Flat module

Template:Short description In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module M over a ring R is flat if 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 Template:Harvs in his paper Géometrie Algébrique et Géométrie Analytique.

DefinitionEdit

A left module Template:Mvar over a ring Template:Mvar is flat if the following condition is satisfied: for every injective linear map <math>\varphi: K \to L</math> of right Template:Mvar-modules, the map

<math>\varphi \otimes_R M: K \otimes_R M \to L \otimes_R M</math>

is also injective, where <math>\varphi \otimes_R M</math> is the map induced by <math>k \otimes m \mapsto \varphi(k) \otimes m.</math>

For this definition, it is enough to restrict the injections <math>\varphi</math> to the inclusions of finitely generated ideals into Template:Mvar.

Equivalently, an Template:Mvar-module Template:Mvar is flat if the tensor product with Template:Mvar is an exact functor; that is if, for every short exact sequence of Template:Mvar-modules <math>0\rightarrow K\rightarrow L\rightarrow J\rightarrow 0,</math> the sequence <math>0\rightarrow K\otimes_R M\rightarrow L\otimes_R M\rightarrow J\otimes_R M\rightarrow 0</math> is also exact. (This is an equivalent definition since the tensor product is a right exact functor.)

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

CharacterizationsEdit

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

A left Template:Mvar-module Template:Mvar is flat if and only if, for every linear relation

with <math> r_i \in R</math> and <math>x_i \in M</math>, there exist elements <math>y_j\in M</math> and <math>a_{i,j}\in R,</math> such thatTemplate:Sfn

<math display=inline>\sum_{i=1}^m r_ia_{i,j}=0\qquad</math> for <math>j=1, \ldots, n ,</math>

and

<math display=inline>x_i=\sum_{j=1}^n a_{i,j} y_j\qquad</math> for <math>i=1, \ldots, m.</math>

It is equivalent to define Template:Mvar elements of a module, and a linear map from <math>R^n</math> to this module, which maps the standard basis of <math>R^n</math> to the Template:Mvar elements. This allows rewriting the previous characterization in terms of homomorphisms, as follows.

An Template:Mvar-module Template:Mvar is flat if and only if the following condition holds: for every map <math>f : F \to M,</math> where <math>F</math> is a finitely generated free Template:Mvar-module, and for every finitely generated Template:Mvar-submodule <math>K</math> of <math>\ker f,</math> the map <math>f</math> factors through a map Template:Mvar to a free Template:Mvar-module <math>G</math> such that <math>g(K)=0:</math>

Factor property of a flat module

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 flat modules are all projective over the rings that are most commonly considered. Moreover, a finitely generated module is flat if and only it is locally free, meaning all the localizations at prime ideals are free modules.

This is partly summarized in the following graphic.

Module properties in commutative algebra

Torsion-free modulesEdit

Every flat module is torsion-free. This results from the above characterization in terms of relations by taking Template:Math.

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 Template:Mvar is projective if and only if there is a free module Template:Mvar and two linear maps <math>i:M\to G</math> and <math>p:G\to M</math> such that <math>p\circ i = \mathrm{id}_M.</math> In particular, every free module is projective (take <math>G=M</math> and Template:Nowrap

Every projective module is flat. This can be proven from the above characterizations of flatness and projectivity in terms of linear maps by taking <math>g=i\circ f</math> and <math>h=p.</math>

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 Template:Mvar surjective and <math>K=\ker f</math> in the above characterization of flatness in terms of linear maps. The condition <math>g(K)=0</math> implies the existence of a linear map <math>i:M\to G</math> such that <math>i\circ f = g,</math> and thus <math>h\circ i \circ f =h\circ g = f. </math> As Template:Mvar is surjective, one has thus <math>h\circ i=\mathrm{id}_M,</math> and Template:Mvar 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.Template:Sfn

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

A finitely generated flat module that is not projective can be built as follows. Let <math>R=F^\mathbb N</math> be the set of the infinite sequences whose terms belong to a fixed field Template:Mvar. It is a commutative ring with addition and multiplication defined componentwise. This ring is absolutely flat (that is, every module is flat). The module <math>R/I,</math> where Template:Mvar 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 Template:Mvar is an ideal in a Noetherian commutative ring Template:Mvar, then <math>R/I</math> is not a flat module, except if Template:Mvar is generated by an idempotent (that is an element equal to its square). In particular, if Template:Mvar is an integral domain, <math>R/I</math> is flat only if <math>I</math> equals Template:Mvar 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 <math>\Q/\Z</math> and all fields of positive characteristics are non-flat <math>\Z</math>-modules, where <math>\Z</math> is the ring of integers, and <math>\Q</math> is the field of the rational numbers.

