= Solution
Let $f:\mathbb P^1_k\to\operatorname{Spec}k$ be the structure map, a <projective morphism>. Let $D$ be the union of the two rational points $[0:1]$ and $[1:0]$. The homogeneous polynomial $t_0t_1$ defines this <effective Cartier divisor> and gives the exact sequence of <coherent sheaves>
$$
0\longrightarrow\mathcal O_{\mathbb P^1}(-2)\xrightarrow{\ t_0t_1\ }\mathcal O_{\mathbb P^1}\longrightarrow\mathcal O_D\longrightarrow0.
$$
The <twisting sheaf on projective space> $\mathcal O(-2)$ has no nonzero <global sections>: the chart calculation of Question 3(iv) now requires $a(t)=t^{-2}b(t^{-1})$, again impossible for nonzero polynomials. We also have $\Gamma(\mathbb P^1,\mathcal O)=k$ and $\Gamma(D,\mathcal O_D)=k\oplus k$.
The <direct image sheaf> under $f$ is the <module> of <global sections> on the one-point <scheme>. Consequently the induced sequence is
$$
0\longrightarrow0\longrightarrow k\xrightarrow{c\mapsto(c,c)}k\oplus k\longrightarrow0.
$$
Its last nonzero map has <cokernel> isomorphic to $k$, since $(a,b)\mapsto b-a$ has the diagonal as its <kernel>. Thus \b[the induced sequence is not exact at $f_*\mathcal O_D$], proving that <projective direct image need not preserve surjections>.
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