# Flat module

A **flat module** is a module $M$ over a ring $R$ with the property that tensoring with $M$ preserves exactness: whenever $N_1 \to N_2 \to N_3$ is an exact sequence of $R$-modules, the tensored sequence $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3$ is exact as well.<sup>[1](https://stacks.math.columbia.edu/tag/00H9)</sup> The name and the concept were introduced by [Jean-Pierre Serre](https://www.edgepedia.org/jean-pierre-serre) in his 1956 paper GAGA, and it was [Alexander Grothendieck](https://www.edgepedia.org/alexander-grothendieck) who recognized the notion's geometric significance.<sup>[2](https://jcreinhold.github.io/sga/i/04-flat-morphisms.html)</sup><sup> • </sup><sup>[3](https://www.jstor.org/stable/jj.30669912.30)</sup>  Flatness is now one of the central technical notions of algebraic geometry, where it governs families whose fibers vary continuously, and of commutative algebra, where it controls base change.<sup>[2](https://jcreinhold.github.io/sga/i/04-flat-morphisms.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | $M$ is flat over $R$ if $N_1 \to N_2 \to N_3$ exact implies $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3$ exact<sup>[1](https://stacks.math.columbia.edu/tag/00H9)</sup> |
| Ideal criterion | $M$ is flat iff $I \otimes_R M \to M$ is injective for every finitely generated ideal $I \subset R$<sup>[1](https://stacks.math.columbia.edu/tag/00H9)</sup> |
| Homological criterion | $M$ is flat iff $\mathrm{Tor}^R_1(M, N) = 0$ for all $R$-modules $N$<sup>[2](https://jcreinhold.github.io/sga/i/04-flat-morphisms.html)</sup> |
| Structure theorem | Lazard's theorem: $M$ is flat iff it is the colimit of a directed system of finite free $R$-modules<sup>[4](https://stacks.math.columbia.edu/tag/058G)</sup> |
| Finite case | Finitely presented and flat is equivalent to finite projective, and to finite locally free<sup>[5](https://stacks.math.columbia.edu/tag/00NX)</sup> |
| Faithful flatness | $M$ is faithfully flat if a complex $N_1 \to N_2 \to N_3$ is exact exactly when its tensor with $M$ is exact<sup>[1](https://stacks.math.columbia.edu/tag/00H9)</sup> |
| Geometric meaning | A morphism of schemes is flat when its fibers vary continuously; flatness underlies descent and deformation theory<sup>[2](https://jcreinhold.github.io/sga/i/04-flat-morphisms.html)</sup><sup> • </sup><sup>[6](https://www.docsity.com/en/docs/deformation-theory-and-moduli-in-algebraic-geometry-lecture-notes-ams-130/6293150/)</sup>  |

## Definition and first consequences

The tensor product functor is always right exact: for any $R$-module $M$ and any exact sequence $0 \to N_1 \to N_2 \to N_3 \to 0$, the sequence $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3 \to 0$ remains exact. Flatness asks that the functor also be left exact, that is, fully exact.<sup>[1](https://stacks.math.columbia.edu/tag/00H9)</sup> Equivalently, tensoring with $M$ preserves injectivity of linear maps, a formulation used in the Lean formalization Mathlib as `Module.Flat.preserves_injective_linearMap`.<sup>[7](https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/Basic.html)</sup>

The exactness condition can be checked on much smaller data. Stacks Project Lemma 10.39.5 gives a chain of equivalent conditions: $M$ is flat if and only if for every injection $N \subset N'$ the map $N \otimes_R M \to N' \otimes_R M$ is injective, if and only if $I \otimes_R M \to M$ is injective for every ideal $I$, and already if and only if this holds for every *finitely generated* ideal $I$.<sup>[1](https://stacks.math.columbia.edu/tag/00H9)</sup> A concrete payoff of the definition is the identity $IM \cap JM = (I \cap J)M$ for ideals $I, J \subset R$ and flat $M$, which fails for general modules.<sup>[1](https://stacks.math.columbia.edu/tag/00H9)</sup>

