Let $X$ be a compact manifold and denote $H^k(X,G)$ the $k$-th cohomology group with coefficients in the abelian group $G$.
Using Cech cohomology one can prove that there is a natural isomorphism $ H^k(X,\mathbf k) \simeq H^k(X,\mathbb Z) \otimes \mathbf k$ for every field $\mathbf k$ of characteristic $0$ (see for instance the book "Hodge Theory and Complex Algebraic Geometry I", C.Voisin p 157.)
Is there a way to proove this fact without using sheaf-cohomology? I have looked around for a "Universal Coefficient Theorem" for cohomology but the only one I know relates $H^k(X,\mathbf k)$ with $H_k(X,\mathbb Z)$ and $H_{k-1}(X,\mathbb Z)$, but not with $ H^k(X,\mathbb Z) $.