I am looking for a few "non-intuitive" counterexamples in the theory of coherent sheaves. In particular:
A ringed space $(X,\mathcal{O}_{X})$ so that $\mathcal{O}_{X}$ is not coherent over itself.
A sheaf $M$ that is finitely generated over $\mathcal{O}_{X}$ but not coherent.
I have read somewhere, but forgotten where, that if $\mathcal{O}_{X}$ is coherent over itself, then coherent and finitely generated are the same for sheaves over $\mathcal{O}_{X}$. Can anyone give me a reference or explain it to me?
Thanks.