Flat module

A left module M over a ring R is flat if the following condition is satisfied: for every injective linear map ${\displaystyle \varphi :K\to L}$ of right R-modules, the map

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

is also injective, where ${\displaystyle \varphi \otimes _{R}M}$ is the map induced by ${\displaystyle k\otimes m\mapsto \varphi (k)\otimes m.}$

For this definition, it is enough to restrict the injections ${\displaystyle \varphi }$ to the inclusions of finitely generated ideals into R.

Equivalently, an R-module M is flat if the tensor product with M is an exact functor; that is if, for every short exact sequence of R-modules ${\displaystyle 0\rightarrow K\rightarrow L\rightarrow J\rightarrow 0,}$ the sequence ${\displaystyle 0\rightarrow K\otimes _{R}M\rightarrow L\otimes _{R}M\rightarrow J\otimes _{R}M\rightarrow 0}$ is also exact. (This is an equivalent definition since the tensor product is a right exact functor.)

These definitions apply also if R is a non-commutative ring, and M is a left R-module; in this case, K, L and J must be right R-modules, and the tensor products are not R-modules in general, but only abelian groups.

Characterizations

Flatness can also be characterized by the following equational condition, which means that R-linear relations in M stem from linear relations in R.

A left R-module M is flat if and only if, for every linear relation

${\textstyle \sum _{i=1}^{m}r_{i}x_{i}=0}$

with ${\displaystyle r_{i}\in R}$ and ${\displaystyle x_{i}\in M}$, there exist elements ${\displaystyle y_{j}\in M}$ and ${\displaystyle a_{i,j}\in R,}$ such that[1]

${\textstyle \sum _{i=1}^{m}r_{i}a_{i,j}=0\qquad }$ for ${\displaystyle j=1,\ldots ,n,}$

and

${\textstyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j}\qquad }$ for ${\displaystyle i=1,\ldots ,m.}$

It is equivalent to define n elements of a module, and a linear map from ${\displaystyle R^{n}}$ to this module, which maps the standard basis of ${\displaystyle R^{n}}$ to the n elements. This allows rewriting the previous characterization in terms of homomorphisms, as follows.

An R-module 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 R-module, and for every finitely generated R-submodule ${\displaystyle K}$ of ${\displaystyle \ker f,}$ the map ${\displaystyle f}$ factors through a map g to a free R-module ${\displaystyle G}$ such that ${\displaystyle g(K)=0:}$

Factor property of a flat module

Flatness is related to various other module properties, such as being free, projective, or torsion-free. In particular, every flat module is torsion-free, every projective module is flat, and every free module is projective.

There are finitely generated modules that are flat and not projective. However, finitely generated flat modules are all projective over the rings that are most commonly considered.

This is partly summarized in the following graphic.

Module properties in commutative algebra

A ring homomorphism ${\displaystyle R\to S}$ is flat if S is a flat R-module for the module structure induced by the homomorphism. For example, the polynomial ring R[t] is flat over R, for any ring R.

For any multiplicative subset ${\displaystyle S}$ of a commutative ring ${\displaystyle R}$, the localization ring ${\displaystyle S^{-1}R}$ is flat over R (it is projective only in exceptional cases). For example, ${\displaystyle \mathbb {Q} }$ is flat and not projective over ${\displaystyle \mathbb {Z} .}$

If ${\displaystyle I}$ is an ideal of a Noetherian commutative ring ${\displaystyle R,}$ the completion ${\displaystyle {\widehat {R}}}$ of ${\displaystyle R}$ with respect to ${\displaystyle I}$ is flat.[6] It is faithfully flat if and only if ${\displaystyle I}$ is contained in the Jacobson radical of ${\displaystyle A.}$ (See also Zariski ring.)[7]

