Editing Hodge theory (section)

===De Rham cohomology===
The Hodge theory references the [[de Rham cohomology|de Rham complex]]. Let ''M'' be a [[smooth manifold]]. For a non-negative integer ''k'', let Ω<sup>''k''</sup>(''M'') be the [[real number|real]] [[vector space]] of smooth [[differential form]]s of degree ''k'' on ''M''. The de Rham complex is the sequence of [[differential operator]]s

:<math>0\to \Omega^0(M) \xrightarrow{d_0} \Omega^1(M)\xrightarrow{d_1} \cdots\xrightarrow{d_{n-1}} \Omega^n(M)\xrightarrow{d_n} 0,</math>

where ''d<sub>k</sub>'' denotes the [[exterior derivative]] on Ω<sup>''k''</sup>(''M''). This is a [[cochain complex]] in the sense that {{nowrap|1=''d''{{sub|''k''+1}} ∘ ''d''{{sub|''k''}} = 0}} (also written {{nowrap|1=''d''{{i sup|2}} = 0}}). De Rham's theorem says that the [[singular cohomology]] of ''M'' with real coefficients is computed by the de Rham complex:

:<math>H^k(M,\mathbf{R})\cong \frac{\ker d_k}{\operatorname{im} d_{k-1}}.</math>