Editing Differential form (section)

=== Exterior differential complex ===
One important property of the exterior derivative is that {{math|1=''d''{{i sup|2}} = 0}}. This means that the exterior derivative defines a [[cochain complex]]:
<math display="block">0\ \to\ \Omega^0(M)\ \stackrel{d}{\to}\ \Omega^1(M)\ \stackrel{d}{\to}\ \Omega^2(M)\ \stackrel{d}{\to}\ \Omega^3(M)\ \to\ \cdots \ \to\ \Omega^n(M)\ \to \ 0.</math>

This complex is called the de Rham complex, and its [[cohomology]] is by definition the [[de Rham cohomology]] of {{math|''M''}}.  By the [[Poincaré lemma]], the de Rham complex is locally [[exact sequence|exact]] except at {{math|Ω<sup>0</sup>(''M'')}}.  The kernel at {{math|Ω<sup>0</sup>(''M'')}} is the space of [[locally constant function]]s on {{math|''M''}}.  Therefore, the complex is a resolution of the constant [[sheaf (mathematics)|sheaf]] {{math|{{underline|'''R'''}}}}, which in turn implies a form of de Rham's theorem: de Rham cohomology computes the [[sheaf cohomology]] of {{math|{{underline|'''R'''}}}}.