Let $X$ be a scheme and $\mathcal{F}$ an $\mathcal{O}_{X}$-module. Suppose that there exists an open affine covering $\{U_{i}\}_{i\in I}$ of $X$ such that $\mathcal{F}|_{U_{i}}\cong \widetilde{M_{i}}$ for some finitely generated $\mathcal{O}_{X}(U_{i})$-module $M_{i}$. Let $f\in \mathcal{F}(X)$ be a global section.
Now consider the quotiënt sheaf $\mathcal{G}:=\mathcal{F}/(f\cdot\mathcal{O}_{X})$.
Question: Are we able to conclude that $\mathcal{F}/(f\cdot\mathcal{O}_{X})$ satisfies this same property?
Best Answer
If we do this in the affine case and take $\operatorname{spec} A=X$, then $\mathcal{F}\cong\widetilde{M}$ for $M$ a finitely generated $A-$module. A global section $f\in\mathcal{F}(X)$ corresponds to an element $m\in M$. Then $f\mathcal{O}_X$ corresponds to $(A m)^{\sim}$. Now, $M/Am$ is a quotient of a finitely generated $A-$module and hence it is finitely generated. In particular, $\mathcal{F}/f\mathcal{O}_X\cong (M/Am)^{\sim}$, which is the sheaf associated to a finitely generated $A-$module.
Really, your question is local. Take an affine open cover $\{U_i=\operatorname{spec} A_i\}_{i\in I}$ of your scheme. Then, if over $U_i$, $\mathcal{F}|_{U_i}\cong \widetilde{M_i}$ for $M_i$ and $A_i$-module, and $f\in \mathcal{F}(X)$ has $f|_{U_i}=m_i\in M_i(U_i)$, we see that $\mathcal{G}|_{U_i}\cong (M_i/A_im_i)^{\sim}$, for $M_i/A_im_i$ is finitely generated as an $A_i-$module.