MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 5 down vote accepted

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).

share|cite|improve this answer
Thanks a lot for your answer. – TonyS Mar 1 '10 at 21:03

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.