Suppose $\pi: X\to Y$ is any quasi-compact separated morphism, $\mathcal{F}$ is a quasi-coherent sheaf on $X$, and $Y$ is a quasi-compact separated $A$-scheme. Describe a natural morphism $H^i(Y,\pi_*\mathcal{F})\to H^i(X, \mathcal{F})$ extending $\Gamma(Y,\mathcal{F})\to\Gamma(X \mathcal{F})$.
We need to use Cech cohomology to prove this.
My plan was to build Cech complexes for $\pi_*\mathcal{F}$ and $\mathcal{F}$ and to create a natural morphism between these two complexes. I could start with an affine covering $\{U_i\}_i$ for $Y$ and this will give me a covering $\{f^{-1}(U_i)\}_i$. Now I can cover the covering $\{f^{-1}(U_i)\}_i$ with an affine covering $V_i$ and use this cover for the Cech complex of $\mathcal{F}$. Is this a correct approach?