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.

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

Definition

A module M over a ring R is called flat if the following condition is satisfied: for any injective map ${\displaystyle \phi :K\to L}$ of R-modules, the map

${\displaystyle K\otimes _{R}M\to L\otimes _{R}M}$

induced by ${\displaystyle k\otimes m\mapsto \phi (k)\otimes m}$ is injective.

In other words, for R-modules K, L, and J, if ${\displaystyle 0\rightarrow K\rightarrow L\rightarrow J\rightarrow 0}$ is a short exact sequence, then M is a flat module over R if ${\displaystyle 0\rightarrow K\otimes _{R}M\rightarrow L\otimes _{R}M\rightarrow J\otimes _{R}M\rightarrow 0}$ is also a short exact sequence.

This definition applies also if R is not necessarily commutative, and M is a left R-module and K and L right R-modules. The only difference is that in this case ${\displaystyle K\otimes _{R}M}$ and ${\displaystyle L\otimes _{R}M}$ are not in general R-modules, but only abelian groups.

Characterizations of flatness

Since tensoring with M is, for any module M, a right exact functor

${\displaystyle {\textbf {Mod}}_{R}\to {\textbf {Ab}}}$

(between the category of R-modules and abelian groups), M is flat if and only if the preceding functor is exact.

It can also be shown in the condition defining flatness as above, it is enough to take ${\displaystyle L=R}$, the ring itself, and ${\displaystyle K}$ a finitely generated ideal of R.

Flatness is also equivalent to the following equational condition, which may be paraphrased by saying that R-linear relations that hold in M stem from linear relations which hold in R: for every linear dependency, ${\displaystyle r^{T}x=\sum _{i=1}^{k}r_{i}x_{i}=0}$ with ${\displaystyle r_{i}\in R}$ and ${\displaystyle x_{i}\in M}$, there exist a matrix ${\displaystyle A\in R^{k\times j}}$ and an element ${\displaystyle y\in M^{j}}$ such that ${\displaystyle Ay=x}$ and ${\displaystyle r^{T}A=0.}$[1] Furthermore, M is flat if and only if the following condition holds: for every map ${\displaystyle f:F\to M,}$ where ${\displaystyle F}$ is a finitely generated free ${\displaystyle R}$-module, and for every finitely generated ${\displaystyle R}$-submodule ${\displaystyle K}$ of ${\displaystyle \ker f,}$ the map ${\displaystyle f}$ factors through a map g to a free ${\displaystyle R}$-module ${\displaystyle G}$ such that ${\displaystyle g(K)=0:}$

Factor property of a flat module

Examples and relations to other notions

Flatness is related to various other conditions on a module, such as being free, projective, or torsion-free. This is partly summarized in the following graphic:

Free or projective modules vs. flat modules

Free modules are flat over any ring R. This holds since the functor

${\displaystyle L\mapsto L\otimes _{R}(\bigoplus _{i\in I}R)\cong \bigoplus _{i\in I}L}$

is exact. For example, vector spaces over a field are flat modules. Direct summands of flat modules are again flat. In particular, projective modules (direct summands of free modules) are flat. Conversely, for a commutative Noetherian ring R, finitely generated flat modules are projective.

Flat vs. torsion-free modules

Any flat module is torsion-free. The converse holds over the integers, and more generally over principal ideal domains. This follows from the above characterization of flatness in terms of ideals. Yet more generally, this converse holds over Dedekind rings.

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

Flatness of completions

Let ${\displaystyle A}$ be a noetherian ring and ${\displaystyle I}$ an ideal. Then the completion ${\displaystyle A\to {\widehat {A}}}$ with respect to ${\displaystyle I}$ is flat.[2] It is faithfully flat if and only if ${\displaystyle I}$ is contained in the Jacobson radical of ${\displaystyle A}$.[3] (cf. Zariski ring.)

