If X is a sub scheme of Y, what can we say about the map O(Y) to O(X). Conversely, what conditions do we need on a map O(Y) to O(X) to ensure that X is a sub scheme of Y.
The natural guess is that (with intuition from Algebraic Sets/Varieties) O(Y) to O(X) should be surjective. However, I'm unable to prove that. I tried using the exactness property of of the Hom(-,T) (over Abelian groups) functor, however, still I'm not able to get a necessary and sufficient condition.
Any thoughts/comments will be helpful.