Editing Flat module (section)

== Homological characterization using Tor functors ==
Flatness may also be expressed using the [[Tor functor]]s, the [[derived functor|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>).{{efn|Similarly, a right <math>R</math>-module <math>M</math> is flat if and only if <math>\operatorname{Tor}_n^R (M, X) = 0</math> for all <math>n \ge 1</math> and all left <math>R</math>-modules <math>X</math>.}}
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 sequence]]s, 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 submodule|pure]] in <math>B</math> and <math>B</math> is flat, then <math>A</math> and <math>C</math> are flat.