How should a beginner learn about arithmetic schemes (interpret this as you wish, or as a regular scheme, proper and flat over Spec(Z))? What are the most important examples of such schemes? Good references? What kind of intuition do people have for such schemes?
5 Answers
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If you are looking for good first examples, Mumford's Red Book and Eisenbud and Harris's 'Geometry of Schemes' have some good pictures and examples.
Its worth playing around with Spec(O_c), where O_c is the ring of integers in the extension of Q by the square root of c, and thinking about it as a scheme over Spec(Z). In particular, several somewhat mysterious number theory terms like 'ramified' and 'split' make geometric sense in this context.
Its also worth thinking about what the p-adics should look like as a scheme - a formal neighborhood of p in Spec(Z) (though to make this precise you need to know what formal schemes are).
Its also not a terrible idea to pick up a book on algebraic number theory and try to translate everything that is said into a geometric statement (the trick is to realize every time talk about a field, they are really talking about the ring of integers in that field).
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I think it's useful to think not only about Spec(Z) but also about Spec(R), where R is Fq[t]; or, better yet, an order in a finite extension of Fq(t). Getting fluent at flipping back and forth between Dedekind rings like this and curves over Fq will be very helpful in understanding why the "geometric" features of Spec Z are so called.
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One example I always found useful was that if you consider an elliptic curve like (the projective model of) y^2=x^3+1, then this equation gives an elliptic curve not only over the complex numbers but over any field of characteristic not 2 or 3. So in fact it's giving an elliptic curve over Spec(Z[1/6]), which will look like a scheme with a map to Spec(Z[1/6]) such that the fibre over the point corresponding to the prime ideal p>3 is just the curve over Z/pZ, and the point corresponding to the fibre over the generic point (0) is just the curve over Q. I think that checking these statements formally gave me some sort of intuition as to what was going on at the time. Note that Spec(Z[1/6]) is just the open subscheme of Spec(Z) that you get by throwing away the two closed points (2) and (3).
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Perhaps this is too special but ramification of primes in number fields is a nice motivation, here you can also draw funny pictures of curves over Spec(Z). The step from here to Dedekind schemes is immediate. I think www.renyi.hu/~szamuely/gal6-7.pdf contains a nice treatment of Dedekind schemes, but perhaps there are better sources.
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One example to think about: Spec O_K, for K a number field and O_K its ring of integers. Then the Picard group of all invertible sheaves is isomorphic to the class group.
The main point though is that for a scheme X over Spec Z, one should think of this as a family of schemes which package, for each prime p, a choice of reduction mod p of the scheme X \otimes_Z \Q. A good place to learn about this is Silverman's two books on elliptic curves, especially the chapter on Neron models in the second. It is very basic with examples and exercises.