Example of a non quasi coherent sheaf whose global section does not vanish.

Question

Can someone give me an example of a non quasi-coherent sheaf on an affine scheme whose global section doesn't vanish ?

Actually the only examples that I can think of are in a two point affine space consisting of one closed and one generic point.

Thanks in advance!!

Do you want an example that is still a sheaf of $\mathcal{O}_X$-modules, or would a sheaf of abelian groups be okay? For the latter, The constant sheaf $\underline{\mathbf{Z}}$ on $X = \operatorname{Spec}(\mathbf{F}_p)$ has global sections $\Gamma(X,\underline{\mathbf{Z}}) = \mathbf{Z}$, but $\mathbf{Z}$ cannot be an $\mathbf{F}_p$-module. — Takumi Murayama, Mar 04 '17 at 20:58

score 3 · Answer 1 · edited Apr 13 '17 at 12:19

I have shown in this post that for the morphism $f:X\to S$ defined by $$f=\coprod_{n\in \mathbb N} \operatorname {Id_{\operatorname {Spec}(\mathbb Z)}}:X=\coprod_{n\in \mathbb N} \operatorname {Spec}(\mathbb Z)\to S=\operatorname {Spec}(\mathbb Z)$$ the direct image sheaf $f_*(\mathcal O_X)$ is a non quasi-coherent $\mathcal O_S$-Module.
Since $\Gamma(S,f_*(\mathcal O_X))=\Gamma(X,\mathcal O_X)=\prod _{n\in \mathbb N} \mathbb Z$ is non-zero, the sheaf $f_*(\mathcal O_X)$ is an example of what you require.

Example of a non quasi coherent sheaf whose global section does not vanish.

1 Answers1