Flat module

Summary

In algebra, 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 Jean-Pierre Serre (1956) in his paper Géometrie Algébrique et Géométrie Analytique. See also flat morphism.

DefinitionEdit

A module M over a ring R is flat if the following condition is satisfied: for every injective linear map   of R-modules, the map

 

is also injective, where   is the map induced by  

For this definition, it is enough to restrict the injections   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   the sequence   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.

CharacterizationsEdit

Flatness can also be characterized by the following equational condition, which means that R-linear relations in M stem from linear relations in R. An R-module M is flat if and only if, for every linear relation

 

with   and  , there exist elements   and   such that[1]

 

It is equivalent to define n elements of a module, and a linear map from   to this module, which maps the standard basis of   to the n elements. This allow 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   where   is a finitely generated free R-module, and for every finitely generated R-submodule   of   the map   factors through a map g to a free R-module   such that  

 

Relations to other module propertiesEdit

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 modules are all projective over the rings that are most commonly considered.

This is partly summarized in the following graphic.

 

Torsion-free modulesEdit

Every flat module is torsion-free. This results from the above characterization in terms of relations by taking m = 1

The converse holds over the integers, and more generally over principal ideal domains and Dedekind rings.

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

Free and projective modulesEdit

A module M is projective if and only if there is a free module G and two linear maps   and   such that   In particular, every free module is projective (take   and  ).

Every projective module is flat. This can be proven from the above characterizations of flatness and projectivity in terms of linear maps by taking   and  

Conversely, finitely generated flat modules are projective under mild conditions that are generally satisfied in commutative algebra and algebraic geometry. This makes the concept of flatness useful mainly for modules that are not finitely generated.

A finitely presented module (that is the quotient of a finitely generated free module by a finitely generated submodule) that is flat is always projective. This can be proven by taking f surjective and   in the above characterization of flatness in terms of linear maps. The condition   implies the existence of a linear map   such that   and thus   As f is surjective, one has thus   and M is projective.

Over a Noetherian ring, every finitely generated flat module is projective, since every finitely generated module is finitely presented. The same result is true over an integral domain, even if it is not Noetherian.[2]

On a local ring every finitely generated flat module is free.

A finitely generated flat module that is not projective can be built as follows. Let   be the set of the infinite sequences whose terms belong to a fixed field F. It is a commutative ring with addition and multiplication defined componentwise. This ring is absolutely flat (that is, every module is flat). The module   where I is the ideal of the sequences with a finite number of nonzero terms, is thus flat and finitely generated (only one generator), but it is not projective.

Non-examplesEdit

  • If I is an ideal in a Noetherian commutative ring R, then   is not a flat module, except if I is generated by an idempotent (that is an element equal to its square). In particular, if R is an integral domain,   is flat only if   equals R or is the zero ideal.
  • Over an integral domain, a flat module is torsion free. Thus a module that contains nonzero torsion elements is not flat. In particular   and all fields of positive characteristics are non-flat  -modules, where   is the ring of integers, and   is the field of the rational numbers.

Direct sums, limits and productsEdit

A direct sum   of modules is flat if and only if each   is flat.

A direct limit of flat is flat. In particular, a direct limit of free modules is flat. Conversely, every flat module can be written as a direct limit of finitely-generated free modules.[3]

Direct products of flat modules need not in general be flat. In fact, given a ring R, every direct product of flat R-modules is flat if and only if R is a coherent ring (that is, every finitely generated ideal is finitely presented).[4]

Flat ring extensionsEdit

A ring homomorphism   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   of a commutative ring  , the localization ring   is flat over R (it is projective only in exceptional cases). For example,   is flat and not projective over  

If   is an ideal of a Noetherian commutative ring   the completion   of   with respect to   is flat.[5] It is faithfully flat if and only if   is contained in the Jacobson radical of   (See also Zariski ring.)[6]

LocalizationEdit

In this section, R denotes a commutative ring. If   is a prime ideal of R, the localization at   is, as usual, denoted with   as an index. That is,   and, if M is an R-module,  

If an R-module M if flat, then   is a flat  -module for every prime ideal  

Conversely, if   is a flat  -module for every maximal ideal  , then M is a flat R-module (and   is a flat  -module for every prime ideal  ).

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.

