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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] $\endgroup$ Commented May 27, 2021 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. $\endgroup$ Commented May 27, 2021 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. $\endgroup$ Commented May 28, 2021 at 16:38
  • $\begingroup$ Sorry for my ignorance, but I do not understand why you previously thought that rigid spaces would be better suited for rigid cohomology? My impression is that the formalism of rigid cohomology does not rely much on the choice of underlying theory of rigid analytic geometry? $\endgroup$ Commented Jun 2, 2022 at 13:08
  • $\begingroup$ @Z.M That's definitely a fair question, and my answer would be - because many authors seem to use it! Primarily this likely comes from the fact Le Stum's book, often a primary reference on the topic, uses that language. I certainly imagine the theory can be developed in other formalism, but especially when I was first learning that theory, it was not so clear to me. $\endgroup$ Commented Jun 2, 2022 at 14:09