Sheaf-theoretic De Rham Isomorphism
The de Rham cohomology is isomorphic to the Čech cohomology H*(U,F), where F is the sheaf of abelian groups determined by F(U) = R for all connected open sets U in M, and for open sets U and V such that U ⊂ V, the group morphism resV,U : F(V) → F(U) is given by the identity map on R, and where U is a good open cover of M (i.e. all the open sets in the open cover U are contractible to a point, and all finite intersections of sets in U are either empty or contractible to a point).
Stated another way, if M is a compact Cm+1 manifold of dimension m, then for each k≤m, there is an isomorphism
where the left-hand side is the k-th de Rham cohomology group and the right-hand side is the Čech cohomology for the constant sheaf with fibre R.
Read more about this topic: De Rham Cohomology