DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-30

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Flat morphism of schemes

Definition

Let $f : X \to S$ be a morphism of schemes (Schemes and morphisms over a base) and let $x \in X$ with image $s = f (x)$ . The morphism induces a homomorphism of local rings $O_{S, s} ⟶ O_{X, x},$ making $O_{X, x}$ an $O_{S, s}$ -module (Flat and faithfully flat modules and ring homomorphisms).

The morphism $f$ is flat at $x$ if $O_{X, x}$ is a flat $O_{S, s}$ -module for this module structure, and $f$ is flat if it is flat at every point of $X$ . A morphism with empty source is flat vacuously, and a morphism whose target is empty has empty source and is flat.

Flatness at $x$ is a condition on the local ring map alone, so a flat morphism is flat at every point of every open subscheme through which it factors, and restricting $f$ to an open subscheme of the source preserves flatness. The affine-local reformulation in terms of an algebra map $A \to B$ is proved separately on this page and is not assumed here.

Depends on

Used by

Deformation cohomology of a smooth scheme: tangent, obstruction and automorphism spaces Corollary
Vanishing of the deformation tangent space forces rigidity of deformation classes Corollary
A flat family with a nodal special fibre is not smooth at the node Counterexample
Flat and finite type is not open without finite presentation Counterexample
Frobenius on the affine line is finite flat but not smooth Counterexample
Unramified of finite presentation does not imply flat or etale Counterexample
Bivariant Chow operations and bivariant classes Definition
Deformations of schemes and the infinitesimal deformation functor Definition
Embedded deformations of a closed subscheme Definition
Étale morphism of schemes Definition
Faithfully flat scheme morphism Definition
Fppf coverings and the fppf site Definition
Pullback of a Cartier divisor Definition
Representable morphisms of presheaves and fibrewise properties Definition
Smooth morphism of schemes Definition
Square-zero extensions, small extensions and first-order thickenings Definition
The rigidified relative Picard functor and the dual abelian variety Definition
Polynomial rings are flat and smooth Example
The family xy=t Example
A flat local map is faithfully flat Lemma
Affine-local flatness Lemma
Cycles of coherent sheaves and of closed subschemes, with flat pullback Lemma
Dense relative-dimension strata in flat finitely presented fibres Lemma
Effective fppf descent for separated locally quasi-finite morphisms Lemma
Étale maps induce completion isomorphisms at equal-residue points Lemma
Fibres, pullbacks and degrees of divisors under a finite morphism of curves Lemma
Flat deformations form a Zariski sheaf of groupoids Lemma
Flat locally finitely presented restrictions give open subquotients Lemma
Flat pullback of cycles and of rational equivalence Lemma
Flatness is stable under arbitrary base change Lemma
Flatness is stable under composition Lemma
Hilbert divisor charts and the Picard diagonal Lemma
Proper pushforward commutes with flat pullback Lemma
Restriction of an etale equivalence relation Lemma
Smooth base change of multiple test blow-ups Lemma
The Picard group and intersection form of a product of projective lines Lemma
The product of two projective lines is an integral smooth projective surface Lemma
Flatness does not force isomorphic or smooth fibres Remark
Etale morphisms are universally open and quasi-finite at every point Theorem
First-order deformations are controlled by Ext¹ of the cotangent complex Theorem

…and 4 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

The Stacks Project, Morphisms of Schemes, Definition 29.25.1 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, 29 August 2022 public draft, Definition 25.1.1 (standard reference, not scraped)