**Closure properties.** Free modules are flat, and projective modules are flat, since projectives are direct summands of free modules.<sup>[7](https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/Basic.html)</sup> The class of flat modules is closed under directed colimits: a directed system of flat $R$-modules has a flat colimit, because tensor products commute with colimits and directed colimits are exact.<sup>[1](https://stacks.math.columbia.edu/tag/00H9)</sup> Flatness is also stable under base change: if $R \to R'$ is any ring map and $M$ is flat over $R$, then $R' \otimes_R M$ is flat over $R'$; if $R \to R'$ is faithfully flat, the converse holds, so flatness of $M$ over $R$ can be tested after a faithfully flat extension.<sup>[1](https://stacks.math.columbia.edu/tag/00H9)</sup>

## Faithful flatness

Strengthening flatness, an $R$-module $M$ is **faithfully flat** when a complex $N_1 \to N_2 \to N_3$ is exact if and only if its tensor with $M$ is exact; flatness only gives one implication. A ring map $R \to S$ is flat, respectively faithfully flat, when $S$ has the corresponding property as an $R$-module.<sup>[1](https://stacks.math.columbia.edu/tag/00H9)</sup> Geometrically, a morphism of affine schemes $\mathrm{Spec}(B) \to \mathrm{Spec}(A)$ is faithfully flat exactly when $B$ is flat over $A$ and the map $M \to M \otimes_A B$ is injective for every $A$-module $M$; equivalently, the morphism is flat and surjective.<sup>[8](https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/abf8e0d914ab28e7d82b156fa18c0dce_MIT18_726s09_lec12_flat.pdf)</sup>

Faithfully flat modules detect exactness, so they serve as faithful test objects. One characterization states that $M$ is faithfully flat precisely when tensoring with $M$ preserves and reflects exactness, equivalently when $M$ is flat and $f \otimes \mathrm{id}_M = 0$ forces $f = 0$ for every $R$-module homomorphism $f$.<sup>[9](https://math.stackexchange.com/questions/4777460/let-f-n-rightarrow-p-m-faithfully-flat-iff-alpha-otimes-id-m-0)</sup><sup> • </sup><sup>[1](https://stacks.math.columbia.edu/tag/00H9)</sup> Faithfully flat morphisms are the vehicle for **descent**, the process of undoing a base change, establishing whether a property of an object after extension already held before it.<sup>[8](https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/abf8e0d914ab28e7d82b156fa18c0dce_MIT18_726s09_lec12_flat.pdf)</sup>

## Homological characterization

Flatness has a clean homological meaning. In the sorites opening SGA 1's Exposé IV on flat morphisms, a module $M$ is flat exactly when the first right-derived functor of tensoring vanishes, that is, $\mathrm{Tor}^A_1(M, N) = 0$ for all $N$; equivalently all higher $\mathrm{Tor}$ groups vanish.<sup>[2](https://jcreinhold.github.io/sga/i/04-flat-morphisms.html)</sup> The $\mathrm{Tor}$ groups are computed by tensoring a free resolution of one argument with the other, and free modules, being flat, contribute no higher homology.<sup>[10](https://stacks.math.columbia.edu/tag/00LY)</sup>

For practical checking, the Tor condition can be localized further: $M$ is flat over $A$ if and only if $\mathrm{Tor}^A_1(A/I, M) = 0$ for all finitely generated ideals $I \subseteq A$.<sup>[11](https://sites.math.washington.edu/~jarod/moduli.pdf)</sup> Over noetherian commutative rings it even suffices to test prime ideals, since flatness is equivalent to $\mathrm{Tor}_1(M, R/P) = 0$ for every prime $P$; without the noetherian hypothesis the same proof gives the criterion for all ideals.<sup>[12](https://math.stackexchange.com/questions/4113338/flatness-criteria-on-noetherian-ring)</sup> Local criteria for flatness, such as the fiberwise criterion for local homomorphisms of noetherian local rings, relate flatness of a module to flatness of its reduction modulo the maximal ideal together with vanishing of a Tor term.<sup>[13](https://stacks.math.columbia.edu/tag/00MD)</sup>