In this section, R denotes a commutative ring. If ${\displaystyle {\mathfrak {p}}}$ is a prime ideal of R, the localization at ${\displaystyle {\mathfrak {p}}}$ is, as usual, denoted with ${\displaystyle {\mathfrak {p}}}$ as an index. That is, ${\displaystyle R_{\mathfrak {p}}=S^{-1}R,}$ and, if M is an R-module, ${\displaystyle M_{\mathfrak {p}}=S^{-1}M=R_{\mathfrak {p}}\otimes _{R}M.}$

If an R-module M is flat, then ${\displaystyle M_{\mathfrak {p}}}$ is a flat ${\displaystyle R_{\mathfrak {p}}}$-module for every prime ideal ${\displaystyle {\mathfrak {p}}.}$

Conversely, if ${\displaystyle M_{\mathfrak {m}}}$ is a flat ${\displaystyle R_{\mathfrak {m}}}$-module for every maximal ideal ${\displaystyle {\mathfrak {m}}}$, then M is a flat R-module (and ${\displaystyle M_{\mathfrak {p}}}$ is a flat ${\displaystyle R_{\mathfrak {p}}}$-module for every prime ideal ${\displaystyle {\mathfrak {p}}}$).

These properties are fundamental in commutative algebra, since they reduce the question of flatness to the case of local rings. They are often expressed by saying that flatness is a local property.

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. Although the concept is defined for modules over a non-necessary commutative ring, it is used mainly for commutative algebras. So, this is the only case that is considered here, even if some results can be generalized to the case of modules over a non-commutaive ring.

In this section, ${\displaystyle f\colon R\to S}$ is a ring homomorphism of commutative rings, which gives to ${\displaystyle S}$ the structures of an ${\displaystyle R}$-algebra and an ${\displaystyle R}$-module. If ${\displaystyle S}$ is a ${\displaystyle R}$-module flat (or faithfully flat), one says commonly that ${\displaystyle S}$ is flat (or faithfully flat) over ${\displaystyle R,}$ and that ${\displaystyle f}$ is flat (or faithfully flat).

If ${\displaystyle S}$ is flat over ${\displaystyle R,}$ the following conditions are equivalent.

• ${\displaystyle S}$ is faithfully flat.
• For each maximal ideal ${\displaystyle {\mathfrak {m}}}$ of ${\displaystyle R}$, one has ${\displaystyle {\mathfrak {m}}S\neq S.}$
• If ${\displaystyle M}$ is a nonzero ${\displaystyle R}$-module, then ${\displaystyle M\otimes _{R}S\neq 0.}$
• For every prime ideal ${\displaystyle {\mathfrak {p}}}$ of ${\displaystyle R,}$ there is a prime ideal ${\displaystyle {\mathfrak {P}}}$ of ${\displaystyle S}$ such that ${\displaystyle {\mathfrak {p}}=f^{-1}({\mathfrak {P}}).}$ In other words, the map ${\displaystyle f^{*}\colon \operatorname {Spec} (S)\to \operatorname {Spec} (R)}$ induced by ${\displaystyle f}$ on the spectra is surjective.
• ${\displaystyle f,}$ is injective, and ${\displaystyle R}$ is a pure subring of ${\displaystyle S;}$ that is, ${\displaystyle M\to M\otimes _{R}S}$ is injective for every ${\displaystyle R}$-module ${\displaystyle M}$.[lower-alpha 1]

The second condition implies that a flat local homomorphism of local rings is faithfully flat. It follows from the last condition that ${\displaystyle I=IS\cap R}$ for every ideal ${\displaystyle I}$ of ${\displaystyle R}$ (take ${\displaystyle M=R/I}$). In particular, if ${\displaystyle S}$ is a Noetherian ring, then ${\displaystyle R}$ is also Noetherian.

The last but one condition can be stated in the following strengthened form: ${\displaystyle \operatorname {Spec} (S)\to \operatorname {Spec} (R)}$ is submersive, which means that the Zariski topology of ${\displaystyle \operatorname {Spec} (R)}$ is the quotient topology of that of ${\displaystyle \operatorname {Spec} (S)}$ (this is a special case of the fact that a faithfully flat quasi-compact morphism of schemes has this property.[10]). See also Flat morphism § Properties of flat morphisms.

