Flat vector bundle

Let ${\displaystyle \pi :E\to X}$ denote a flat vector bundle, and ${\displaystyle \nabla$ :\Gamma (X,E)\to \Gamma \left(X,\Omega _{X}^{1}\otimes E\right)} be the covariant derivative associated to the flat connection on E.

Let ${\displaystyle \Omega _{X}^{*}(E)=\Omega _{X}^{*}\otimes E}$ denote the vector space (in fact a sheaf of modules over ${\displaystyle {\mathcal {O}}_{X}}$) of differential forms on X with values in E. The covariant derivative defines a degree-1 endomorphism d, the differential of ${\displaystyle \Omega _{X}^{*}(E)}$, and the flatness condition is equivalent to the property ${\displaystyle d^{2}=0}$.

In other words, the graded vector space ${\displaystyle \Omega _{X}^{*}(E)}$ is a cochain complex. Its cohomology is called the de Rham cohomology of E, or de Rham cohomology with coefficients twisted by the local coefficient system E.