I'm trying to solve following problem:
Let $X$ be a noetherian scheme, $U$ an open subspace of $X$, $\mathcal F \in Qcoh(X), \mathcal G\in Coh(U), \mathcal G \subset \mathcal F|_{U},$ then there is a sheaf $ \mathcal G'\in Coh(X), \mathcal G' \subset \mathcal F,$ such that $\mathcal G'|_{U}=\mathcal G.$
I proved that there is a subsheaf of $\mathcal F$ $-$ $\mathcal H\in Qcoh(X),$ such that $\mathcal H|_{U}=\mathcal G.$ I want to say further that any quasicoherent sheaf is direct limit("union") of its coherent subsheaves, so $\mathcal G$ is union of finite number of them since X is noetherian, and this union will be a coherent sheaf on X. But I can't prove it rigorously.(Namely what is union? and why can I choose finite number of this sheaves?)
Best Answer
Your argument is ok. Union means sum or equally colimit here. It is a fact that every quasi-coherent sheaf on a quasi-compact quasi-separated is the union of its quasi-coherent subsheaves of finite type. Noetherian schemes are quasi-compact and quasi-separated, and "finite type = coherent" here. You can find all this in EGA I (1970), Section 6.9.