Let $X=\operatorname{Spec}k[x]$ and $U=D(x)$. A global section of $j_!\mathcal O_U$ is a regular function on the integral scheme $U$ whose support is closed in $X$ and contained in $U$. Every nonzero regular function on $U$ has support dense in $X$, so
$$
\Gamma(X,j_!\mathcal O_U)=0.
$$
If this sheaf were <quasi-coherent sheaf>[quasi-coherent], then on the affine scheme $X$ it would be the sheaf associated with this zero module and hence would vanish. Its stalks at points of $U$ are instead $\mathcal O_{U,P}\ne0$ by part a. This contradiction proves that extension by zero need not preserve quasi-coherence.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 113 / 1 / d / Solution

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A sheaf of modules on a scheme is quasi-coherent when every affine chart $\operatorname{Spec}A$ restricts it to the sheaf associated with an $A$-module. On an affine scheme it is determined by its module of global sections.


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