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Thanks a lot! For some reason, I haven't thought of reading research papers. I will look into them, as well as Mumford's Lectures on curves on an algebraic surface!
I searched on Amazon and Springer.com for "Introduction to EGA I" but I don't think they have it. Is that the correct title? Could you please tell me where I can get it?
@Harry: Thanks a lot! I will look over what you posted. I'm actually reading Hartshorne (doing as many exercises as possible). I'm looking for another approach so I can get good intuitions from both sides (hopefully).
The proof for the case when $A=k$ (a field) is easy, which is in the 3rd paragraph of Emerton's answer. I'm interested in seeing the proof for the present case or even better, the even more general one mentioned by Emerton in the last paragraph (in the EDIT part).
Thanks! I also started to think about the local vs. global like you. In the "local case", we do actually learn something about the local property when we look at "maps in," like the map $I\to M$ (in differentiable manifolds) and $\mathrm{Spec}k[x]/(x^2) \to X$ in Algebraic geometry. I am wondering if there is any other example along these lines.
