If we define invertible sheaf as a locally free sheaf of rank 1, which is the most common definition. I saw it is true that an invertible sheaf must be quasi-coherent. But why is it also coherent?
Update: My concerns are related to the fact that $\mathcal{O}_X$, the structure sheaf, can be NOT coherent if it is not locally-Noetherian. See Is locally free sheaf of finite rank coherent? please.
Question: "If we define invertible sheaf as a locally free sheaf of rank 1, which is the most common definition. I saw it is true that an invertible sheaf must be quasi-coherent. But why is it also coherent?"
Answer: The definition says that a sheaf $E \in Qcoh(X)$ on $(X, \mathcal{O}_X)$ is coherent iff there is an affine open cover $U_i:=Spec(A_i)$ such that the restriction $E_{U_i}$ of $E$ to $U_i$ is isomorphic to $\tilde{E}_i$ where $E_i$ is a finitely generated $A_i$-module. If $L$ is an invertible sheaf, there is an open affine cover $U_i$ with $L_{U_i}\cong \tilde{A}_i$ and $A_i$ is a finitely generated $A_i$-module.