Quotients of flat modules

Quotients of flat modules are not in general flat. For example, for each integer ${\displaystyle n>1,\mathbb {Z} /n\mathbb {Z} }$ is not flat over ${\displaystyle \mathbb {Z} ,}$ because ${\displaystyle n:\mathbb {Z} \to \mathbb {Z} ,x\mapsto nx}$ is injective, but tensored with ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ it is not. Similarly, ${\displaystyle \mathbb {Q} /\mathbb {Z} }$ is not flat over ${\displaystyle \mathbb {Z} .}$

Finite field

Another interesting case comes from looking at the finite field ${\displaystyle \mathbb {F} _{p}}$ as a ${\displaystyle \mathbb {Z} }$-module. In this case, there is a flat resolution

${\displaystyle \mathbb {Z} \xrightarrow {\cdot p} \mathbb {Z} }$

whose cokernel is isomorphic to ${\displaystyle \mathbb {F} _{p}}$ as a ${\displaystyle \mathbb {Z} }$-module. This is an important result because it is used to compute the Hochschild homology of ${\displaystyle \mathbb {F} _{p}}$ over ${\displaystyle \mathbb {Z} }$.

Further permanence properties

In general, arbitrary direct sums and filtered colimits (also known as direct limits) of flat modules are flat, a consequence of the fact that the tensor product commutes with direct sums and filtered colimits (in fact with all colimits), and that both direct sums and filtered colimits are exact functors. In particular, this shows that all filtered colimits of free modules are flat.

Lazard (1969) proved that the converse holds as well: 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. The direct sum ${\displaystyle \bigoplus \nolimits _{i\in I}M_{i}}$ is flat if and only if each ${\displaystyle M_{i}}$ is flat.

Products of flat R-modules need not in general be flat. In fact, Chase (1960) showed a ring R is coherent (i.e., any finitely generated ideal is finitely presented) if and only if arbitrary products of flat R-modules are again flat.[4]

Flat ring extensions

If ${\displaystyle \phi :R\to S}$ is a ring homomorphism, S is called flat over R (or a flat R-algebra) if it is flat as an R-module. For example, the polynomial ring R[t] is flat over R, for any ring R. Moreover, for any multiplicatively closed subset ${\displaystyle S}$ of a commutative ring ${\displaystyle R}$, the localization ring ${\displaystyle S^{-1}R}$ is flat over R. For example, ${\displaystyle \mathbb {Q} }$ is flat over ${\displaystyle \mathbb {Z} }$ (though not projective).

Let ${\displaystyle S=R[x_{1},\dots ,x_{r}]}$ be a polynomial ring over a noetherian ring ${\displaystyle R}$ and ${\displaystyle f\in S}$ a nonzerodivisor. Then ${\displaystyle S/fS}$ is flat over ${\displaystyle R}$ if and only if ${\displaystyle f}$ is primitive (the coefficients generate the unit ideal).[5] This yields an example of a flat module that is not free.

Kunz (1969) showed that a noetherian local ring ${\displaystyle R}$ of positive characteristic p is regular if and only if the Frobenius morphism ${\displaystyle R\to R,r\mapsto r^{p}}$ is flat and ${\displaystyle R}$ is reduced.

Flat ring extensions are important in algebra, algebraic geometry and related areas. A morphism ${\displaystyle f:X\to Y}$ of schemes is a flat morphism if, by one of several equivalent definitions, the induced map on local rings

${\displaystyle {\mathcal {O}}_{Y,f(x)}\to {\mathcal {O}}_{X,x}}$

is a flat ring homomorphism for any point x in X. Thus, the above-mentioned properties of flat (or faithfully flat) morphisms established by methods of commutative algebra translate into geometric properties of flat morphisms in algebraic geometry.

