Let $X=\mathrm{Spec}(A)$ be an affine integral scheme of finite type over $\mathbb{C}$ and $\phi:\mathcal{F} \to \mathcal{G}$ be a surjective morphism of coherent sheaves on $X$. Let $f \in A$, $U:=D(f)$ the open set defined by $f$ and $s \in H^0(\mathcal{G})$. Let $s' \in \Gamma(U,\mathcal{F})$ be a section such that $\phi(s')=s|\_U$. Does there exist an open set $V \subset X$ containing $\mathrm{Supp}(s)$ and a section $s\_0 \in \Gamma(V,\mathcal{F})$ such that $\phi|\_V(s\_0)=s|\_V$ and $s\_0|\_{V \cap U} =s'|\_{V \cap U}$?



Extending a section of a coherent sheaf and homomorphism
No, that is not true as posed. Let $A=\mathbb{C}[x]$, let $f$ be $x$, let $\mathcal{F}$ be $\widetilde{A}$, and let $\mathcal{G}$ be the zero sheaf. Let $s'$ be $1/x$. Do you want to allow modifications of $s'$ and $s$ by multiplying by powers of $f$?


