I have often read that Grothendieck's insight was to put emphasis on studying the morphisms between schemes as opposed to just the schemes by themselves. What do we gain from this point of view? Why is it important that we study S-schemes, change of base, fibered products, and the like? Are there any specific concrete examples of this point of view in action?
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4$\begingroup$ $S$-schemes are families of schemes over $S$. This is the starting point for understanding things like moduli spaces, deformation theory, etc. $\endgroup$– Qiaochu YuanJan 24, 2016 at 6:11
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$\begingroup$ Thank you very much for this answer. A moduli space is something I can get a mental picture of, so this greatly helps. There seems to be a lot more to it though, as AreaMan's answer below seems to be of a quite different nature. Would you mind giving some more examples, or pointing me towards some references? $\endgroup$– Anton HiladoJan 24, 2016 at 8:31
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$\begingroup$ @AntonHilado If you are interested in learning scheme theory Ravi Vakil's notes are really great. I don't know another way to learn about this stuff other than getting into the dirty details. $\endgroup$– AreapersonJan 25, 2016 at 23:55
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Here are some examples of the strength of this point of view, taken from Ravi chapter 18.
A classic theorem: The space of global sections of a coherent sheaf on a projective $k$ variety (regular functions, regular differentials, for example) is finite dimensional over $k$. (The finiteness of the space of differentials is one way that genus can be defined in algebraic geometry.)
A similar result holds in the relative situation, in the following way: If $X$ is a projective $A$-scheme, where $A$ is a Noetherian ring, and $F$ is a coherent sheaf on $X$, then $\Gamma(X, F)$ is finitely generated as an $A$ module.
A useful consequence of this is: A projective and affine morphism to a locally Noetherian scheme is finite (meaning, the induced maps on coordinate rings of affine patches are integral). More generally, the pushforward along a projective morphism to a locally Noetherian scheme sends coherent sheaves to coherent sheaves.
If you like, this is the relative version of the statement that affine and projective varieties over $\operatorname{Spec} k$ are finite over $\operatorname{Spec} k$, i.e. just a bunch of points.
A nice classical geometry fact follows from this: A morphism between two projective varieties with finite fibers is finite.
The proof of this works in the relative projective space of $\operatorname{Spec}(A)$, and uses projective geometry intuition there (complement of a hyperplane being affine) to reach this conclusion by appealing to the aforementioned affine and projective maps are finite, which was also a consequence of the relative point of view.
answered Jan 24, 2016 at 5:51
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$\begingroup$ Thank you very much for your answer, although I admit I'm still trying to digest this stuff. I'll check back with Ravi's notes (don't you mean chapter 17 instead of 18? Chapter 9 seems like it would be useful too). Further examples would be most welcome. $\endgroup$ Jan 24, 2016 at 8:35
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$\begingroup$ @AntonHilado Chapter 17 deals with relative proj and relative spec, which is certainly relevant. But these results about finite generation of global sections, and the corollaries I described, are cohomological results from chapter 18. $\endgroup$ Jan 24, 2016 at 15:17