Descent Theory

I’m going to do a change in plan.

Galois Theory: Let F be a field. In some sense the “universal” Galois group is Gal(\overline{F}/F) where \overline{F} is the algebraic closure, since given any algebraic extension K/F we have that Gal(\overline{F}/K) < Gal(\overline{F}/F). In fact, there is a bijective correspondence between subgroups of the Galois group and algebraic extensions (this is just loosely speaking to show a connection later on, I’m not being careful about finiteness and things). In this case the we have an inverse corrolation. As the fields get bigger, the groups get smaller.

Covering spaces: For suggestive notation, let’s denote Y/X to mean Y is a covering of X. Then if X has sufficiently nice conditions, we have that there is a universal cover \overline{X}/X with covering map q: \overline{X}\to X. Then we have that Aut_q(\overline{X})\cong \pi_1(X) where Aut_q(\overline{X}) is the group of “deck transformations,” i.e. the automorphisms \phi: \overline{X}\to\overline{X} such that q\circ \phi= q. Now any other cover will “sit below” the universal one, in that the covering p: Y\to X will have a factoring \overline{X}\to Y\to X. Moreover Aut_p(Y)\cong H<\pi_1(X). Just as in the Galois case, there is a bijective correspondence between conjugacy classes of subgroups of \pi_1(X) and isomorphism classes of coverings. This time in a sense it is not reversed, though it depends on how you want to look at it.

I found the similarities of these two situations very strange. There must be something deeper. All field extensions are in bijective correspondence to subgroups of the Galois group of the “largest one,” and all (iso classes of) coverings are in bijective correspondence with (up to conjugacy) subgroups of the fundamental group.

It turns out that after some hunting, there is a huge deep field called “the theory of descent” or something similar. It all looks so fascinating, but it is just too far astray from what I’m studying for me to actually learn right now. I thought I could dip a toe in or something and report back my findings, but there doesn’t seem to be any good introductions to the subject or any hope for quickly seeing some of the ideas. So, after a few days of hunting, I’m changing my plans and am going to look for something new to go on about (possibly back to the prime and localization that I built up, then left for dead?).

Actually, if anyone knows of a place to learn some of this stuff, it would be greatly appreciated if you let me know!


4 thoughts on “Descent Theory

  1. My understanding is that Descent Theory as a subject deals with how things glue together, and I usually think of it as a subfield of algebraic geometry. It does include a subject called “Galois Descent”, which is related to all this and Theorem 90 from which you take your name.

    However, my understanding is also that when you’re talking about is just called Galois Theory. I’ve definitely heard people refer to the second as the Galois Theory of Covering Spaces. For good places to learn about this, I’m not really sure beyond the stuff for specific cases. But centered here at Penn is this group, which also has a seminar. They SEEM to me to focus more on the field extensions part, but there’s some covering space stuff going on too, if you look through the papers on the first link.

    Of course, keep in mind, I’m commenting from a neighboring field, not the same one, so I might be wrong about any part of this.

  2. Thanks for the links. You renewed my hope that it really isn’t as abstract as Descent theory, so I tried searching again.

    I found precisely what I was looking for, so I won’t rehash their post on a post of my own. I’ll just link it here. Both are just special cases of a (finite?) etale cover of schemes.

    Galois descent may be one more generalization, or may not be related…at least I found a short readable introduction (by Keith Conrad) that I might try in the near future .

  3. @hilbetthm90 There is a book called “Galois groups and fundamental groups” by Tszamuley freely available online. It talks about many deep things in this direction.

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