Editing Flat module (section)

=== Characterizations ===

Flatness can also be characterized by the following equational condition, which means that  {{mvar|R}}-[[linear relation]]s in {{mvar|M}} stem from linear relations in {{mvar|R}}. 

A left {{mvar|R}}-module {{mvar|M}} is flat if and only if, for every linear relation
: <math display=inline>\sum_{i=1}^m r_i x_i = 0</math>
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 that{{sfn|Bourbaki|loc=Ch. I, § 2. Proposition 13, Corollary 1|ps=none}}
: <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 {{mvar|n}} 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 {{mvar|n}} elements. This allows rewriting the previous characterization in terms of homomorphisms, as follows.

An  {{mvar|R}}-module {{mvar|M}} 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 {{mvar|R}}-module, and for every finitely generated {{mvar|R}}-submodule <math>K</math> of <math>\ker f,</math> the map <math>f</math> factors through a map {{mvar|g}} to a free {{mvar|R}}-module <math>G</math> such that <math>g(K)=0:</math>

[[Image:FlatModule-01.png|center|Factor property of a flat module]]