Local aspects of flatness over commutative rings

In this section, the ring R is supposed to be commutative. In this situation, flatness of R-modules is related in several ways to the notion of localization: M is flat if and only if the module ${\displaystyle M_{\mathfrak {p}}}$ is a flat ${\displaystyle R_{\mathfrak {p}}}$-module for all prime ideals ${\displaystyle {\mathfrak {p}}}$ of R. In fact, it is enough to check the latter condition only for the maximal ideals, as opposed to all prime ideals. This statement reduces the question of flatness to the case of (commutative) local rings.

If R is a local (commutative) ring and either M is finitely generated or the maximal ideal of R is nilpotent (e.g., an artinian local ring) then the standard implication "free implies flat" can be reversed: in this case M is flat if and if only if its free.[6]

The local criterion for flatness states:[7]

Let R be a local noetherian ring, S a local noetherian R-algebra with ${\displaystyle {\mathfrak {m}}_{R}S\subset {\mathfrak {m}}_{S}}$, and M a finitely generated S-module. Then M is flat over R if and only if ${\displaystyle \operatorname {Tor} _{1}^{R}(M,R/{\mathfrak {m}}_{R})=0.}$

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:[8]

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 ${\displaystyle \dim S=\dim R+\dim S/{\mathfrak {m}}_{R}S.}$

Faithfully flat ring homomorphism

Let A be a ring (assumed to be commutative throughout this section) and B an A-algebra, i.e., a ring homomorphism ${\displaystyle f:A\to B}$. Then B has the structure of an A-module. Then B is said to be flat over A (resp. faithfully flat over A) if it is flat (resp. faithfully flat) as an A-module.

There is a basic characterization of a faithfully flat ring homomorphism: given a flat ring homomorphism ${\displaystyle f:A\to B}$, the following are equivalent.

1. ${\displaystyle f}$ is faithfully flat.
2. For each maximal ideal ${\displaystyle {\mathfrak {m}}}$ of ${\displaystyle A}$, ${\displaystyle {\mathfrak {m}}B\neq B.}$
3. If ${\displaystyle N\neq 0}$ is a nonzero ${\displaystyle A}$-module, then ${\displaystyle N\otimes _{A}B\neq 0.}$
4. Every prime ideal of A is the inverse image under f of a prime ideal in B. In other words, the induced map ${\displaystyle f^{*}:\operatorname {Spec} (B)\to \operatorname {Spec} (A)}$ is surjective.
5. A is a pure subring of B (in particular, a subring); here, "pure subring" means that ${\displaystyle N\to N\otimes _{A}B}$ is injective for every ${\displaystyle A}$-module ${\displaystyle N}$.[9]

Condition 2 implies a flat local homomorphism between local rings is faithfully flat. It follows from condition 5 that ${\displaystyle I=IB\cap A}$ for every ideal ${\displaystyle I\subset A}$ (take ${\displaystyle N=A/I}$); in particular, if ${\displaystyle B}$ is a Noetherian ring, then ${\displaystyle A}$ is a Noetherian ring.

Condition 4 can be stated in the following strengthened form: ${\displaystyle \operatorname {Spec} (B)\to \operatorname {Spec} (A)}$ is submersive: the topology of ${\displaystyle \operatorname {Spec} (A)}$ is the quotient topology of ${\displaystyle \operatorname {Spec} (B)}$ (this is a special case of the fact that a faithfully flat quasi-compact morphism of schemes has this property.[10]) It compares to an integral extension of an integrally closed domain. See also flat morphism#Properties of flat morphisms for further information.

