# Flat morphism

A **flat morphism** is a morphism of schemes \( f : X \to S \) such that, for every point \( x \in X \), the induced map of local rings \( \mathcal{O}_{S, f(x)} \to \mathcal{O}_{X, x} \) makes \( \mathcal{O}_{X, x} \) a flat module over \( \mathcal{O}_{S, f(x)} \).<sup>[1](https://mathworld.wolfram.com/FlatMorphism.html)</sup> The definition imports the commutative-algebra notion of a [flat module](https://www.edgepedia.org/flat-module), one for which the tensor functor \( M \mapsto M \otimes_A M' \) is exact, into geometry, and it has become one of the standard technical tools of algebraic geometry.<sup>[2](https://stacks.math.columbia.edu/tag/0250)</sup><sup> • </sup><sup>[3](https://stacks.math.columbia.edu/tag/01U2)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( f : X \to S \) is flat at \( x \) if \( \mathcal{O}_{X,x} \) is a flat \( \mathcal{O}_{S,f(x)} \)-module; flat means flat at every point of \( X \)<sup>[1](https://mathworld.wolfram.com/FlatMorphism.html)</sup> |
| Faithful flatness | Flat plus surjective on underlying topological spaces; for affine schemes \( \mathrm{Spec}\,B \to \mathrm{Spec}\,A \) this is equivalent to \( B \) being faithfully flat as an \( A \)-module<sup>[4](https://kskedlaya.org/18.726/flat.pdf)</sup> |
| Stability | Compositions of flat morphisms are flat, and every base change of a flat morphism is flat<sup>[1](https://mathworld.wolfram.com/FlatMorphism.html)</sup> |
| Relation to smoothness | Every smooth morphism is flat; for \( f \) locally of finite type, smoothness at \( x \) is equivalent to flatness at \( x \) together with smoothness of the fiber \( f^{-1}(y) \) over \( \kappa(y) \) at \( x \)<sup>[1](https://mathworld.wolfram.com/FlatMorphism.html)</sup><sup> • </sup><sup>[5](https://jcreinhold.github.io/sga/i/02-smooth-morphisms-generalities-differential-properties.html)</sup> |
| Openness | A flat morphism locally of finite presentation is universally open<sup>[3](https://stacks.math.columbia.edu/tag/01U2)</sup> |
| Families | In a projective flat family over a connected Noetherian base, the Hilbert polynomial, degree, and arithmetic genus of the fibers are constant<sup>[6](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)</sup> |
| Generic flatness | If \( S \) is integral (or reduced), \( f \) is of finite type, and \( \mathcal{F} \) is a finite type \( \mathcal{O}_X \)-module, then \( X_U \to U \) and \( \mathcal{F}|_{X_U} \) are flat and of finite presentation over some open dense \( U \subset S \)<sup>[7](https://stacks.math.columbia.edu/tag/0529)</sup> |

## Algebraic origin

The underlying algebra is the notion of a flat module: an \( A \)-module \( M \) is flat when the functor \( M \otimes_A \cdot \) is exact, and a ring map \( A \to B \) is flat when \( B \) is flat as an \( A \)-module.<sup>[2](https://stacks.math.columbia.edu/tag/0250)</sup> Free and projective modules are flat, and localizations are flat because localization is exact.<sup>[6](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)</sup> Flatness is a local property: \( M \) is flat over \( A \) if and only if \( M_{\mathfrak p} \) is flat over \( A_{\mathfrak p} \) for every prime \( \mathfrak p \).<sup>[2](https://stacks.math.columbia.edu/tag/0250)</sup>

The history is unusual. [Jean-Pierre Serre](https://www.edgepedia.org/jean-pierre-serre) has stated that he introduced flatness purely for reasons of algebra in his landmark "GAGA" paper, and that it was [Alexander Grothendieck](https://www.edgepedia.org/alexander-grothendieck) who recognized its geometric significance; Serre did not anticipate that it would interest anyone beyond pure algebra.<sup>[6](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)</sup><sup> • </sup><sup>[8](https://mathoverflow.net/questions/46541/how-to-introduce-notions-of-flat-projective-and-free-modules)</sup> [David Mumford](https://www.edgepedia.org/david-mumford) described the concept as "a riddle that comes out of algebra, but which technically is the answer to many prayers."<sup>[6](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)</sup> The systematic development appeared in Grothendieck and Dieudonné's *Éléments de géométrie algébrique*, volume IV, published in 1965.<sup>[1](https://mathworld.wolfram.com/FlatMorphism.html)</sup>

