A roadmap for understanding perfectoid spaces

Question

Perfectoid spaces are this year's subject for the Arizona Winter School (link) and, as preparation, I am currently trying to understand the subject better. There are wonderful explanatory accounts (What is a perfectoid space, What are "perfectoid spaces"?) but I would like to learn the technical details. An additional problem would be that I don't know much (anything) about rigid analytic geometry. I have found many lecture notes on the internet treating non-archimedean geometry, adic-spaces, Berkovich spaces etc but I don't know where to start learning. Moreover, I don't know which things are important and which things can be skipped at first reading. I am familiar with schemes and local fields at the least.

Is there a natural way to progress through the material?

Answer 1

I can tell from personal experience that it is possible to learn perfectoid spaces without knowing rigid geometry, just like it is possible to learn schemes or even stacks without knowing much about varieties over complex numbers. In fact, it's even possible to successfully transition to research with this approach. Of course, for the approach to be meaningful/successful you need some "mathematical maturity" (in the sense of being able to clearly distinguish easy/formal parts of the theory from the real meat); for instance, you need a good command of commutative algebra/algebraic geometry (to say the least).

If you want to pursue this, I suggest reading (more or less line by line) Wedhorn's "Adic spaces" for the basics and then (or in parallel) Scholze's "Perfectoid spaces."

Before wise elders start reprimanding me for giving such "irresponsible" advice, let me issue a couple of caveats:

1. This approach is not for everyone. Specifically, if you feel you need "motivating examples" at every step and find it difficult to swallow dry theories for the mere sake of their own intrinsic beauty, then I think you're better off studying in a more linear fashion (i.e., learning rigid geometry first).
2. You'll need to take some parts of section 2 of "Perfectoid spaces" on faith because you will not know much about Berkovich spaces (but the point is, they are not really needed!). For instance, you will need to ignore the description of the types of points on $(\mathbb{P}^1)^{ad}$ (an overrated example anyway, from my somewhat limited experience).
3. If you take this approach and want to at some later point transition to research, you have to be acutely aware of the fact that for a long time your intuition will derive from a good command of dry aspects of the theory (or from analogies with related theories that you know well) and not from examples or of concrete computations. In particular, you need to be aware of the danger of making big blunders when forming intuitions in this way about what should be true or how to prove something. Try to continuously bridge the gap as you move on.

I do agree with the others though that it may be a little bit too late for the AWS to be entirely meaningful. Even if you do not completely understand the lectures, try to isolate the contact points with the material that you've been studying in order to get something out of them.

Answer 2

Apart from all the written sources, there are six videos of Peter Scholze's lectures at the IHÉS in 2011 on Perfectoid Spaces and the Weight-Monodromy Conjecture :

http://www.dailymotion.com/video/xm4b56_perfectoid-spaces-and-the-weight-monodromy-conjecture-1-6-peter-scholze_tech

http://www.dailymotion.com/video/xmes49_perfectoid-spaces-and-the-weight-monodromy-conjecture-2-6-peter-scholze_tech

http://www.dailymotion.com/video/xmet75_perfectoid-spaces-and-the-weight-monodromy-conjecture-3-6-peter-scholze_tech

http://www.dailymotion.com/video/xmev2o_perfectoid-spaces-and-the-weight-monodromy-conjecture-4-6-peter-scholze_tech

http://www.dailymotion.com/video/xmewlx_perfectoid-spaces-and-the-weight-monodromy-conjecture-5-6-peter-scholze_tech

http://www.dailymotion.com/video/xmip3g_perfectoid-spaces-and-the-weight-monodromy-conjecture-6-6-peter-scholze_tech

Answer 3

I agree with nfdc23, notwithstanding here is some kind of 'roadmap' but I am by no means an expert.

I would begin with Brian Conrad's "Several approaches to Non-Archimedean geometry" chapter in a set of lecture notes of a previous AWS http://www.ams.org/mathscinet-getitem?mr=MR2482345 and there are many useful exercises there. Also see the second half of Tate's foreword and Berkovich's foreword to this book for quite an enlightening historical perspective.

This doesn't treat adic geometry at all, but should give a good idea of why one might want a 'rigid analytic geometry' and the many many many subtleties that arise in the theories.

For all the details in say a concrete case like the rigid projective line there is an e.g. first chapter in excellent textbook of Fresnel and van der Put http://www.ams.org/mathscinet-getitem?mr=MR2014891 and the rest of the book has a lot of things that are difficult to find elsewhere. Note this isn't just a translation of the French version, there is a lot more in it than in the 1981 book.

Then I suppose it is a matter of what you are interested in. For me it was things related to Picard--Fuchs, so the book of André was a treasure trove http://www.ams.org/mathscinet-getitem?mr=MR1978691 using the Berkovich dialect.

Then I guess for the adic case which isn't such a massive jump from Berkovich spaces, there are excellent lecture notes of Wedhorn (his webpage) and also its worth noting Huber was an extremely clear writer of articles, and for the perfectoid theory well you are lucky because Scholze is also a remarkably clear writer.