Here is one characterization of a faithfully flat homomorphism for a not-necessarily-flat homomorphism. Given an injective local homomorphism ${\displaystyle (A,{\mathfrak {m}})\hookrightarrow (B,{\mathfrak {n}})}$ such that ${\displaystyle {\mathfrak {m}}B}$ is an ${\displaystyle {\mathfrak {n}}}$-primary ideal, ${\displaystyle A\to B}$ is faithfully flat if and only if the theorem of transition holds for it; i.e., for each ${\displaystyle {\mathfrak {m}}}$-primary ideal ${\displaystyle Q}$ of ${\displaystyle A}$, ${\displaystyle \operatorname {length} _{B}(B/QB)=\operatorname {length} _{B}(B/{\mathfrak {m}}B)\operatorname {length} _{A}(A/Q).}$[11]

Example. For a ring ${\displaystyle A,A\to A[x]}$ is faithfully flat. More generally, an ${\displaystyle A}$-algebra that is free of positive rank as an ${\displaystyle A}$-module is faithfully flat. Thus, for example, for a monic polynomial ${\displaystyle f\in A[t]}$, the inclusion ${\displaystyle A\hookrightarrow A[t]/(f)}$ is faithfully flat.

Example. Let ${\displaystyle A}$ be a ring and ${\displaystyle f_{1},\ldots ,f_{r}}$ elements in it. Then those elements generate the unit ideal ${\displaystyle (1)}$ of ${\displaystyle A}$ if and only if

${\displaystyle A\to B=\prod _{i}A[f_{i}^{-1}]}$

is faithfully flat, since localizations are flat, their direct sums are then flat and

${\displaystyle \operatorname {Spec} B=\bigcup _{i}\operatorname {Spec} A[f_{i}^{-1}]\to \operatorname {Spec} A}$

is surjective if and only if the elements generate the unit ideal.[12]

For a given ring homomorphism ${\displaystyle f:A\to B,}$ there is an associated complex called the Amitsur complex:[13]

${\displaystyle 0\to A{\overset {f}{\to }}B{\overset {\delta ^{0}}{\to }}B\otimes _{A}B{\overset {\delta ^{1}}{\to }}B\otimes _{A}B\otimes _{A}B\to \cdots }$

where the coboundary operators ${\displaystyle \delta ^{n}}$ are the alternating sums of the maps obtained by inserting 1 in each spot; e.g., ${\displaystyle \delta ^{0}(b)=b\otimes 1-1\otimes b}$. Then (Grothendieck) this complex is exact if ${\displaystyle f}$ is faithfully flat.

Homological characterization using Tor functors

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

${\displaystyle \operatorname {Tor} _{n}^{R}(X,M)=0}$ for all ${\displaystyle n\geq 1}$ and all right R-modules X).[14]

In fact, it is enough to check that the first Tor term vanishes, i.e., M is flat if and only if

${\displaystyle \operatorname {Tor} _{1}^{R}(N,M)=0}$

for any R-module N or, even more restrictively, when ${\displaystyle N=R/I}$ and ${\displaystyle I\subset R}$ is any finitely generated ideal.

Using the Tor functor's long exact sequences, one can then easily prove facts about a short exact sequence

${\displaystyle 0\to A{\overset {f}{\longrightarrow }}B{\overset {g}{\longrightarrow }}C\to 0}$

If A and C are flat, then so is B. Also, if B and C are flat, then so is A. If A and B are flat, C need not be flat in general, as is shown by the above non-example ${\displaystyle \mathbb {Z} /n}$. However, 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

${\displaystyle \cdots \to F_{2}\to F_{1}\to F_{0}\to M\to 0,}$

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 > 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[15] 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.[16]

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) harv error: no target: CITEREFEnochs1981 (help). The conjecture turned out to be true, resolved positively and proved simultaneously by L. Bican, R. El Bashir and E. Enochs.[17] 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) harv error: no target: CITEREFMacLane1963 (help) 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.[18]

See also

• Generic flatness
• Flat morphism
• von Neumann regular ring – those rings over which all modules are flat.
• Normally flat ring

