Global section defines a map from structure sheaf

Question

Let $X$ be a smooth projective scheme over an algebraically closed field. Let $F$ be a coherent torsion-free sheaf on $X$. A global section $f$ of $F$ defines a morphism $O_X\rightarrow F$ given by: on each open set $U$, $O_X(U)\rightarrow F(U)$, $c\mapsto c.f|_U$. Is this morphism injective? When is this morphism injective?

Answer 1

Let $(X,\mathcal O_X)$ is a locally ringed space, let $\mathcal F$ be a torsion-free sheaf of $\mathcal O_X$- Modules, let $0\neq s\in \mathcal F(X)$ be a non-zero section and let $\phi=\phi_s:\mathcal O_X\to \mathcal F$ be the associated morphism.

a) If all $\mathcal O_{X,x}$ are domains, then $\phi:\mathcal O_X\to \mathcal F$ is injective, since all $\phi_x:\mathcal O_{X,x}\to \mathcal F_x, \; x\in X$ are injective by the torsion-freeness hypothesis on $\mathcal F$.
This implies that the maps $\phi(U):\mathcal O_X(U)\to \mathcal F(U)$ are injective for all open $U\subset X$.

b) Else $\phi_s$ needn't be injective:
Take for $X=\{x\}$ a single point endowed with the ring $\mathcal O_X(X)=A=\mathbb Q[\epsilon]=\mathbb Q[T]/(T^2)$ and for $\mathcal F$ take $\mathcal F(X)=A$ too.
This sheaf is torsion-free but for the section $s=\epsilon\in \mathcal F(X) $ the associated map $$\phi_s: \mathcal O_X(X)=A\to \mathcal F(X)=A:c\mapsto c\epsilon $$ is not injective since $\phi_s(\epsilon)=0$