Direct sums, limits and productsEdit

A direct sum <math>\textstyle\bigoplus_{i \in I} M_i</math> of modules is flat if and only if each <math>M_i</math> 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.Template:Sfn

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

Flat ring extensionsEdit

A ring homomorphism <math>R \to S</math> is flat if Template:Mvar is a flat Template:Mvar-module for the module structure induced by the homomorphism. For example, the polynomial ring Template:Math is flat over Template:Mvar, for any ring Template:Mvar.

For any multiplicative subset <math>S</math> of a commutative ring <math>R</math>, the localization <math>S^{-1}R</math> is a flat Template:Math-algebra (it is projective only in exceptional cases). For example, <math>\Q</math> is flat and not projective over <math>\Z.</math>

If <math>I</math> is an ideal of a Noetherian commutative ring <math>R,</math> the completion <math>\widehat{R}</math> of <math>R</math> with respect to <math>I</math> is flat.Template:Sfn It is faithfully flat if and only if <math>I</math> is contained in the Jacobson radical of <math>A.</math> (See also Zariski ring.)Template:Sfn

Local propertyEdit

In this section, Template:Mvar denotes a commutative ring. If <math>\mathfrak p</math> is a prime ideal of Template:Mvar, the localization at <math>\mathfrak p</math> is, as usual, denoted with <math>\mathfrak p</math> as an index. That is, <math>R_{\mathfrak p} = (R\setminus \mathfrak p)^{-1}R,</math> and, if Template:Mvar is an Template:Mvar-module, <math>M_{\mathfrak p} = (R\setminus \mathfrak p)^{-1}M = R_{\mathfrak p}\otimes_R M.</math>

If Template:Mvar is an Template:Mvar-module the three following conditions are equivalent:

<math>M</math> is a flat <math>R</math>-module;
<math>M_\mathfrak p</math> is a flat <math>R_\mathfrak p</math>-module for every prime ideal <math>\mathfrak p;</math>
<math>M_\mathfrak m</math> is a flat <math>R_\mathfrak m</math>-module for every maximal ideal <math>\mathfrak m.</math>

This property is fundamental in commutative algebra and algebraic geometry, since it reduces the study of flatness to the case of local rings. They are often expressed by saying that flatness is a local property.

Flat morphisms of schemesEdit

The definition of a flat morphism of schemes results immediately from the local property of flatness.

A morphism <math>f: X \to Y</math> of schemes is a flat morphism if the induced map on local rings

<math>\mathcal O_{Y, f(x)} \to \mathcal O_{X,x}</math>

is a flat ring homomorphism for any point Template:Mvar in Template:Mvar.

Thus, properties of flat (or faithfully flat) ring homomorphisms extends naturally to geometric properties of flat morphisms in algebraic geometry.

For example, consider the flat <math>\mathbb{C}[t]</math>-algebra <math>R = \mathbb{C}[t,x,y]/(xy-t)</math> (see below). The inclusion <math>\mathbb{C}[t] \hookrightarrow R</math> induces the flat morphism

<math>\pi : \operatorname{Spec}(R) \to \operatorname{Spec}(\mathbb C[t]).</math>

Each (geometric) fiber <math>\pi^{-1}(t)</math> is the curve of equation <math>xy = t.</math> (See also flat degeneration and deformation to normal cone.)

Let <math>S = R[x_1, \dots, x_r]</math> be a polynomial ring over a commutative Noetherian ring <math>R</math> and <math>f \in S</math> a nonzerodivisor. Then <math>S/fS</math> is flat over <math>R</math> if and only if <math>f</math> is primitive (the coefficients generate the unit ideal).Template:Sfn An example isTemplate:Sfn <math>\mathbb{C}[t,x,y]/(xy-t),</math> which is flat (and even free) over <math>\mathbb{C}[t]</math> (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, <math>f\colon R \to S</math> is a ring homomorphism of commutative rings, which gives to <math>S</math> the structures of an <math>R</math>-algebra and an <math>R</math>-module. If <math>S</math> is a <math>R</math>-module flat (or faithfully flat), one says commonly that <math>S</math> is flat (or faithfully flat) over <math>R, </math> and that <math>f</math> is flat (or faithfully flat).

If <math>S</math> is flat over <math>R,</math> the following conditions are equivalent.

