# Prerequisites for Rigid Geometry

I'm very interested in learning modern Rigid Geometry, but I'm not sure about the prerequisites for learning it.

I am a 3rd undergrad student majoring in Algebra and Topology. According to you, what is the ideal background I should have before diving into Rigid Geometry? What are some great resources (books, videos,..) about this topic that you would prefer to me?

Thanks

A bit about my current knowledge: Commutative Algebra up to Michale Atiyah, Algebraic Topology up to Hatcher, Fields and Galois theory up to Steven Roman, Manifolds (I haven't learned anything about smooth manifolds, I just finished Topological Manifold by John Lee), a bit about Homological Algebra and nothing about Algebraic Geometry.

• “3rd year undergrad” is unfortunately difficult to assess as a measure of what you know (it depends on your country, your institution, your own curiosity, etc). Nonetheless, I’d recommend commutative algebra (including Galois theory; Hensel’s lemma looks like another good starting point, but you’ll probably need the whole thing eventually), some algebraic number theory (about valuations, ramifications, local fields). It could be a good idea to get a grasp of differential geometry as a “toy model” (for charts, sheaves, cohomology), and then maybe some algebraic geometry? Nov 25, 2021 at 8:26
• @Mindlack Thanks for your answer, I just added my current background. According to you, what should I learn next in order to fulfil the gap? Nov 25, 2021 at 8:35
• What does “up to” mean in this case (ie have you studied all of Hatcher, Atiyah, etc, or is that in progress)? I don’t know enough about most of the other books. Smooth manifolds may have been better than topological manifolds (de Rham cohomology is a useful viewpoint), but Hatcher covered a cohomology already. How much do you know about local fields (they’re important)? You may want to learn the basics of category theory (although one might argue that it’ll come with the rest). I’m not sure how useful algebraic geometry actually is (more for the viewpoint than the precise results?). Nov 25, 2021 at 9:20
• @Mindlack Thanks for your reply! Upto in this case means I've studied the book. The only thing in progress is Homological Algebra, and I also started to learn Algbraic Geometry for a few days. For category theory, I think I know the basics (which Lang has covered in his famous Algebra book). Unfortunately, I've never touched anything in Algebraic Number theory, and really don't know anything about local fields. Would you recommend me some resources on local fields? Nov 25, 2021 at 11:14
• These are references for the content more than the pedagogy – what I mean is, they might be hard. The first two parts of Serre, Local fields should be quite sufficient. The first chapter of Cassels and Frohlich’s Algebraic Number Theory, or the second chapter of Neukrich’s ANT. After that, I think you get into more of an “uncharted” territory where your only option is basically to find a text – or preferably a few different ones – on the topic (see the references on ncatlab for instance) and work out from there what else you need to understand. Nov 25, 2021 at 13:09

With your comment in mind, working towards perfectoid spaces, I can make a few suggestions.

The best source I can suggest are the Berkley lecture notes Lecture 1-7, maybe also 8-10 to actually see some strong tools around this theory.

Now while there is nothing stopping you from jumping right into these notes and trying to digest the material, I would add a small caveat. Just as you can immediately start learning about algebra (field extensions, Galois theory...) without learning linear algebra before, you can also learn about rigid geometry without prior knowledge of algebraic geometry.

The problem that you might be facing is that often things might seem unmotivated, lacking intuition and difficult to understand. Things like sheaf theory are important, if not concretely, then at the very least on a conceptual level. These ideas came to life initially in algebraic geometry, when people introduced schemes, étale morphisms/topology etc. To a certain degree rigid geometry (whatever that is supposed to mean) was developed after algebraic geometry, in parts it aims to mimic it.

Thus let me suggest that it is very helpful to first learn about algebraic geometry (along the lines of Hartshorne Chapter II or Götz and Wedhorn's Algebraic geometry and other possible sources) and also something about non-archimedean fields/algebraic number theory (along the lines of Serre's Local fields). Certainly there is no need to learn everything up to the last detail, but working with these objects will give you an indispensible intuition.

Afterwards you can think, if you first want to read about rigid-analytic spaces à la Tate (Bosch's Lectures on Formal and Rigid Geometry is a great resource here) or jump right into action in the aforementioned source. I personally did the latter and didn't find it to be such a big hassle. A great supplement to the Berkeley notes are the notes from this Number theory seminar. You can of course also read about perfectoid spaces straight from the source in the survey paper, but I found this to be more technical. In any case it is a great back-up source. Also you can find more references in these study group notes.

• Thanks for your great answer! But is it necessary to have a prior basic background in algebraic number theory before jumping into Serre's Local fields? Nov 28, 2021 at 8:30
• I will leave this up to you. In any case jump into it and see how you do! Nov 28, 2021 at 22:04
• Just to comment, this is very much from the perspective (the 'start with the Berkeley lectures part') that 'modern rigid geometry=things that show up in the work of Scholze'. While I won't comment on whether that's the goal or not, that's up to you, I would say that is rigid geometry from an end-user perspective mostly (with the exception of foundational work by people like Hansen/Scholze). If you're interested in rigid geometry for rigid geometry's sake, I would definitely suggest looking at some other sources. In particular, Bosch's lecutres suggested by Notone are a very good place to Dec 21, 2021 at 6:19
• start, with follow-up by looking at the papers of Bosch--Luhtkebohmmert--Raynaud, Berkovich's two published books (Spectral theory... and Etale cohomology...), Huber's book on etale cohomology adic spaces, and Fujiwara--Kato's Foundations of Rigid Geometry I (although maybe even II will be out by the time you get to them...). This will give you a more broad taste of rigid geometry. Dec 21, 2021 at 6:22
• I agree with your comment! Well, maybe except for the work of Fujiwara--Kato. I found it as beginner quite impenetrable (which is the price for being very exhaustive I guess). Dec 22, 2021 at 16:58