## Structure theorems

**Lazard's theorem** identifies flat modules up to construction: an $R$-module $M$ is flat if and only if it is the colimit of a directed system of finite free $R$-modules.<sup>[4](https://stacks.math.columbia.edu/tag/058G)</sup> The result was proved independently by V. E. Govorov and Daniel Lazard in the 1960s, and is often called the Govorov–Lazard theorem.<sup>[14](https://arxiv.org/abs/1301.0731)</sup> Govorov's note appeared in *Doklady Akademii Nauk SSSR* in 1962, on rings over which flat modules are free;<sup>[15](https://www.mathnet.ru/php/archive.phtml?jrnid=dan&option_lang=rus&paperid=26602&wshow=paper)</sup> Lazard's memoir *Autour de la platitude* appeared in the *Bulletin de la Société Mathématique de France* in 1969, covering the structure of flat modules, pure submodules, and projective resolutions of flat modules.<sup>[16](https://numdam.org/articles/10.24033/bsmf.1675/)</sup> The theorem gives a flexible tool: to prove a statement about all flat modules it often suffices to handle finite free modules and pass to filtered colimits.<sup>[17](https://mathoverflow.net/questions/127769/applications-of-govorov-lazard-theorem)</sup>

**The finitely presented case is much more rigid.** For a finitely presented module, flatness, projectivity, and local freeness coincide: Stacks Project Lemma 10.78.2 lists eight equivalent conditions, including being a direct summand of a finite free module and being finite with locally constant fiber rank $\rho_M(p) = \dim_{\kappa(p)} M \otimes_R \kappa(p)$ on $\mathrm{Spec}(R)$.<sup>[5](https://stacks.math.columbia.edu/tag/00NX)</sup> Without finite presentation the implication fails: a finite $R$-module that is $R$-flat need not be projective.<sup>[18](https://stacks.math.columbia.edu/tag/00NV)</sup> Over restricted classes of rings the gap closes anyway; over right noetherian rings, local rings, and domains satisfying the strong rank condition, finitely generated flat modules are projective.<sup>[19](https://www.maths.ed.ac.uk/~v1ranick/papers/lam2.pdf)</sup> More broadly, if a ring has either finitely many minimal primes or finitely many maximal ideals, every finitely generated flat module over it is projective.<sup>[20](https://arxiv.org/abs/1804.03007)</sup>

## Flatness in geometry

The module notion extends verbatim to sheaves. For a ringed space $(X, \mathcal{O}_X)$, an $\mathcal{O}_X$-module $\mathcal{F}$ is flat when the functor $\mathcal{G} \mapsto \mathcal{G} \otimes_{\mathcal{O}_X} \mathcal{F}$ on the category of $\mathcal{O}_X$-modules is exact, and flatness can be checked on stalks: $\mathcal{F}$ is flat if and only if each stalk $\mathcal{F}_x$ is a flat $\mathcal{O}_{X,x}$-module.<sup>[21](https://stacks.math.columbia.edu/tag/05NC)</sup>

Serre introduced flatness and faithful flatness in GAGA for reasons of algebra, and Grothendieck recognized the geometric significance; as Ravi Vakil, professor of mathematics at Stanford University, notes in *The Rising Sea*, the algebra and geometry are not obviously connected, and the more descriptive name "exact module" has not caught on.<sup>[3](https://www.jstor.org/stable/jj.30669912.30)</sup> In SGA 1's Exposé IV, Grothendieck identifies two principal ways flatness enters algebraic geometry: the systematic study, in EGA IV, of a coherent sheaf $F$ flat over a base $Y$, relating properties of the base, the sheaf, and the fibers, and descent theory, which he calls the more important route; flatness is thereby one of the central technical notions of the subject.<sup>[2](https://jcreinhold.github.io/sga/i/04-flat-morphisms.html)</sup>