## Definition and basic properties

For a morphism of schemes \( f : X \to S \), flatness at \( x \in X \) means that \( \mathcal{O}_{X,x} \) is flat over \( \mathcal{O}_{S,f(x)} \), and \( f \) is flat when this holds at every point.<sup>[2](https://stacks.math.columbia.edu/tag/0250)</sup> Equivalently, \( f \) is flat if and only if the structure sheaf \( \mathcal{O}_X \) is flat over \( S \), and the condition can be checked on affine open covers: \( f \) is flat exactly when the ring maps \( \mathcal{O}_S(V) \to \mathcal{O}_X(U) \) are flat for affine opens \( U \subset X \), \( V \subset S \) with \( f(U) \subset V \).<sup>[3](https://stacks.math.columbia.edu/tag/01U2)</sup>

**Stability properties.** The class of flat morphisms is closed under composition, and every base change of a flat morphism is flat.<sup>[1](https://mathworld.wolfram.com/FlatMorphism.html)</sup><sup> • </sup><sup>[3](https://stacks.math.columbia.edu/tag/01U2)</sup> Open immersions are flat, since they induce isomorphisms on local rings.<sup>[2](https://stacks.math.columbia.edu/tag/0250)</sup> [Polynomial](https://www.edgepedia.org/polynomial) rings give the basic positive example: \( A[T_1, \dots, T_n] \) is a free, hence flat, \( A \)-module, so affine \( n \)-space \( \mathbf{A}^n_A \to \mathrm{Spec}\,A \) is flat, smooth of relative dimension \( n \), and étale for \( n = 0 \).<sup>[9](https://alphabetamath.cc/scheme-theory/flat-smooth-and-etale-morphisms-examples)</sup> A basic negative example is \( \mathrm{Spec}\,k \to \mathrm{Spec}\,k[t]/(t^2) \), which is not flat.<sup>[6](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)</sup>

Flatness also has a homological consequence on the source: if \( f : X \to Y \) is flat, the pullback functor on quasi-coherent sheaves \( \mathrm{QCoh}(\mathcal{O}_Y) \to \mathrm{QCoh}(\mathcal{O}_X) \) is exact, and conversely, when \( Y \) is quasi-compact and quasi-separated, exactness of \( f^* \) implies \( f \) is flat.<sup>[2](https://stacks.math.columbia.edu/tag/0250)</sup>

## Faithfully flat morphisms

A morphism that is both flat and surjective is called **faithfully flat**.<sup>[2](https://stacks.math.columbia.edu/tag/0250)</sup><sup> • </sup><sup>[4](https://kskedlaya.org/18.726/flat.pdf)</sup> In module terms, \( A \to B \) is faithfully flat when \( M \mapsto M \otimes_A B \) is exact and faithful, equivalently \( B \) is flat over \( A \) and \( M \to M \otimes_A B \) is injective for every \( A \)-module \( M \); this means tensoring with \( B \) detects isomorphisms.<sup>[2](https://stacks.math.columbia.edu/tag/0250)</sup><sup> • </sup><sup>[4](https://kskedlaya.org/18.726/flat.pdf)</sup><sup> • </sup><sup>[10](https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/meier-TMF-Lecture.pdf)</sup> For ring maps, faithful flatness is equivalent to flatness plus surjectivity of the induced map on spectra.<sup>[2](https://stacks.math.columbia.edu/tag/0250)</sup>

A useful special case is that a flat local homomorphism of nonzero local rings \( (A, \mathfrak{m}) \to (B, \mathfrak{n}) \) is automatically faithfully flat; the proof uses the Axiom of Choice to place each proper ideal of \( A \) inside a maximal ideal.<sup>[11](https://alphabetamath.cc/scheme-theory/flat-smooth-and-etale-morphisms)</sup> Consequently, if \( f : X \to S \) is flat at \( x \), the local ring map \( \mathcal{O}_{S,s} \to \mathcal{O}_{X,x} \) is faithfully flat.<sup>[11](https://alphabetamath.cc/scheme-theory/flat-smooth-and-etale-morphisms)</sup>

Faithfully flat morphisms are the building blocks of the flat topologies on schemes. The two standard choices are the fppf topology, from French *fidèlement plat de présentation finie*, faithfully flat and of finite presentation, and the fpqc topology, from *fidèlement plat et quasicompact*, faithfully flat and quasi-compact.<sup>[12](https://ncatlab.org/nlab/show/flat%20morphism)</sup> These topologies support [faithfully flat descent](https://www.edgepedia.org/faithfully-flat-descent), which generalizes Galois descent and allows gluing in the same way as the [Zariski topology](https://www.edgepedia.org/zariski-topology) but for a larger class of covers.<sup>[6](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)</sup>

