I'm currently in need an explicit formula in classical cohomology which I'm pretty sure is well known, but which I've been unable to find in the references I am aware of.

Let $X$ be a smooth manifold and let $\mathcal{U}=\{U_\alpha\}$ be a fixed open cover of $X$ such that all the finite intersections $U_{\alpha_1}\cap\cdots U_{\alpha_n}$ are contractible. Consider the following two cochain complexes:

the de Rham complex $\Omega^\bullet(X)$ of $X$

the total complex of the Cech-de Rham bicomplex $\Omega^\bullet(\mathcal{U}_\bullet)$.

The restriction of a global form on $X$ to the open sets $U_\alpha$ gives a linear map

$j: \Omega^\bullet(X) \to Tot^\bullet(\Omega^\bullet(\mathcal{U}_\bullet))$

which, if I'm not wrong here, is a injective quasi-isomorphism of cochain complexes. I've been able to prove this (if I've not made mistakes), by brute force: i.e. by showing that $j$ is bijective in cohomology. But I'd like to have a fancier proof by writing an explicit "globalization" morphism

$\pi : Tot^\bullet(\Omega^\bullet(\mathcal{U}_\bullet)) \to \Omega^\bullet(X)$

such that

$\pi j= id_{\Omega^\bullet(X)}$

$j \pi = id_{Tot^\bullet(\Omega^\bullet(\mathcal{U}_\bullet))} + [d_{tot},K]$

with $K$ some explicit morphism of graded vector spaces $Tot^\bullet(\Omega^\bullet(\mathcal{U}_\bullet)) \to Tot^\bullet(\Omega^\bullet(\mathcal{U}_\bullet))[-1]$.

I guess one should be able to build $K$ by using a partition of the unit subordinate to the cover $\mathcal{U}$, but somehow I got lost in the computation. Since I feel this should be a well known fact, I'm asking here for direct references before attempting back to write $K$ myself.