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Classically, formal schemes were invented to study completions of schemes along closed subschemes. Eventually, people started using them for more arithmetical reasons. (I.e., to study non-archimedean geometry.) Now, in modern non-archimedean geometry, a lot of progress has been made in the framework of adic spaces. This leads to the question:

Can we see the completion of a scheme along a closed subscheme as an adic space? If so, what are the (dis)advantages of doing so?

(To be precise: this question is not about the advantages or disadvantages of doing non-archimedean geometry using formal schemes or adic spaces. Rather, it's about using adic spaces to understand "classical" "geometric" questions.)

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    $\begingroup$ Yes, reasonable formal schemes embed fully faithfully into adic spaces. This is in Huber's original article. $\endgroup$
    – Satan's Minion
    Commented Feb 14, 2023 at 13:50
  • $\begingroup$ Dear @Satan'sMinion, I'm aware of this. What I'm not aware of is if there's any advantage or disadvantage of seeing this completion as an adic space. $\endgroup$
    – Gabriel
    Commented Feb 14, 2023 at 14:11

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