Direct image of a quasi-coherent sheaf under an affine morphism

For an affine morphism $f:X\to Y$, the direct image of every quasi-coherent sheaf on $X$ is quasi-coherent. On affine charts this is the fact that restriction of scalars turns the module sheaf $\tilde{M}$ into the module sheaf associated with the same underlying module over the target ring.

Let $j:{\mathbb{A}}_k^1\hookrightarrow {\mathbb{P}}_k^1$ be a standard affine chart and $\mathcal{G}=\tilde{M}$ a quasi-coherent sheaf there. Its direct image is quasi-coherent because $j$ is an affine morphism. The standard two-chart acyclic cover has Čech cochain complex $M\oplus M_x\to M_x$ in degrees zero and one, with differential $(m,n)\mapsto n-m/1$. Surjectivity gives $H^i({\mathbb{P}}_k^1,j_{\ast }\mathcal{G})=0$ for $i>0$. This does not assert that the direct image itself is flasque.