= Infinite negative-twist sum with zero global sections
{title2=$\bigoplus_{n\geq1}\mathcal O_{\mathbb P^1}(-1)$}
On the <projective line>, an infinite direct sum of copies of $\mathcal O(-1)$ is a <quasi-coherent sheaf> but not a <coherent sheaf>: at any point its residue-vector-space dimension is infinite. Its <global sections> vanish because each summand has no sections, and sections commute with <direct sums> for the finite two-chart equalizer. Its <direct image sheaf> on $\operatorname{Spec}k$ is consequently zero and coherent. This shows that a nonaffine direct image can lose information needed to detect coherence.
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