DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Fibre product of schemes

Definition

Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes. A fibre product is a scheme $P$ , with projections $p : P \to X$ and $q : P \to Y$ , such that $f p = g q$ and, for every scheme $T$ and morphisms $a : T \to X$ , $b : T \to Y$ with $f a = g b$ , there is exactly one $h : T \to P$ satisfying $p h = a$ and $q h = b$ . Thus, naturally in every test scheme $T$ , $Hom (T, P) ≅ Hom (T, X) \times_{Hom (T, S)} Hom (T, Y) .$ Write $P = X \times_{S} Y$ . The commutative square with edges $p, q, f, g$ is Cartesian when it has this universal property. Morphisms here are morphisms of locally ringed spaces, as in Morphisms of schemes. No existence assertion is part of the definition.

Depends on

Morphisms of schemes

Used by

Euler characteristic in a proper flat family is locally constant Corollary
The orbit set of k-points need not be the k-points of the fppf quotient sheaf Counterexample
Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers Definition
Cohomology and base-change map Definition
Crossed homomorphisms, principal crossed homomorphisms and Hochschild extensions Definition
Deformations of schemes and the infinitesimal deformation functor Definition
Descent data for schemes over an fppf covering Definition
Descent data, prestacks and stacks in groupoids over the fppf site Definition
Embedded deformations of a closed subscheme Definition
Fppf coverings and the fppf site Definition
Fppf sheaves of sets and sheafification Definition
Group schemes over a base scheme Definition
Groupoids in schemes, relations and etale equivalence relations Definition
Morphisms representable by algebraic spaces Definition
Morphisms, products and fibre products of algebraic spaces Definition
Presentations of algebraic spaces Definition
Quotient sheaves and representable quotients for pre-relations and group actions Definition
Representable morphisms of presheaves and fibrewise properties Definition
Split augmented simplicial schemes and matching objects Definition
Square-zero extensions, small extensions and first-order thickenings Definition
The derived subgroup, the derived series and solvable algebraic groups Definition
The fppf quotient sheaf of a pre-relation Definition
A proper singular curve with weight-zero cohomology Example
The affine line is an algebraic space Example
A linear representation induces an action on projective space with the same line stabilizers Lemma
A tangent direction is realized by a local smooth curve Lemma
A transverse hyperplane slice is smooth at the chosen point Lemma
Chow lemma for proper Noetherian schemes Lemma
Criterion for a scheme to represent an fppf quotient sheaf Lemma
Effective fppf descent for separated locally quasi-finite morphisms Lemma
Every representable functor is an algebraic space Lemma
Fibres of the orbit map and the scheme-theoretic stabilizer as a closed subgroup scheme Lemma
Flat deformations form a Zariski sheaf of groupoids Lemma
Flat locally finitely presented restrictions give open subquotients Lemma
Homogeneous curves and automorphisms of P¹ Lemma
Lie algebras of subspace stabilizers and Lie-stable subspaces Lemma
Monomorphisms and diagonals Lemma
Proper pushforward commutes with flat pullback Lemma
Pushouts of closed immersions exist Lemma
Restricting fibre products to open subschemes Lemma

…and 12 more results.

Dependency tree · two levels

3 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 26.17.1 (standard reference, not scraped)