Editing Flat module (section)

== Relations to other module properties ==

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

There are [[finitely generated module]]s 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 [[Localisation of a module|localizations]] at [[prime ideal]]s are free modules.

This is partly summarized in the following graphic.

[[File:Module properties in commutative algebra.svg|center|Module properties in commutative algebra]]

=== Torsion-free modules ===

Every flat module is [[torsion-free module|torsion-free]].<!-- with the appropriate definition of torsion-free, the ring need not be a domain.--> This results from the above characterization in terms of relations by taking {{math|1=''m'' = 1}}.

The converse holds over the integers, and more generally over [[principal ideal domain]]s and [[Dedekind ring]]s.

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

=== Free and projective modules ===

A module {{mvar|M}} is [[projective module|projective]] if and only if there is a [[free module]] {{mvar|G}} 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 {{nowrap|<math>i=p=\mathrm{id}_M</math>).}}

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 module|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 {{mvar|f}} 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 {{mvar|f}} is surjective, one has thus <math>h\circ i=\mathrm{id}_M,</math> and {{mvar|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.{{sfn|Cartier|1958|loc=Lemme 5, p. 249|ps=none}}

On a [[local ring]] every finitely generated flat module is free.{{sfn|Matsumura|1986|loc=Theorem 7.10|ps=none}}

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 sequence]]s whose terms belong to a fixed field {{mvar|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 <math>R/I,</math> where {{mvar|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-examples ===
* If {{mvar|I}} is an ideal in a Noetherian commutative ring {{mvar|R}}, then <math>R/I</math> is not a flat module, except if {{mvar|I}} is generated by an [[idempotent]] (that is an element equal to its square). In particular, if {{mvar|R}} is an [[integral domain]], <math>R/I</math> is flat only if <math>I</math> equals {{mvar|R}} or is the [[zero ideal]].
* Over an integral domain, a flat module is [[torsion-free module|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 products ===
A [[direct sum of modules|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 module]]s is flat. Conversely, every flat module can be written as a direct limit of [[finitely generated module|finitely-generated]] free modules.{{sfn|Lazard|1969|ps=none}}

[[Direct product]]s of flat modules need not in general be flat. In fact, given a ring {{mvar|R}}, every direct product of flat {{mvar|R}}-modules is flat if and only if {{mvar|R}} is a [[coherent ring]] (that is, every finitely generated ideal is finitely presented).{{sfn|Chase|1960|ps=none}}