## Flatness as a well-behaved family

The geometric meaning of flatness is that \( f : X \to S \) describes a family of fibers \( X_s \) that varies continuously, without sudden jumps.<sup>[6](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)</sup> The quantitative statement is constancy of numerical invariants. For a projective flat family over a connected Noetherian base, the Hilbert polynomial of the fibers is constant, and with it the degree in projective space, the arithmetic genus, the degree of a line bundle on curve fibers, and intersections of divisors and line bundles.<sup>[6](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)</sup><sup> • </sup><sup>[13](https://people.math.harvard.edu/~bejleri/teaching/math259xfa19/math259x_lecture4.pdf)</sup> Conversely, for a projective family over a Noetherian integral base, flatness is equivalent to the Hilbert polynomial of the fibers being constant.<sup>[14](https://math-jh.com/en/math/scheme_theory/flat_morphisms)</sup>

The hypothesis is deliberately weaker than smoothness. Branched covers are not fibrations in any traditional sense, yet they behave well: the double cover \( \mathbf{A}^1 \to \mathbf{A}^1 \) given by \( y \mapsto x^2 \) has constant degree 2, and a family of smooth curves degenerating to a nodal curve, such as \( y^2 = x^3 + x^2 + tx \), changes the topology of the underlying analytic curve while the arithmetic genus remains constant.<sup>[6](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)</sup> This behavior underlies degeneration techniques, in which a statement is proved first for a special fiber and then transported through flat families.<sup>[6](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)</sup>

Flatness is also the hypothesis that makes moduli problems well-posed. Maps from a scheme \( B \) to the [Hilbert scheme](https://www.edgepedia.org/hilbert-scheme) of \( \mathbf{P}^n \) correspond to finitely presented closed subschemes of \( \mathbf{P}^n_B \) flat over \( B \), and the moduli space of smooth projective curves is characterized by maps to it corresponding to projective flat families whose geometric fibers are smooth curves; flatness is likewise central to deformation theory.<sup>[6](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)</sup>

## Criteria for flatness

Several standard criteria reduce flatness to checkable conditions.

**Fiber criteria.** EGA's fiber criterion for flatness, stated in \( \mathrm{IV}_3 \), 11.3.10 of *Éléments de géométrie algébrique*, requires certain finiteness assumptions on the objects involved.<sup>[15](https://arxiv.org/abs/2401.02006)</sup> Later work has extended fiber criteria to settings where the finiteness conditions are dropped: for a noetherian ring \( R \), a flat ring map \( f : R \to A \), and an \( A \)-module \( M \) flat over \( R \), \( M \) is flat over \( A \) if the fiber modules \( M \otimes_R k(\mathfrak{p}) \) are flat over \( A \otimes_R k(\mathfrak{p}) \) for all \( \mathfrak{p} \in \mathrm{Spec}\,R \).<sup>[15](https://arxiv.org/abs/2401.02006)</sup>

**Local criterion.** For local noetherian rings \( B \to C \) both flat over a local noetherian ring \( (A, \mathfrak{m}) \), the map \( B \to C \) is flat if and only if the induced map on fibers \( B/\mathfrak{m}B \to C/\mathfrak{m}C \) is flat.<sup>[16](https://mathoverflow.net/questions/99266/on-the-geometric-interpretation-of-the-local-criterion-for-flatness)</sup>

**Finite morphisms.** A finite morphism \( f : X \to Y \) is flat if and only if it is locally free.<sup>[17](https://people.maths.bris.ac.uk/~matyd/AG15/ag.pdf)</sup>

**Generic flatness.** Flatness often fails only on a small locus. If \( S \) is integral, \( f : X \to S \) is of finite type, and \( \mathcal{F} \) is a finite type \( \mathcal{O}_X \)-module, there is an open dense subscheme \( U \subset S \) over which \( X_U \to U \) is flat and of finite presentation and \( \mathcal{F}|_{X_U} \) is flat over \( U \); the same holds for an arbitrary reduced base \( S \).<sup>[7](https://stacks.math.columbia.edu/tag/0529)</sup> The Stacks Project notes these results as a first step toward the flattening techniques developed by Raynaud and Gruson.<sup>[7](https://stacks.math.columbia.edu/tag/0529)</sup>

**Flat base change.** Flatness of the base change map makes cohomology commute with pullback: if \( f \) is quasi-compact and quasi-separated and \( u \) is flat, the base change morphism \( u^* R^i f_*(\mathcal{F}) \to R^i f'_*(u'^* \mathcal{F}) \) is an isomorphism for all \( i \ge 0 \).<sup>[13](https://people.math.harvard.edu/~bejleri/teaching/math259xfa19/math259x_lecture4.pdf)</sup>

