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

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Existence of all scheme fibre products

Statement

Every diagram $X \to S \leftarrow Y$ of schemes has a fibre product. Given an affine cover $S = ⋃_{i} Spec A_{i}$ and affine covers $f^{- 1} (Spec A_{i}) = ⋃_{j} Spec B_{i j}$ and $g^{- 1} (Spec A_{i}) = ⋃_{k} Spec C_{i k}$ , the product has open affine cover $Spec (B_{i j} \otimes_{A_{i}} C_{i k}) .$

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]

Let $X \to S \leftarrow Y$ be given. If $Y = ⋃_{i} Y_{i}$ is an open cover and every $P_{i} = X \times_{S} Y_{i}$ exists, these products glue along their inverse images over $Y_{i} \cap Y_{j}$ to a fibre product $X \times_{S} Y$ . There is also a base-cover version: if $S = ⋃_{i} S_{i}$ and $f^{- 1} (S_{i}) \times_{S_{i}} g^{- 1} (S_{i})$ exists for every $i$ , these products glue to $X \times_{S} Y$ . No overlap is required to be affine. (Gluing fibre products along open covers)

Proof

1.1

When $X, S$ are affine, cover $Y$ by affines. Each local product exists by F1, so F2 glues them to the desired product, with the displayed affine charts.

givenF1F2

2.1

When only $S$ is affine, cover $X$ by affine opens. The preceding construction gives each product of one of these opens with $Y$ . Interchanging the roles of the two projections in F2 glues them. This interchange is justified directly by the symmetric compatible-pair condition, without needing a later associativity result.

F2step 1.1

3.1

For arbitrary $S$ , apply the preceding result over each affine $Spec A_{i}$ , and then use the base-cover clause of F2. Its overlap construction makes the stated tensor spectra an open cover. Empty schemes and zero tensor rings give empty charts, and the empty-base case forces both factors to be empty.

F1F2step 2.1∎

Depends on

Used by

Products and initial and terminal S-schemes Corollary
Base change of objects, morphisms and properties Definition
The diagonal morphism Definition
The graph morphism over a base Definition
Affine charts after extension of the ground field Lemma
Local finiteness conditions under base change Lemma
Points of a fibre product via residue-field tensors Lemma
Quasi-compactness is local on the target and survives base change Lemma
Symmetry, associativity and units Lemma
Classical products and scheme products Theorem

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

Vakil Theorem 10.1.1, complete proof; Stacks 26.17.4 (standard reference, not scraped)