Let F be a non zero coherent sheaf on a non affine scheme X; can F(X) be zero?

Answer 1

i skimmed through EGA I and II and didn't find anything that looked very helpful. maybe it would help to try and use some kind of cohomological trick, like serre duality, but, honestly, i'm not even sure whether i expect this to be true or false. it's a good question. if i think of anything, i'll come back and edit my post.

Edit: ok, i was overthinking that one. let n be a strictly negative integer, let S be a noetherian graded ring, let X = Proj S and let F be the sheaf associated to the graded S-module S(n) (in standard notation, F = O_X(n)). you can see that F is coherent, and you can check that F(X)=0.