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In Huber's theory of adic spaces, as utilized in Scholze's theory of perfectoid spaces, we consider the situation of a Huber (or f-adic) ring $A$, which is a topological ring with an open subring $A_0$ that is $I$-adic for a finitely generated ideal $I\subset A_0$, and a Huber (or affinoid) pair $(A,A^+)$ where $A$ is Huber and $A^+$ is an integrally closed open subring. We then take the space

$$\mathrm{Spa}(A,A^+)=\{\text{continuous valuations $x$ on $A$ such that $|f(x)|\le1$ for all $f\in A^+$}\}.$$

My question is, why do we want this particular definition of a Huber ring? Or in other words, what is lost (or gained but not wanted) if we allow $A$ to be any topological ring and $A^+$ an integrally closed open subring? I understand that a "motivating example" is the ring $k\langle T_1,\dots,T_n\rangle$ from classical rigid geometry, which has these properties, but is there anything else in particular?

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    $\begingroup$ As you may know, there are many other theories of p-adic geometry: Bourbaki's "woobly" geometry, Tate's rigid geometry, Raynaud's approach through formal schemes, and Berkovich spaces. The category of adic spaces (amazingly) provides a natural home for these theories; in particular, this category contains the category of schemes, formal schemes, and their generic fibers (which are realizable as rigid analytic spaces and Berkovich spaces) as full subcategories. In my opinion, if one allows A to be any topolgoical ring, one would lose the intuition for how adic spaces relate to formal geometry. $\endgroup$
    – Jackson Morrow
    Commented Nov 3, 2017 at 20:14
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    $\begingroup$ We want this definition because it is useful to prove real theorems, such as reconstructing $A^+$ from it (an "adic Nullstellensatz"): $A^+$ is precisely the set of elements $a \in A$ such that $v(a) \le 1$ for all $v \in {\rm{Spa}}(A,A^+)$. The details of the proofs in Huber's foundational papers (such as his initial two, "Continuous valuations..." which shows ${\rm{Spa}}(A,A^0)$ is a sober space realizing the Tate topos for $k$-affinoid $A$, and the one linking it to formal schemes) with upon which the entire theory depends convey very nicely why the definitions are chosen. $\endgroup$
    – nfdc23
    Commented Nov 3, 2017 at 22:32
  • $\begingroup$ Although @nfdc23 's comment answers the question, let me elaborate further for whoever might be interested in the future. For a subring $A^+$ of $A$ to be a ring of integral elements, we require $A^+$ to be open, integrally closed in $A$ and $A^+\subset A^o$. From the latter assumptions, only the last one is necessary. The other two are needed to assure that we can recover $A^+$ from $Spa(A,A^+)$ and $A$ using the "adic Nullstellensatz", just like the comment above says. $\endgroup$
    – RumDiary
    Commented Jan 5, 2021 at 17:55

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