Flat module

In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact.

Vector spaces over a field are flat modules. Free modules, or more generally projective modules, are also flat, over any R. For finitely generated modules over a Noetherian local ring, flatness, projectivity, and freeness are all equivalent.

Flatness was introduced by Serre (1956) in his paper Géometrie Algébrique et Géométrie Analytique. See also flat morphism.

Definition

Commutative rings

There are many ways to define flatness over a commutative ring $R$.

• A flat $R$-module is an $R$-module $M$ such that the functor

$F_M : Mod(R) \to Mod(R), \quad N \mapsto M \otimes_R N$

is exact, where $Mod(R)$ is the category of $R$-modules.

• A flat $R$-module is an $R$-module $M$ such that for every injective morphism $\phi : K \to L$ of $R$-modules $K$ and $L$, the induced map,

$F_M(\phi) : M \otimes_R K \to M \otimes_R L$,

is injective.

• A flat $R$-module is an $R$-module $M$ such that for every finitely generated ideal $I \hookrightarrow R$, the induced morphism $I \otimes_R M \to R \otimes_R M \cong M$ is injective.
• A flat $R$-module is an $R$-module $M$ such that there exists a directed system of $R$-modules $\{ F_\alpha \}_\alpha$ with the following properties:

1. For all $\alpha$, $F_\alpha$ is a finitely generated, free $R$-module.
2. The direct limit is $M$: $\varinjlim_\alpha F_\alpha= M$.

• A flat $R$-module is an $R$-module $M$ such that for every linear dependency in $M$,

$r^T m = \sum_{i=1}^k r_i m_i = 0$,

where $r_i \in R, m_i \in M$, there exists a matrix $A \in R^{k \times k}$ such that

1. $An = m$ has a solution for some $n \in M^k$.
2. $r^T A = 0$.

• A flat $R$-module is an $R$-module $M$ such that for every $R$-module $N$,

$\mathrm{Tor}_1^R (N, M) = 0$

• A flat $R$-module is an $R$-module $M$ such that for every finitely generated ideal $I \subset R$,

$\mathrm{Tor}_1^R (R/I, M) = 0$.

• A flat $R$-module is an $R$-module $M$ such that for every map $f : F \to M$, where $F$ is a finitely generated free $R$-module, and for every finitely generated $R$-submodule $K \leq \ker f$, $f$ factors through a map to a free $R$-module $G$ that kills $K$:

General rings

When R isn't commutative one needs the more careful statement, that (if M is a left R-module) the tensor product with M maps exact sequences of right R-modules to exact sequences of abelian groups.

Taking tensor products (over arbitrary rings) is always a right exact functor. Therefore, the R-module M is flat if and only if for any injective homomorphism K → L of R-modules, the induced homomorphism K$\otimes$M → L$\otimes$M is also injective.

Case of commutative rings

For any multiplicatively closed subset S of R, the localization ring $S^{-1}R$ is flat as an R-module.

When R is Noetherian and M is a finitely-generated R-module, being flat is the same as being locally free in the following sense: M is a flat R-module if and only if for every prime ideal (or even just for every maximal ideal) P of R, the localization $M_P$ is free as a module over the localization $R_P$.

If S is an R-algebra, i.e., we have a homomorphism $f \colon R \to S$, then S has the structure of an R-module, and hence it makes sense to ask if S is flat over R. If this is the case, then S is faithfully flat over R if and only if every prime ideal of R is the inverse image under f of a prime ideal in S. In other words, if and only if the induced map $f^* \colon \mathrm{Spec}(S) \to \mathrm{Spec}(R)$ is surjective.

Categorical colimits

In general, arbitrary direct sums and direct limits of flat modules are flat, a consequence of the fact that the tensor product commutes with direct sums and direct limits (in fact with all colimits), and that both direct sums and direct limits are exact functors. Submodules and factor modules of flat modules need not be flat in general. However we have the following result: the homomorphic image of a flat module M is flat if and only if the kernel is a pure submodule of M.

Daniel Lazard proved in 1969 that a module M is flat if and only if it is a direct limit of finitely-generated free modules. As a consequence, one can deduce that every finitely-presented flat module is projective.

An abelian group is flat (viewed as a Z-module) if and only if it is torsion-free.

