Question

Given a scheme $X$ with generic point p and a quasi-coherent sheaf $F$ on $X$. Viewing $X$ as a scheme over $Spec(\mathbb{Z})$, let us assume $f: X \rightarrow Spec(\mathbb{Z})$ is a proper map.

What conditions have $X$ and $F$ to satisfy, so that one can embed the $\mathbb{Z}$-module $F(X)=H^0(X,F)$ in $F_p$, respectively when is the restriction map $h: F(X) \rightarrow F_p$ injective?

Are there some mild conditions, like $X$ integral and $F$ coherent or torsion free?

Answer 1

If $X$ is integral and $F$ is torsion-free, then for any non-empty affine open subset $U$ of $X$, the canonical map $F(U)\to F_p$ is injective. So $F(X)\to F_p$ is injective. You don't need hypothesis on $X \to Spec(\mathbb Z)$. If $X$ is not necessarily reduced, then the flatness of $F$ over $X$ is also enough (same proof).