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Recently in a seminar the following question was raised and, despite my familiarity with theory, I couldn't come up with a good answer:

Are there any good reasons to use Tate's theory of rigid-analytic spaces, given that Huber's theory of adic spaces seems to be superior in all regards?

The only possible advantage I can think of is that of simplicity — only having classical points to worry about may be conceptually simpler. This would be similar to treatments of classical algebraic geometry using maximal spectra (as done e.g. by Milne), although having to work with the G-topology seems to offset any pedagogical benefit to me.

For some time I also thought the development of rigid cohomology is where rigid spaces would be advantageous, but as discussed in Lazda and Pál, "Rigid Cohomology over Laurent Series Fields", rigid cohomology can be, and for their purposes has to be, developed with adic generic fibers instead.

For contrast, let me mention that I am aware of some ways in which say Berkovich spaces have advantages despite also being subsumed by adic spaces, coming from their "Euclidean" nature, for instance the theory of integration on them, the theory of skeleta or some relations to tropical geometry. Are there any contexts like that where classical rigid varieties shine?

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  • $\begingroup$ [I changed "if" to "since" in the title. Of course I'm ok with you reverting back the edit, if you deem appropriate] $\endgroup$
    – Qfwfq
    May 27 at 21:50
  • $\begingroup$ @Qfwfq Thank you, I was actually unsure about exactly how it should be written. "since" does sound more natural in hindsight. $\endgroup$
    – Wojowu
    May 27 at 23:54
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    $\begingroup$ In my experience, for lots of applications of rigid geometry, e.g. to eigenvarieties, p-adic automorphic forms, Galois deformation spaces..., there's no real advantage to switching over to adic spaces. However, anything involving etale cohomology is radically easier in the adic space language. $\endgroup$ May 28 at 16:38
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There are two questions here: which version of the theory is easiest to get off the ground axiomatically, and which version is more convenient to work with in applications?

For the first question, it's very much a matter of taste. Adic spaces are genuinely topological spaces, whereas Tate's G-topological spaces aren't; but there are many more, and weirder, points in them whose geometric significance takes some getting used to.

At risk of treading on some toes, I'm going to point out that Huber's adic spaces were available in the literature for at least 20 years before they started to become really popular. If there were a decisive advantage to setting up the theory in terms of adic spectra rather than G-topologies, then number theorists would have abandoned Tate's theory en masse in 1995 or so; and they didn't. The benefits offered by Huber's foundations weren't persuasive enough to to outweigh the "first-mover advantage" given by 30 years' worth of literature written in Tate's language. Adic spaces caught on because Scholze showed they could be used to do radically new things that were impossible in Tate's rigid geometry -- not because they allowed you to re-prove or re-visualize existing theorems in nicer ways.

As for the second question, I think the correct answer is "both". There are some (mostly younger) mathematicians who, whenever rigid spaces are mentioned, smile indulgently at the folly of their elders and assume that anything written in this language is obsolete or misguided, a bit like teenagers laughing at their parents' CD collection. This is a misconception, since rigid spaces over a nonarchimedean field K can be identified with a subcategory of adic spaces over K, and a rigid space and its corresponding adic space have the same sheaf theory (equivalent as topoi). Hence, when you want to apply p-adic analytic geometry to actually do something, it very often doesn't matter whether you write $Max(A)$ or $Spa(A, A^+)$ -- their underlying sets are hugely different, but that's usually not relevant if you're writing a research paper as opposed to a textbook. Indeed, in recent literature it seems to be quite common to simply redefine "rigid space over K" to mean an adic space which is locally of finite type over K. So the large corpus of work written in Tate's language remains useful, and a working number theorist nowadays who isn't familiar with the older language is at risk of needlessly reinventing the wheel. [I'm sure Wojowu already knows everything I've written in this paragraph, but I'm putting it in for the benefit of other readers of this question.]

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  • $\begingroup$ Thanks for the answer. This is an interesting pragmatic point, and does answer a somewhat implicit question whether it is still worth it to learn the theory, or at least the language, of rigid spaces. $\endgroup$
    – Wojowu
    May 28 at 13:30
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My opinion (knowing that this is somehow a matter of taste) is that the rigid-analytic viewpoint is more or less obsolete, and that one should better use Berkovich, Huber or Raynaud instead (each of them having its advantages).

Indeed, working with rigid spaces is really not convenient, even if the points are simpler. Indeed when you want to glue things, you have to use the G-topology, but proving purely in the rigid setting that something is G-local is, to the best of my knowledge, almost impossible. Usually one does that by proving that it is local in the usual sense on adic spaces or on Berkovich spaces, or that it is Zariski-local on the special fibres while looking on formal schemes (Raynaud’s approach).

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    $\begingroup$ Thanks for the answer. This is definitely not a reply I was hoping for, but I suppose it's good to see a more senior researcher share the sentiment. I wonder if anyone will come up with a counter-answer :) $\endgroup$
    – Wojowu
    May 26 at 21:24
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    $\begingroup$ Yes, this would be interesting to have other (possibly totally opposite) viewpoints. I mainly work with Berkovich spaces, so I might be considered as having a “conflict of interest” in this discussion :-) $\endgroup$ May 27 at 20:27

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