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 $s_0_{V \cap U} =s'_{V \cap U}$?
1 Answer
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 $f$?


$\begingroup$ @Ron. The answer is still no, even with the edits. Let $\mathcal{F}$ be $\widetilde{A}\oplus \widetilde{A}$, let $\mathcal{G}$ be $\widetilde{A}$, let $\phi$ be projection onto the second factor, let $s$ be $1$ and let $s'$ be $(1/x,1)$. If you hope for something like this to be true, you typically allow modifications of $s$ and $s'$ by multiplying by powers of $f$, cf. Lemma 5.3 of Chapter II of Hartshorne's "Algebraic Geometry". $\endgroup$ May 4, 2015 at 20:20