References

1. Bourbaki, Ch. I, § 2. Proposition 13, Corollary 1. harvnb error: no target: CITEREFBourbaki (help)
2. Matsumura 1970, Corollary 1 of Theorem 55, p. 170
3. Matsumura 1970, Theorem 56
4. "Flatness of Power Series Rings". mathoverflow.net.
5. Eisenbud, Exercise 6.4. harvnb error: no target: CITEREFEisenbud (help)
6. Matsumura, Prop. 3.G harvnb error: no target: CITEREFMatsumura (help)
7. Eisenbud 1994, Theorem 6.8 harvnb error: no target: CITEREFEisenbud1994 (help)
8. Eisenbud 1994, Theorem 18.16 harvnb error: no target: CITEREFEisenbud1994 (help)
9. Proof: Suppose ${\displaystyle f:A\to B}$ is faithfully flat. For an A-module N, the map ${\displaystyle B=A\otimes _{A}B\to B\otimes _{A}B}$ exhibits B as a pure subring and so ${\displaystyle N\otimes _{A}B\to N\otimes _{A}(B\otimes B)\simeq (N\otimes _{A}B)\otimes _{A}B}$ is injective. Hence, ${\displaystyle N\to N\otimes _{A}B}$ is injective. Conversely, if ${\displaystyle N\neq 0}$ is a module over ${\displaystyle A}$, then ${\displaystyle 0\neq N\subset N\otimes _{B}A}$.
10. SGA 1, Exposé VIII., Corollay 4.3. harvnb error: no target: CITEREFSGA_1 (help)
11. Matsumura 1986, Ch. 8, Exercise 22.1.
12. Artin, Exercise (3) after Proposition III.5.2. harvnb error: no target: CITEREFArtin (help)
13. "Amitsur Complex". ncatlab.org.
14. Similarly, a right R-module M is flat if and only if ${\displaystyle \operatorname {Tor} _{n}^{R}(M,X)=0}$ for all ${\displaystyle n\geq 1}$ and all left R-modules X.
15. Lam 1999, p. 183.
16. A module isomorphic to an flat module is of course flat.
17. Bican, El Bashir & Enochs 2001.
18. Richman 1997.

• Artin, Michael (1999). "Noncommutative Rings" (PDF).
• 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, MR 1832549
• N. Bourbaki, Commutative Algebra
• Chase, Stephen U. (1960), "Direct products of modules", Transactions of the American Mathematical Society, 97: 457–473, doi:10.2307/1993382, MR 0120260
• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 978-0-387-94268-1, MR 1322960, ISBN 978-0-387-94269-8
• Enochs, Edgar E. (1981), "Injective and flat covers, envelopes and resolvents", Israel J. Math., 39 (3): 189–209, doi:10.1007/BF02760849, ISSN 0021-2172, MR 0636889
• Enochs, Edgar E.; Jenda, Overtoun M. G. (2000), Relative homological algebra, de Gruyter Expositions in Mathematics, 30, Berlin: Walter de Gruyter & Co., doi:10.1515/9783110803662, ISBN 978-3-11-016633-0, MR 1753146
• Kunz, Ernst (1969), "Characterizations of regular local rings of characteristic p", American Journal of Mathematics, 91: 772–784, doi:10.2307/2373351, MR 0252389
• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0525-8, ISBN 978-0-387-98428-5, MR 1653294
• Lazard, D. (1969), "Autour de la platitude", Bulletin de la Société Mathématique de France, 97: 81–128
• Mac Lane, Saunders (1963), Homology, Die Grundlehren der mathematischen Wissenschaften, Bd. 114, Boston, MA: Academic Press, MR 0156879
• Matsumura, Hideyuki (1970), Commutative algebra
• Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. 8. Cambridge University Press. ISBN 0-521-36764-6. MR 0879273. Zbl 0603.13001.CS1 maint: ref=harv (link)
• Mumford, David, The red book of varieties and schemes
• 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, MR 1601663
• Serre, Jean-Pierre (1956), "Géométrie algébrique et géométrie analytique", Annales de l'Institut Fourier, 6: 1–42, doi:10.5802/aif.59, ISSN 0373-0956, MR 0082175