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Let $k$ be a perfectoid field of zero characteristic. Recall that a Tate $k$-algebra is called uniform if the set of power-bounded elements is bounded. Let $(A, A^+)$ be a uniform complete affinoid $k$-algebra such that $Spa(A, A^+)$ has a cover by rational subsets which are perfectoid. Is $A$ perfectoid?

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  • $\begingroup$ This is an open problem, even if you assume that $A$ is sheafy too. $\endgroup$
    – David Hansen
    Commented Feb 26, 2019 at 21:02

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