LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Projections on primes, stalks and residue fields

Statement

For $A \to B, A \to C$ , put $D = B \otimes_{A} C$ . Given $r \in Spec D$ , its projections are $q = {b : b \otimes 1 \in r}$ and $q^{'} = {c : 1 \otimes c \in r}$ . Their contractions to $A$ coincide at $p$ . The stalk maps are $B_{q} ⟶ D_{r}, b / s ⟼ (b \otimes 1) / (s \otimes 1), C_{q^{'}} ⟶ D_{r}, c / t ⟼ (1 \otimes c) / (1 \otimes t) .$ They are local and induce embeddings $κ (q), κ (q^{'}) \to κ (r)$ agreeing on $κ (p)$ .

Facts & Assumptions

Given: The objects, hypotheses and conventions in the statement above.

[F1]

Let $A \to B$ and $A \to C$ be maps of commutative unital rings, allowing the zero ring. In the category of all schemes, $Spec B \times_{Spec A} Spec C ≅ Spec (B \otimes_{A} C) .$ The projections correspond to $b \mapsto b \otimes 1$ and $c \mapsto 1 \otimes c$ . (Affine fibre products are spectra of tensor products)

[F2]

For $p \in Spec A$ , there is a canonical isomorphism $O_{Spec A, p} ≅ A_{p}$ . (The stalk of the affine structure sheaf at a prime is A_p)

[F3]

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. (The stalk maps induced by a ring map are local)

Proof

1.1

F1 identifies the projection ring maps with the two tensor inclusions. Contraction therefore gives the stated primes, and their contractions to $A$ agree because $a \otimes 1 = 1 \otimes a$ .

givenF1

2.1

Elements outside the contracted primes map outside $r$ and hence become units in $D_{r}$ . F2 identifies these localizations as stalks, and F3 shows that the displayed maps are local.

F2F3step 1.1

3.1

Quotient each local map by maximal ideals. The resulting unital maps between fields are injective: their kernels are proper ideals of a field, hence zero. The maps agree on $A$ and then on its residue field after localization and quotient. If $D = 0$ there is no prime $r$ , so the pointwise assertion is vacuous.

step 1.1step 2.1algebra∎

Depends on

Used by

Points of a fibre product via residue-field tensors Lemma

Dependency tree · two levels

14 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

Stacks 26.17.2 and 26.17.5 (standard reference, not scraped)