extension (A flat ring map is faithfully flat exactly when it detects proper ideals and is surjective on spectra).

Verification

technique · direct

1.1

By [L1], $Z [1 / 2]$ is flat over $Z$ .

L1given

1.2

The proper ideal $(2) \subset Z$ becomes the unit ideal after localization, since $2$ is invertible in $Z [1 / 2]$ . Thus $(2) Z [1 / 2] = Z [1 / 2]$ . By [L2], the map is not faithfully flat.

2.1

So a proper localization can be flat without being faithfully flat.

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

J. S. Milne, A Primer of Commutative Algebra, §11 (standard reference, not scraped)