Editing Flat module (section)

=== 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.