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

1.1givenF1

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

2.1F2F3step 1.1

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.

3.1step 1.1step 2.1algebra∎

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.

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)