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 ring, flatness and projectivity are equivalent. For finitely generated modules over local rings, flatness, projectivity and freeness are all equivalent.^[1] The field of quotients of an integral domain, and, more generally, any localization of a commutative ring are flat modules. The product of the local rings of a commutative ring is a faithfully flat module.

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

Definition

Commutative rings

Let M be an R-module. The following conditions are all equivalent, so M is flat if it satisfies any (thus all) of them:

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.

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.

For every finitely generated ideal $I \hookrightarrow R$ , the induced morphism $I \otimes_R M \to R \otimes_R M \cong M$ is injective.
There exists a directed system of $R$ -modules $\{ F_\alpha \}_\alpha$ with the following properties:

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

^[2] For every linear dependency in $M$ ,

r^{T}x=\sum _{{i=1}}^{k}r_{i}x_{i}=0

where

r_{i}\in R,x_{i}\in M

, there exists a matrix

A\in R^{{k\times j}}

such that

$Ay=x$ has a solution for some $y\in M^{j}$ .
$r^T A = 0$ .

For every $R$ -module $N$ ,

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

For every finitely generated ideal $I \subset R$ ,

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

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 flat 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.

Examples

For any multiplicatively closed subset $S$ of a commutative ring $R$ , the localization ring $S^{-1}R$ is flat as an $R$ -module. For example, $\mathbb {Q}$ is flat over $\mathbb {Z}$ (though not projective).
Every free $\mathbb {Z}$ -module (in fact any free module over a ring) is a flat module.
For each integer $n$ > 1, $\mathbb {Z} /n\mathbb {Z}$ is not flat over $\mathbb {Z}$ , because $n: \mathbb{Z} \to \mathbb{Z}, \, x \mapsto nx$ is injective, but tensored with $\mathbb {Z} /n\mathbb {Z}$ it is not.
Similarly, ${\mathbb {Q}}/{\mathbb {Z}}$ is not flat over $\mathbb {Z}$ .
Let $R = k[t], k$ a field, and $S = R[x]/(tx - 1)$ . Since $S$ is the same thing as the localization $R[t^{-1}]$ , it is flat over $R$ . On the other hand, $R[x]/(tx-t)$ is not flat over $R$ since $t$ is a torsion element on it (so it is not torsion-free).
Let $A$ be a noetherian ring and $I$ an ideal. Then the completion $A \to \widehat{A}$ with respect to $I$ is flat.^[3] It is faithfully flat if and only if $I$ is contained in the Jacobson radical of $A$ .^[4] (cf. Zariski ring.)
The direct sum $\bigoplus _{{i\in I}}M_{i}$ is flat if and only if each $M_{i}$ is flat.
Every product of flat $A$ -modules is flat if and only if $A$ is a coherent ring.^[5]
(Kunz) A noetherian ring containing a field of characteristic $p$ is regular if and only if the Frobenius morphism $R$ → $R$ is flat and $R$ is reduced.

Case of commutative rings

When 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 $R_{P}$ .

Let R be a local ring with nilpotent maximal ideal (e.g., an artinian local ring) and M a module over it. Then M flat implies M free.^[1]

The local criterion for flatness states:^[6]

Let R be a local noetherian ring, S a local noetherian R-algebra with

\mathfrak{m}_R S \subset \mathfrak{m}_S

, and M a finitely generated S-module. Then M is flat over R if and only if

\operatorname{Tor}_1^R(M, R/\mathfrak{m}_R) = 0.

The significance of this is that S need not be finite over R and we only need to consider the maximal ideal of R instead of an arbitrary ideal of R.

The next criterion is also useful for testing flatness:^[7]

Let R, S be as in the local criterion for flatness. Assume S is Cohen–Macaulay and R is regular. Then S is flat over R if and only if

\operatorname{dim}S = \operatorname{dim}R + \operatorname{dim}S/\mathfrak{m}_R S

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. In this setting we have:

If S is flat over R, 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.

Flat modules over commutative rings are always torsion-free (that is, the multiplication by any regular element of the ring in injective in the module). Projective modules (and thus free modules) are always flat. For certain common classes of rings, these statements can be reversed (for example, every torsion-free module over a Dedekind ring is automatically flat and flat modules over perfect rings are always projective), as is subsumed in the following diagram of module properties:

An integral domain is called a Prüfer domain if every torsion-free module over it is flat.

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 (e.g. Z/nZ is not a flat Z-module for n>1). 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.^[8] 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 $n\geq 1$ (i.e., if and only if Tor_n^R(X, M) = 0 for all $n\geq 1$ 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 $n\geq 1$ 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^[9] 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.^[10]

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.

Flat covers

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 M would be the epimorphic image of a flat module F such that every map from a flat module onto M factors through F, and any endomorphism of F over M is an automoprhism. 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.^[11] This 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.^[12]

References

1 2 Matsumura 1970, Proposition 3.G
↑ Bourbaki, Ch. I, § 2. Proposition 13, Corollay 1.
↑ Matsumura 1970, Corollary 1 of Theorem 55, p. 170
↑ Matsumura 1970, Theorem 56
↑ http://mathoverflow.net/questions/120403/flatness-of-power-series-rings/
↑ Eisenbud 1994, Theorem 6.8
↑ Eisenbud 1994, Theorem 18.16
↑ Lazard, D. (1969), "Autour de la platitude", Bulletin de la Société Mathématique de France, 97: 81–128
↑ Lam 1999, p. 183.
↑ A module isomorphic to an flat module is of course flat.
↑ Bican, El Bashir & Enochs 2001.
↑ Richman 1997.

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