## Relation to smooth and étale morphisms

Flatness sits below smoothness in the standard hierarchy. Every smooth morphism is flat.<sup>[1](https://mathworld.wolfram.com/FlatMorphism.html)</sup> For a morphism \( f : X \to Y \) locally of finite type, smoothness at \( x \) is necessary and sufficient for \( f \) to be flat at \( x \) and the fiber \( f^{-1}(y) \) to be smooth over \( \kappa(y) \) at \( x \).<sup>[5](https://jcreinhold.github.io/sga/i/02-smooth-morphisms-generalities-differential-properties.html)</sup> An étale morphism is a smooth morphism of relative dimension 0, so étaleness combines flatness with a vanishing differential condition.<sup>[18](https://web.mnstate.edu/fulghesu/Lecture3-Topologies.pdf)</sup> Flatness alone does not guarantee geometric regularity: flat morphisms can fail to be smooth and can have fibers with varying singularities.<sup>[19](https://kindatechnical.com/math-algebraic-geometry/lesson-35-flat-morphisms-and-the-flat-topology.html)</sup>

## References

1. [Weisstein, Eric W. "Flat Morphism." MathWorld--A Wolfram Resource.](https://mathworld.wolfram.com/FlatMorphism.html)
2. [The Stacks Project Authors. "Flat morphisms," Section 41.9 (Tag 0250), The Stacks Project.](https://stacks.math.columbia.edu/tag/0250)
3. [The Stacks Project Authors. "Flat morphisms," Section 29.26 (Tag 01U2), The Stacks Project.](https://stacks.math.columbia.edu/tag/01U2)
4. [Kedlaya, Kiran. 18.726 Algebraic Geometry, lecture notes on flat morphisms, MIT, Spring 2009.](https://kskedlaya.org/18.726/flat.pdf)
5. [SGA 1, Exposé II, "Smooth Morphisms: Generalities, Differential Properties," English translation by J. C. Reinhold.](https://jcreinhold.github.io/sga/i/02-smooth-morphisms-generalities-differential-properties.html)
6. [Vakil, Ravi. *The Rising Sea: Foundations of Algebraic Geometry*, pre-publication version, 2024, Chapter 24.](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)
7. [The Stacks Project Authors. "Generic flatness," Section 29.28 (Tag 0529), The Stacks Project.](https://stacks.math.columbia.edu/tag/0529)
8. [MathOverflow. "How to introduce notions of flat, projective and free modules?" (account of Serre's recollection of the history).](https://mathoverflow.net/questions/46541/how-to-introduce-notions-of-flat-projective-and-free-modules)
9. [Alphabeta Math. "Flat, Smooth and Etale Morphisms — Examples."](https://alphabetamath.cc/scheme-theory/flat-smooth-and-etale-morphisms-examples)
10. [Meier, Lennart. TMF lecture notes, University of Rochester.](https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/meier-TMF-Lecture.pdf)
11. [Alphabeta Math. "Flat, Smooth and Etale Morphisms."](https://alphabetamath.cc/scheme-theory/flat-smooth-and-etale-morphisms)
12. [nLab. "Flat morphism."](https://ncatlab.org/nlab/show/flat%20morphism)
13. [Bejleri, Dori. Math 259x lecture notes, "Flat morphisms and Hilbert polynomials," Harvard, 2019.](https://people.math.harvard.edu/~bejleri/teaching/math259xfa19/math259x_lecture4.pdf)
14. ["Flat Morphisms," Blackbox scheme theory notes.](https://math-jh.com/en/math/scheme_theory/flat_morphisms)
15. [Hà, Phùng Hô, Nguyen, Hop D., and dos Santos, João Pedro. "Fiber criteria for flatness and homomorphisms of flat affine group schemes," arXiv:2401.02006.](https://arxiv.org/abs/2401.02006)
16. [MathOverflow. "On the Geometric Interpretation of the Local Criterion for Flatness."](https://mathoverflow.net/questions/99266/on-the-geometric-interpretation-of-the-local-criterion-for-flatness)
17. [Dolan, Tom. "Topics in Algebraic Geometry," University of Bristol, 2015.](https://people.maths.bris.ac.uk/~matyd/AG15/ag.pdf)
18. [Fulghesu, Damiano. "Introduction to Algebraic Stacks," Lecture 3: Grothendieck topologies.](https://web.mnstate.edu/fulghesu/Lecture3-Topologies.pdf)
19. [kindatechnical(). "A Guide to Algebraic Geometry: Flat Morphisms and the Flat Topology."](https://kindatechnical.com/math-algebraic-geometry/lesson-35-flat-morphisms-and-the-flat-topology.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Algebraic geometry › Schemes, stacks, and morphisms › Properties of morphisms*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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