Coherent sheaf

Remarks

On $\mathbb{A}^1_k = \operatorname{Spec} k[t]$ over a field $k$, the skyscraper sheaf $\widetilde{M}$ with $M = k[t]/(t)$ is coherent, since $M$ is generated by one element. On $\operatorname{Spec} \Z$, the sheaf $\widetilde{\Q}$ is quasi-coherent but not coherent. For it to be coherent, $\Q = \widetilde{\Q}(\operatorname{Spec} \Z)$ would have to be a finitely generated $\Z$-module, and it is not. If $\mathcal{F}$ is coherent on a locally Noetherian scheme, then $\mathcal{F}(U)$ is finitely generated over $A$ for every affine open $U = \operatorname{Spec} A$. On a general ringed space, Serre's definition asks that $\mathcal{F}$ is locally finitely generated and that every finitely generated subsheaf over an open set is locally finitely presented. On a locally Noetherian scheme the two definitions agree.