Flat morphisms of schemesEdit

A morphism   of schemes is a flat morphism if the induced map on local rings

 

is a flat ring homomorphism for any point x in X. Thus, properties of flat (or faithfully flat) ring homomorphisms extends naturally to geometric properties of flat morphisms in algebraic geometry. For example, consider the previous example of  . The inclusion   then determines the flat morphism

 

Each (geometric) fiber   is the curve of equation   See also flat degeneration and deformation to normal cone.

Let   be a polynomial ring over a commutative Noetherian ring   and   a nonzerodivisor. Then   is flat over   if and only if   is primitive (the coefficients generate the unit ideal).[7] An example is[8]pg 3   which is flat (and even free) over   (see also below for the geometric meaning). Such flat extensions can be used to yield examples of flat modules that are not free and do not result from a localization.

Faithful flatnessEdit

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,   is a ring homomorphism of commutative rings, which gives to   the structures of an  -algebra and an  -module. If   is a  -module flat (or faithfully flat), one says commonly that   is flat (or faithfully flat) over   and that   is flat (or faithfully flat).

If   is flat over   the following conditions are equivalent.

  •   is faithfully flat.
  • For each maximal ideal   of  , one has  
  • If   is a nonzero  -module, then  
  • For every prime ideal   of   there is a prime ideal   of   such that   In other words, the map   induced by   on the spectra is surjective.
  •   is injective, and   is a pure subring of   that is,   is injective for every  -module  .[9]

The second condition implies that a flat local homomorphism of local rings is faithfully flat. It follows from the last condition that   for every ideal   of   (take  ). In particular, if   is a Noetherian ring, then   is also Noetherian.

The last but one condition can be stated in the following strengthened form:   is submersive, which means that the Zariski topology of   is the quotient topology of that of   (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.

ExamplesEdit

  • A ring homomorphism   such that   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   is a monic polynomial, the inclusion   is faithfully flat.
  • Let   The direct product   of the localizations at the   is faithfully flat over   if and only if   generate the unit ideal of   (that is, if   is a linear combination of the  ).[11]
  • The direct sum of the localizations   of   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   there is an associated complex called the Amitsur complex:[12]
     
    where the coboundary operators   are the alternating sums of the maps obtained by inserting 1 in each spot; e.g.,  . Then (Grothendieck) this complex is exact if   is faithfully flat.

Faithfully flat local homomorphismsEdit

Here is one characterization of a faithfully flat homomorphism for a not-necessarily-flat homomorphism. Given an injective local homomorphism   such that   is an  -primary ideal, the homomorphism   is faithfully flat if and only if the theorem of transition holds for it; that is, for each  -primary ideal   of  ,  [13]

Homological characterization using Tor functorsEdit

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

  for all   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

 

for any R-module N or, even more restrictively, when   and   is any finitely generated ideal.

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. Also, if B and C are flat, then so is A. If A and B are flat, C need not be flat in general. However, if A is pure in B and B is flat, then A and C are flat.

Flat resolutionsEdit

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

 

where the Fi 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   is nonzero and   for  . 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 → F0M → 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 coversEdit

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

In constructive mathematicsEdit

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 alsoEdit

ReferencesEdit

  1. ^ Bourbaki, Ch. I, § 2. Proposition 13, Corollary 1.
  2. ^ Cartier, Pierre (1958). "Questions de rationalité des diviseurs en géométrie algébrique". Bulletin de la Société Mathématique de France (in French). Lemme 5, page 249. 86: 177–251. doi:10.24033/bsmf.1503.{{cite journal}}: CS1 maint: location (link)
  3. ^ (Lazard 1969)
  4. ^ (Chase 1960)
  5. ^ Matsumura 1970, Corollary 1 of Theorem 55, p. 170
  6. ^ Matsumura 1970, Theorem 56
  7. ^ Eisenbud 1995, Exercise 6.4.
  8. ^ Artin. "Deformation theory" (PDF). Archived (PDF) from the original on 28 Oct 2020.
  9. ^ Proof: Suppose   is faithfully flat. For an R-module   the map   exhibits   as a pure subring and so   is injective. Hence,   is injective. Conversely, if   is a module over  , then  
  10. ^ SGA 1, Exposé VIII., Corollay 4.3.
  11. ^ Artin, Exercise (3) after Proposition III.5.2.
  12. ^ "Amitsur Complex". ncatlab.org.
  13. ^ Matsumura 1986, Ch. 8, Exercise 22.1.
  14. ^ Similarly, a right R-module M is flat if and only if   for all   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.