Let $S$ be a graded ring. I was wondering if someone could please explain me how I can interpret structure sheaf of $Proj \ S$ in terms of compatible stalks? Thank you!
Edit: This is Exercise 4.5.M. on Ravi Vakil's notes on Algebraic Geometry. I have been trying to work it out for a while with some of the comments I received. I still could not figure it out and I would appreciate an explanation/answer. Thank you!
On
I just want to say that you already know how to do this, in the sense that if you specify a sheaf on a base (or something even smaller) for the topology then you know the stalks. Then the sheaf is isomorphic to the one whose sections over $U$ are elements of $\prod_{p \in U} \mathscr{F}_p$ that are locally compatible, using the base.
Yes, see Hartshorne's book, Section II.5. It's like for affine schemes, but localizations are replaced by homogeneous localizations.