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A scheme is a ringed space locally isomorphic to an affine scheme. An affine scheme is the spectrum of a commutative unital ring together with a structure sheaf.

What are examples of "interesting" affine schemes wich are not (finitely generated) $k$-algebras? Put differently, the definition of a scheme seems very reasonable, but why not take an affine scheme to be something more handy?

I'd be most interested in operations one can perform on schemes modelled on spectra of (f.g.) $k$-algebras whose result is a scheme not modelled on a (f.g.) $k$-algebra. In particular something starting with finitely generated $k$-algebras (truely geometric objects) and going to something which isn't even modelled on spectra of $k$-algebras would be cool.

A number theoretic example would only be "interesting" with an explanation why it is advantageous to consider it in the context of schemes. An advantage would be, for example, a scheme theoretic theorem which may be applied to both, the number theoretic and the geometric context, and yields non-trivial results in both cases.

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  • $\begingroup$ Boy, there are just so many ways to answer this question. Perhaps one of the most obvious examples of your last desire (a scheme theorem which has non-trivial applications to the classical and geometric setting) is that the degree of morphism of curves can be computed by summing up the local degrees. In classical geometry, say even in the land of Riemann surfaces this manifests itself with the formula $\displaystyle \sum_{x\in f^{-1}(y)}e_x=n$ and in number theory gives the formula $\displaystyle \sum_\mathfrak{p}e_\mathfrak{p}f_\mathfrak{p}=n$. Does that type of thing make you happy? $\endgroup$ – Alex Youcis Mar 29 '14 at 12:24
  • $\begingroup$ Another obvious one would be that the Picard group of a scheme, the group of invertible line bundles, gives you the normal Picard group in the classical setting (e.g. the group of holomorphic line bundles on a Riemann surface) and the class group of a number ring in the number theoretic setting. I don't know if these are the type of things you're interested in though. $\endgroup$ – Alex Youcis Mar 29 '14 at 12:26
  • $\begingroup$ Your first example makes me happy. What is the significance (and name) of the number theoretic formula? And from which scheme theoretic argument can I conclude both? $\endgroup$ – jds Mar 29 '14 at 12:40
  • $\begingroup$ I don't know if the formula has a specific name--people usually just say "sum of the ef's are n". But it's a pivotal formula in number theory. Many arguments are hinged on this basic fact. The number theory statement is theorem 3.34 here: <jmilne.org/math/CourseNotes/ANT.pdf> The general scheme theory fact is exercise 17.4.D(b) here: <math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf> In fact, if you want to use some ever-so-slightly more sophisticated machinery, you can easily prove the second part of that NT theorem about Galois extensions in the context of schemes. $\endgroup$ – Alex Youcis Mar 29 '14 at 12:45
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    $\begingroup$ I'd rather wait. I would hate to prematurely end this question. Perhaps someone else has a more enlightening answer--there are some really smart people on here :) $\endgroup$ – Alex Youcis Mar 29 '14 at 13:02
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This question is a little broad. There are a huge number of theorems about schemes, and number theorists use them all the time. Open any of Mazur's papers from the 70's, any of Ribet's papers, any of Gross's papers from the 80's, ... .

One example that I find interesting is that the theory of integral models of curves over a number field is parallel to the theory of surfaces over a field (because a curve over Spec $\mathcal O_K$ is two-dimensional as a scheme), and one has the theory of minimal models, Castelnuovo's criterion, etc., as tools for analyzing the integral models, in complete parallel to the theory of surfaces.


Regarding constructions that pass from finite-type $k$-schemes to more general schemes, note that there are many results saying that various processes preserve being finite-type, etc., so it is not completely trivial to escape the world of finite-type $k$-schemes if that is where you start.

One way is as follows: given $X \to S$ a morphism of finite-type $k$-schemes, and $s \in S$ a point, one can form the base-change $X' \to $ Spec $\mathcal O_{s,S}$ over the canonical morphism Spec $\mathcal O_{s,S} \to S$, where $\mathcal O_{s,S}$ is the local ring of $S$ at $s$.

The base-change $X'$ is finite-type over the local ring $\mathcal O_{s,S}$, but is typically not finite-type over $k$.

One can also make the analogous construction, but with the completed local ring $\widehat{\mathcal O_{s,S}}$.

This is a standard manner by which one leaves the world of finite-type schemes over a field. It is convenient for studying properties of the original morphism locally around the point $s$. Note that Spec of a local ring is typically not Jacobson, so from a technical point of view, this puts one in a scheme-theoretic context which is pretty far from that of finite-type schemes over a field.

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