Examples

• A ring homomorphism ${\displaystyle R\to S}$ such that ${\displaystyle S}$ is a nonzero free R-module is faithfully flat. For example:
  • Every field extension is faithfully flat. This property is implicitly behind the use of complexification for proving results on real vector spaces.
  • A polynomial ring is a faithfully flat extension of its ring of coefficients.
  • If ${\displaystyle p\in R[x]}$ is a monic polynomial, the inclusion ${\displaystyle R\hookrightarrow R[t]/\langle p\rangle }$ is faithfully flat.
• Let ${\displaystyle t_{1},\ldots ,t_{k}\in R.}$ The direct product ${\displaystyle \textstyle \prod _{i}R[t_{i}^{-1}]}$ of the localizations at the ${\displaystyle t_{i}}$ is faithfully flat over ${\displaystyle R}$ if and only if ${\displaystyle t_{1},\ldots ,t_{k}}$ generate the unit ideal of ${\displaystyle R}$ (that is, if ${\displaystyle 1}$ is a linear combination of the ${\displaystyle t_{i}}$).[11]
• The direct sum of the localizations ${\displaystyle R_{\mathfrak {p}}}$ of ${\displaystyle R}$ at all its prime ideals is a faithfully flat module that is not an algebra, except if there are finitely many prime ideals.

The two last examples are implicitly behind the wide use of localization in commutative algebra and algebraic geometry.

• For a given ring homomorphism ${\displaystyle f:A\to B,}$ there is an associated complex called the Amitsur complex:[12]

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

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.

Flatness may also be expressed using the Tor functors, the left derived functors of the tensor product. A left ${\displaystyle R}$-module ${\displaystyle M}$ is flat if and only if

${\displaystyle \operatorname {Tor} _{n}^{R}(X,M)=0}$ for all ${\displaystyle n\geq 1}$ and all right ${\displaystyle R}$-modules ${\displaystyle X}$).[lower-alpha 2]

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 ${\displaystyle R}$-module ${\displaystyle N}$ 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 ${\displaystyle A}$ and ${\displaystyle C}$ are flat, then so is ${\displaystyle B}$. Also, if ${\displaystyle B}$ and ${\displaystyle C}$ are flat, then so is ${\displaystyle A}$. If ${\displaystyle A}$ and ${\displaystyle B}$ are flat, ${\displaystyle C}$ need not be flat in general. However, if ${\displaystyle A}$ is pure in ${\displaystyle B}$ and ${\displaystyle B}$ is flat, then ${\displaystyle A}$ and ${\displaystyle C}$ are flat.

A flat resolution of a module ${\displaystyle M}$ is a resolution of the form

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

where the ${\displaystyle F_{i}}$ 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 ${\displaystyle F_{n}}$ is nonzero and ${\displaystyle F_{i}=0}$ for ${\displaystyle i>n}$. If a module ${\displaystyle M}$ admits a finite flat resolution, the minimal length among all finite flat resolutions of ${\displaystyle M}$ is called its flat dimension[14] and denoted ${\displaystyle \operatorname {fd} (M)}$. If ${\displaystyle M}$ does not admit a finite flat resolution, then by convention the flat dimension is said to be infinite. As an example, consider a module ${\displaystyle M}$ such that ${\displaystyle \operatorname {fd} (M)=0}$. In this situation, the exactness of the sequence ${\displaystyle 0\to F_{0}\to M\to 0}$ indicates that the arrow in the center is an isomorphism, and hence ${\displaystyle M}$ itself is flat.[lower-alpha 3]

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 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 automorphism. 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.[15] 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 Mac Lane (1963) and in more recent works focussing on flat resolutions such as Enochs and Jenda (2000).

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.[16]

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