ag.algebraic geometry - Is the mapping from a scheme to its global sections a closed map?

[Added: I misread the question, and in fact this answer does not answer the OP's question,
but rather the following question: is $phi(T)$ closed in Spec $Gamma(T)$, which is
a different question. Probably the upvotes can be attributed to the link to the stacks project!]

If $T$ is quasi-affine (i.e. admits an open immersion into affine space),
then the map $phi$ is an open immersion, and in fact Spec $Gamma(T)$
is the initial object in the category of affine schemes containing $T$
as an open subscheme.

In particular, in this case $phi$ has closed image
if and only if and only if $T$ is in fact affine. [Added: As Qing Liu points out in a comment below, in this quasi-affine situation, $phi$ is in fact a closed map onto its image.]

Thus if we take $T$ to be ${mathbb A}^2_k setminus {0}$ for some
field $k$, i.e. affine $2$-space with the origin removed,
then we get an example of $T$ where this map is open with non-closed
image (since this $T$ is quasi-affine but not affine). Note that
Spec $Gamma(T) = {mathbb A^2}_k$.

(This is a geometric analogue of Qing Lui's more arithmetic example;
what both have in common is that a closed point was removed from a 2-dimensional
affine scheme, so as to make a quasi-affine scheme that is not affine.[Added: I also misread Qing Liu's example; my remark would apply to the affine line over ${mathbb Z}$ with a closed point removed; Qing's example is more complicated, since it is actually dealing with the OP's question. One can make a geometric analogue of Qing's example by deleting a closed point from ${mathbb A}^1times {mathbb P}^1$; more geometrically still, remove one of the lines of a ruling from a projective quadric surface, and then remove an additional point.])

EDIT: In the definition of quasi-affine, one should also require that $T$
be quasi-compact. (The stacks project
is a terrific resource for these foundational definitions in scheme theory,
particularly with regard to finiteness and separation issues.)

Note that if $T$ is any quasi-compact scheme, then the map $T to$ Spec $Gamma(T)$ has dense image. (If $f in Gamma(T)$ and $D(f)$ is the usual affine open in Spec $Gamma(T)$,
i.e. Spec $Gamma(T)_f$, then if $phi^{-1}(D(f))$ is empty, it must be
that $f$ is locally nilpotent on $T$. Since $T$ is quasi-compact this implies
that $f$ is actually nilpotent, and hence that $D(f)$ is empty.) As Martin notes
in his answer, this is similarly true if $T$ is reduced.

It need not be true if $T$ is non-reduced and non-quasi-compact (since $T$
may then admit locally nilpotent sections of $mathcal O_T$ that are not
globally nilpotent, e.g. $T = coprod_n$ Spec $k[x]/(x^n)$).

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