DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-07

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Schemes and morphisms over a base

Definition

An $S$ -scheme is a scheme $X$ equipped with a morphism $X \to S$ . An $S$ -morphism $X \to Y$ is a scheme morphism commuting with the maps to $S$ . For $n \geq 0$ , choose an affine open cover $S = ⋃_{i} Spec A_{i}$ . Over each chart take $Spec A_{i} [t_{1}, \dots, t_{n}]$ . On overlaps, localization in the coefficients gives canonical isomorphisms that fix the variables; these satisfy the cocycle condition and glue by thm-gluing-affine-schemes. The result, independent of the cover up to the unique $S$ -isomorphism respecting the coefficient maps and the ordered coordinate functions $t_{1}, \dots, t_{n}$ , is the relative affine space $A_{S}^{n}$ . Its structure morphism is affine, although its total scheme need not be affine when $S$ is not. For $n = 0$ it is $S$ ; for $S = \emptyset$ the construction gives the empty scheme. The uniqueness assertion concerns these coordinate-compatible identifications, not arbitrary $S$ -isomorphisms.

Depends on

Used by

Maps from proper integral schemes to the affine line have closed-point image Corollary
Products and initial and terminal S-schemes Corollary
A nonclosed open immersion is not proper Counterexample
A nonseparated affine cover can have nonaffine intersection Counterexample
The affine line is not proper Counterexample
Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
Abelian varieties over a field Definition
Affine-local quasi-coherent algebras before general sheaf theory Definition
Affine-overlap separation condition Definition
Algebraic cycles and the cycle group of a scheme of finite type over a field Definition
Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers Definition
Base change of objects, morphisms and properties Definition
Deformations of schemes and the infinitesimal deformation functor Definition
Descent data for schemes over an fppf covering Definition
Etale neighbourhoods and elementary etale neighbourhoods of a point Definition
Flat morphism of schemes Definition
Formally etale morphism Definition
Formally smooth morphism Definition
Formally unramified morphism Definition
Fppf coverings and the fppf site Definition
Geometric vector bundle with the sections convention Definition
Group schemes of finite type over a field Definition
Groupoids in schemes, relations and etale equivalence relations Definition
Open and universally open morphisms of schemes Definition
Relative Proj of a graded quasi-coherent algebra Definition
Relative projective space from standard charts Definition
Scheme-theoretic varieties Definition
Separated S-scheme Definition
Sheaf of relative Kähler differentials Definition
The cotangent complex of a morphism of schemes Definition
The graph morphism over a base Definition
The intrinsic Zariski tangent space Definition
The scheme-theoretic tangent cone at a point Definition
Affine n-space over an arbitrary base Example
An irreducible curve can have arbitrarily large tangent dimension Example
Classical bilinear-form equations linearize to matrix spaces Example
Dual numbers compute the tangent spaces of the general and special linear groups Example
Finite power map of the affine line Example
Incidence projection has closed determinantal image Example
The affine line is an algebraic space Example

…and 19 more results.

Dependency tree · two levels

7 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

Ravi Vakil, Foundations of Algebraic Geometry, Section 7.3 (standard reference, not scraped)