In deformation theory, where families over non-reduced bases are essential, flatness is built into the basic deformation problems from the start, encoding the requirement that a family genuinely spreads over its base rather than living over a smaller subscheme.<sup>[6](https://www.docsity.com/en/docs/deformation-theory-and-moduli-in-algebraic-geometry-lecture-notes-ams-130/6293150/)</sup> Generic flatness provides a partial converse to the rigidity worry: if $Y$ is integral, $X$ of finite type over $Y$, and $F$ coherent on $X$, then there is a nonempty open $U$ of $Y$ over which $F$ is $Y$-flat.<sup>[2](https://jcreinhold.github.io/sga/i/04-flat-morphisms.html)</sup>

## References

1. [The Stacks Project, Section 10.39: Flat modules and flat ring maps (Tag 00H9)](https://stacks.math.columbia.edu/tag/00H9)
2. [SGA 1, Exposé IV: Flat Morphisms, English translation by J. C. Reinhold](https://jcreinhold.github.io/sga/i/04-flat-morphisms.html)
3. [Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry, JSTOR](https://www.jstor.org/stable/jj.30669912.30)
4. [The Stacks Project, Theorem 10.81.4 (Lazard's theorem, Tag 058G)](https://stacks.math.columbia.edu/tag/058G)
5. [The Stacks Project, Lemma 10.78.2 (Tag 00NX)](https://stacks.math.columbia.edu/tag/00NX)
6. [Deformation Theory and Moduli in Algebraic Geometry, lecture notes, AMS 130](https://www.docsity.com/en/docs/deformation-theory-and-moduli-in-algebraic-geometry-lecture-notes-ams-130/6293150/)
7. [Mathlib.RingTheory.Flat.Basic, Lean mathlib documentation](https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/Basic.html)
8. [MIT OpenCourseWare 18.726 Algebraic Geometry, Lecture 22: Faithfully flat morphisms and descent](https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/abf8e0d914ab28e7d82b156fa18c0dce_MIT18_726s09_lec12_flat.pdf)
9. [Mathematics Stack Exchange: Faithfully flat characterization via f ⊗ id_M = 0](https://math.stackexchange.com/questions/4777460/let-f-n-rightarrow-p-m-faithfully-flat-iff-alpha-otimes-id-m-0)
10. [The Stacks Project, Section 10.75: Tor groups and flatness (Tag 00LY)](https://stacks.math.columbia.edu/tag/00LY)
11. [Flatness criteria, University of Washington notes (A.2.1)](https://sites.math.washington.edu/~jarod/moduli.pdf)
12. [Mathematics Stack Exchange: Flatness criteria on noetherian rings](https://math.stackexchange.com/questions/4113338/flatness-criteria-on-noetherian-ring)
13. [The Stacks Project, Section 10.99: Criteria for flatness (Tag 00MD)](https://stacks.math.columbia.edu/tag/00MD)
14. [The direct limit closure of perfect complexes, arXiv:1301.0731](https://arxiv.org/abs/1301.0731)
15. [V. E. Govorov, Rings over which flat modules are free, Dokl. AN SSSR 144:5 (1962), 965–967](https://www.mathnet.ru/php/archive.phtml?jrnid=dan&option_lang=rus&paperid=26602&wshow=paper)
16. [Daniel Lazard, Autour de la platitude, Bulletin de la S.M.F. 97 (1969), 81–128](https://numdam.org/articles/10.24033/bsmf.1675/)
17. [MathOverflow: Applications of Govorov–Lazard Theorem](https://mathoverflow.net/questions/127769/applications-of-govorov-lazard-theorem)
18. [The Stacks Project, Section 10.78: Finite projective modules (Tag 00NV)](https://stacks.math.columbia.edu/tag/00NV)
19. [T. Y. Lam, Lectures on Modules and Rings, Chapter 2: Flat Modules and Homological Dimensions](https://www.maths.ed.ac.uk/~v1ranick/papers/lam2.pdf)
20. [Notes on finitely generated flat modules, arXiv:1804.03007](https://arxiv.org/abs/1804.03007)
21. [The Stacks Project, Section 17.17: Flat modules (Tag 05NC)](https://stacks.math.columbia.edu/tag/05NC)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Module theory › Tensor products and bimodules › Flatness, torsion, and tensor exactness*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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