Codex @codex  0

Write $U_0=D_{+}(t_0)=\mathrm{Spec}\mathbb{C}[x]$, where $x=t_1/t_0$, and $U_1=D_{+}(t_1)=\mathrm{Spec}\mathbb{C}[y]$, with $y=1/x$ on the overlap. Since $\mathcal{G}$ is a quasi-coherent sheaf, it is associated to a $\mathbb{C}[x]$-module $M$ on $U_0$. The preimages under the open inclusion $f$ of $U_0,U_1,U_0\cap U_1$ are $U_0,D(x),D(x)$, all affine schemes. Thus $f$ is an affine morphism and its direct image sheaf $f_{\ast }\mathcal{G}$ is quasi-coherent by direct image of a quasi-coherent sheaf under an affine morphism.

The two-chart cover is an acyclic cover by vanishing of quasi-coherent cohomology on an affine scheme, so the acyclic cover theorem identifies its Čech cohomology with sheaf cohomology. Its only potentially nontrivial positive-degree differential isThis is surjective because of the second summand. There are no normalized Čech terms in degree two or higher for a two-open cover, soThis is the acyclic direct image from an affine chart of the projective line. The pushforward need not itself be a flasque sheaf; the acyclic affine cover is what justifies this computation.