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May 13, 2020 at 20:21 comment added Pedro Juan Soto Sheaves allow one to perform the topological analog of the algebraic operation of localization. Indeed this was the intended point; that is why it is called the localization. There are many different viewpoints here are some: 1) ideals and coordinate rings, 2) function fields, and 3) germs, stalks, and sheaves. In number theory, the object is discrete, and its not entirely obvious how to apply geometric notions like locality and closeness but sheaves allow us to apply topological/geometric techniques to something non-continous. This is the easy Babylonian/Greeks/Persians way from ancient time.
May 13, 2020 at 20:20 comment added Asvin Finitely many solutions is Falting's theorem. The proof is immensely complicated! Here is an outline of the recent paper: web.stanford.edu/~aaronlan/assets/…
May 13, 2020 at 20:10 comment added Pedro Juan Soto The idea is the following: sheaves were originally invented for other more general problems. Serre had the good idea to apply this theory to algebraic geometry (which is roughly "polynomial geometry") and Grothendieck further refined sheaves to locally ringed spaces which are isomorphic to the spectrum of a ring; i.e. affine schemes. Why sheaves? The stalks of the sheaves carry local information about the "manifold" defined by the polynomial. This is exactly the point in most of geometry (for example in Differential Geometry); i.e. to study the interplay between the local and global geometry.
May 13, 2020 at 20:01 comment added Pedro Juan Soto A scheme is a sheaf; you need sheaves to define schemes. I think another text that will help you a lot is Hartshorne springer.com/gp/book/9780387902449; his advisor was Zariski who invented the Zariski topology which eventually lead to sheaves, which eventually lead to schemes. The book starts with sheaves and then defines schemes in terms of sheaves, "The Geometry of Schemes" does this as well. The language of sheaves was first used by Jean Pierre Serre to attack the problems of algebraic geometry and commutative algebra in a famous paper "Faisceaux algébriques cohérents."
May 13, 2020 at 20:00 comment added Asvin I just saw your edit: Yes, you can often construct sheaves (on the etale site!) that tell you about solutions to equations. This is the idea behind the proof of the weak Mordell-Weil theorem which effectively says that the torsion points of an elliptic curve inject into some Galois/etale cohomology group. There are really too many examples to list but here's one more - a recent paper proves that the average rank of elliptic curves over finite fields is some number by computing the cohomology of an explicit sheaf.
May 13, 2020 at 19:55 comment added Asvin Yes, they will let you solve things. Does proving that any high degree equation has only finitely many rational solutions count as "solving things"? Even if you object that it isn't very explicit, there are other ways of bounding solutions like the Chabauty method which provides very explicit bounds on how many solutions such an equation can have. Surely that counts as solving an equation!
May 13, 2020 at 19:50 comment added Asvin @ReginaldAnderson Well, even if all you wanted was to study affine schemes, you will often have to study non affine ones first - for instance the proof of Falting's theorem (or Mordell's conjecture) reduces to statements about abelian varieties and to study projective schemes, you definitely need sheaves (in the simplest case - maps to projective space are given by line bundle+sections but it goes much deeper of course). There seems to be no way around it - sheaves and schemes are crucial to algebraic number theory.
May 13, 2020 at 19:04 history answered Pedro Juan Soto CC BY-SA 4.0