= Acyclic direct image from an affine chart of the projective line
Let $j:\mathbb A^1_k\hookrightarrow\mathbb P^1_k$ be a standard affine chart and $\mathcal G=\widetilde M$ 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 $H^i(\mathbb P^1_k,j_*\mathcal G)=0$ for $i>0$. This does not assert that the direct image itself is flasque.
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