This is a question posed to me in private communication by [this user](https://mathoverflow.net/users/3582/amaanush).


Given a scheme $T$, let $\Gamma (T) = Mor (T, \mathbb{A}^1)$ be the ring
of global sections. Note that there is a canonical map
$\phi : T \rightarrow Spec (\Gamma (T))$.


Is $\phi$ a closed mapping onto the image, ie. is $im\ (Z)$ a
closed subset of $im(T)$ for all closed subsets $Z$ of $T$ ?.



Is the mapping from a scheme to its global sections a closed map?
Here is a ''natural'' example as expected by Martin. Let $T$ be the projective line over ${\mathbb Z}$, minus a rational point $x\_0$ of the closed fiber at some prime $p$. Then $O(T)=\mathbb Z$ (direct computation or use Zariski's extension theorem for normal schemes), $\phi$ is just the structural morphism and is onto. Let $Z'$ be a section of the projective line passing through $x\_0$. Then $Z=Z'\cap T$ is closed in $T$, but $\phi(Z)$ is not closed in $\phi(T)$ because it is the image of 
the composition $Z\to {\rm Spec} O(Z)\to {\rm Spec} O(T)$, and as $Z$ is affine, the image is just the complementary of the closed point $p$. 


