LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

The stalk maps induced by a ring map are local

Statement

Let $φ : A \to B$ , let $q \in Spec B$ , and put $p = φ^{- 1} (q)$ . The induced stalk homomorphism $A_{p} \to B_{q}$ is local.

Facts & Assumptions

Given: A ring map $φ : A \to B$ and a prime $q$ of $B$ .

[F1]

The induced map of affine spectra has, on stalks, the localization map at $p = φ^{- 1} (q)$ (The map of affine spectra induced by a ring homomorphism).

[F2]

The maximal ideals of $A_{p}$ and $B_{q}$ are $p A_{p}$ and $q B_{q}$ , respectively ( $R_{p}$ is local with unique maximal ideal $p R_{p}$ ).

Proof

technique · direct

1.1F1

By [F1], the stalk map sends $a / s$ to $φ (a) / φ (s)$ , for $s \notin p$ .

1.2givenalgebra

This image lies in $q B_{q}$ exactly when $φ (a) \in q$ , because its denominator is outside the prime ideal.

2.1F2step 1.2∎

Thus [F2] identifies the inverse image of $q B_{q}$ with $p A_{p}$ , so the map is local.

Depends on

Used by

Dependency tree · two levels

16 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, Lemma 26.6.4 (standard reference, not scraped)