<math>S</math> is faithfully flat.
For each maximal ideal <math>\mathfrak{m}</math> of <math>R</math>, one has <math>\mathfrak{m}S \ne S.</math>
If <math>M</math> is a nonzero <math>R</math>-module, then <math>M \otimes_R S \ne 0.</math>
For every prime ideal <math>\mathfrak{p}</math> of <math>R,</math> there is a prime ideal <math>\mathfrak{P}</math> of <math>S</math> such that <math>\mathfrak{p} = f^{-1}(\mathfrak P).</math> In other words, the map <math>f^*\colon \operatorname{Spec}(S) \to \operatorname{Spec}(R)</math> induced by <math>f</math> on the spectra is surjective.
<math>f,</math> is injective, and <math>R</math> is a pure subring of <math>S;</math> that is, <math>M \to M \otimes_R S</math> is injective for every <math>R</math>-module <math>M</math>.Template:Efn

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

The last but one condition can be stated in the following strengthened form: <math>\operatorname{Spec}(S) \to \operatorname{Spec}(R)</math> is submersive, which means that the Zariski topology of <math>\operatorname{Spec}(R)</math> is the quotient topology of that of <math>\operatorname{Spec}(S)</math> (this is a special case of the fact that a faithfully flat quasi-compact morphism of schemes has this property.Template:Sfn). See also Template:Slink.

ExamplesEdit

A ring homomorphism <math>R\to S</math> such that <math>S</math> is a nonzero free Template:Mvar-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 <math>p\in R[x]</math> is a monic polynomial, the inclusion <math>R \hookrightarrow R[t]/\langle p \rangle</math> is faithfully flat.
Let <math>t_1, \ldots, t_k\in R.</math> The direct product <math>\textstyle\prod_i R[t_i^{-1}]</math> of the localizations at the <math>t_i</math> is faithfully flat over <math>R</math> if and only if <math>t_1, \ldots, t_k</math> generate the unit ideal of <math>R</math> (that is, if <math>1</math> is a linear combination of the <math>t_i</math>).Template:Sfn
The direct sum of the localizations <math>R_\mathfrak p</math> of <math>R</math> 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 <math>f: A \to B,</math> there is an associated complex called the Amitsur complex:<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref> <math display="block">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</math> where the coboundary operators <math>\delta^n</math> are the alternating sums of the maps obtained by inserting 1 in each spot; e.g., <math>\delta^0(b) = b \otimes 1-1 \otimes b</math>. Then (Grothendieck) this complex is exact if <math>f</math> 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 <math>(R, \mathfrak m) \hookrightarrow (S, \mathfrak n)</math> such that <math>\mathfrak{m} S</math> is an <math>\mathfrak{n}</math>-primary ideal, the homomorphism <math>S \to B</math> is faithfully flat if and only if the theorem of transition holds for it; that is, for each <math>\mathfrak m</math>-primary ideal <math>\mathfrak q</math> of <math>R</math>, <math>\operatorname{length}_S (S/ \mathfrak q S) = \operatorname{length}_S (S/ \mathfrak{m} S) \operatorname{length}_R(R/\mathfrak q).</math>Template:Sfn

Homological characterization using Tor functorsEdit

Flatness may also be expressed using the Tor functors, the left derived functors of the tensor product. A left <math>R</math>-module <math>M</math> is flat if and only if

<math>\operatorname{Tor}_n^R (X, M) = 0</math> for all <math>n \ge 1</math> and all right <math>R</math>-modules <math>X</math>).Template:Efn

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

<math>\operatorname{Tor}_1^R (N, M) = 0</math>

for any <math>R</math>-module <math>N</math> or, even more restrictively, when <math>N=R/I</math> and <math>I\subset R</math> is any finitely generated ideal.

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

<math>0 \to A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} C \to 0</math>

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

Flat resolutionsEdit

A flat resolution of a module <math>M</math> is a resolution of the form

<math>\cdots \to F_2 \to F_1 \to F_0 \to M \to 0,</math>

where the <math>F_i</math> 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 <math>F_n</math> is nonzero and <math>F_i=0</math> for <math>i>n</math>. If a module <math>M</math> admits a finite flat resolution, the minimal length among all finite flat resolutions of <math>M</math> is called its flat dimension Template:Sfn and denoted <math>\operatorname{fd}(M)</math>. If <math>M</math> does not admit a finite flat resolution, then by convention the flat dimension is said to be infinite. As an example, consider a module <math>M</math> such that <math>\operatorname{fd}(M)=0</math>. In this situation, the exactness of the sequence <math>0 \to F_0 \to M \to 0</math> indicates that the arrow in the center is an isomorphism, and hence <math>M</math> itself is flat.Template:Efn

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 automorphism. This flat cover conjecture was explicitly first stated in Template:Harvs. The conjecture turned out to be true, resolved positively and proved simultaneously by L. Bican, R. El Bashir and E. Enochs.Template:Sfn 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 Template:Harvs and in more recent works focussing on flat resolutions such as Template:Harvs.

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.Template:Sfn

NotesEdit

Template:Notelist

CitationsEdit

Template:Reflist

ReferencesEdit

Template:Refbegin

{{#invoke:citation/CS1|citation

|CitationClass=web }}

{{#invoke:citation/CS1|citation