Homological algebra

Flatness may also be expressed using the Tor functors, the left derived functors of the tensor product. A left R-module M is flat if and only if Tor_n^R(–, M) = 0 for all $n \ge 1$ (i.e., if and only if Tor_n^R(X, M) = 0 for all $n \ge 1$ and all right R-modules X). Similarly, a right R-module M is flat if and only if Tor_n^R(M, X) = 0 for all $n \ge 1$ and all left R-modules X. Using the Tor functor's long exact sequences, one can then easily prove facts about a short exact sequence

• If A and C are flat, then so is B
• If B and C are flat, then so is A

If A and B are flat, C need not be flat in general. However, it can be shown that

• If A is pure in B and B is flat, then A and C are flat.

Flat resolutions

A flat resolution of a module M is a resolution of the form

... → F₂ → F₁ → F₀ → M → 0

where the F_i are all flat modules. Any free or projective resolution is necessarily a flat resolution. Flat resolutions can be used to compute the Tor functor.

The length of a finite flat resolution is the first subscript n such that F_n is nonzero and F_i=0 for i greater than n. If a module M admits a finite flat resolution, the minimal length among all finite flat resolutions of M is called its flat dimension and denoted fd(M). If M does not admit a finite flat resolution, then by convention the flat dimension is said to be infinite. As an example, consider a module M such that fd(M) = 0. In this situation, the exactness of the sequence 0 → F₀ → M → 0 indicates that the arrow in the center is an isomorphism, and hence M itself is flat .

In some areas of module theory, a flat resolution must satisfy the additional requirement that each map is a flat pre-cover of the kernel of the map to the right. For projective resolutions, this condition is almost invisible: a projective pre-cover is simply an epimorphism from a projective module. These ideas are inspired from Auslander's work in approximations. These ideas are also familiar from the more common notion of minimal projective resolutions, where each map is required to be a projective cover of the kernel of the map to the right. However, projective covers need not exist in general, so minimal projective resolutions are only of limited use over rings like the integers.

While projective covers for modules do not always exist, it was speculated that for general rings, every module would have a flat cover, that is, every module would be the epimorphic image of a flat module under a homomorphism with superfluous kernel. This flat cover conjecture was explicitly first stated in (Enochs 1981, p 196). The conjecture turned out to be true, resolved positively and proved simultaneously by L. Bican, R. El Bashir and E. Enochs. The was preceded by important contributions by P. Eklof, J. Trlifaj and J. Xu.

Since flat covers exist for all modules over all rings, minimal flat resolutions can take the place of minimal projective resolutions in many circumstances. The measurement of the departure of flat resolutions from projective resolutions is called relative homological algebra, and is covered in classics such as (MacLane 1963) and in more recent works focussing on flat resolutions such as (Enochs & Jenda 2000).

In constructive mathematics

Flat modules have increased importance in constructive mathematics, where projective modules are less useful. For example, that all free modules are projective is equivalent to the full axiom of choice, so theorems about projective modules, even if proved constructively, do not necessarily apply to free modules. In contrast, no choice is needed to prove that free modules are flat, so theorems about flat modules can still apply.

References

• Bican, L.; El Bashir, R.; Enochs, E. (2001), "All modules have flat covers", Bull. London Math. Soc. 33 (4): 385–390, doi:10.1017/S0024609301008104, ISSN 0024-6093, MR1832549 
• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1; 978-0-387-94269-8, MR1322960 
• Enochs, Edgar E. (1981), "Injective and flat covers, envelopes and resolvents", Israel J. Math. 39 (3): 189–209, doi:10.1007/BF0276084, ISSN 0021-2172, MR(83a:16031) 636889 (83a:16031) 
• Enochs, Edgar E.; Jenda, Overtoun M. G. (2000), Relative homological algebra, de Gruyter Expositions in Mathematics, 30, Berlin: Walter de Gruyter & Co., ISBN 978-3-11-016633-0, MR1753146 
• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR1653294 
• Mac Lane, Saunders (1963), Homology, Die Grundlehren der mathematischen Wissenschaften, Bd. 114, Boston, MA: Academic Press, MR0156879 
• Northcott, D. G. (1984), Multilinear algebra, Cambridge University Press, ISBN 978-0-521-26269-9  - page 33
• Richman, Fred (1997), "Flat dimension, constructivity, and the Hilbert syzygy theorem", New Zealand Journal of Mathematics 26 (2): 263–273, ISSN 1171-6096, MR1601663 
• Serre, Jean-Pierre (1956), "Géométrie algébrique et géométrie analytique", Université de Grenoble. Annales de l'Institut Fourier 6: 1–42, ISSN 0373-0956, MR0082175, http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0 

See also

• localization of a module
• flat morphism
• von Neumann regular ring: those rings over which all modules are flat.