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In Lemma 7.1.6 of his lecture notes on perfectoid spaces, Bhatt states that every complete characteristic p perfect Tate ring $A$ is uniform. In the proof he uses the Banach open mapping theorem on the Frobenius.

My problem is that the Frobenius is not a $A$-linear map and so all forms of Banach open mapping theorems that I am aware of do not apply (the closest related one that I am aware of is Theorem 6.16 in Wedhorn's notes on adic spaces).

Does anyone know a Banach open mapping theorem that applies here? Or can one prove that the inverse of Frobenius is bounded via other means? I would also be happy about a proof for $K$-Banach algebras because this is all I care about.

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    $\begingroup$ Can't you consider the target of the Frobenius (i.e., $A$) as an $A$-module via the Frobenius map? Then the map in question will become $A$-linear. You will lose the finite generation of the target as an $A$-module, but Theorem 0.1 from arxiv.org/abs/1407.5647v2 should still apply and give the claim. $\endgroup$
    – Kestutis Cesnavicius
    Commented Jun 21, 2018 at 11:02
  • $\begingroup$ Great, thanks, I wasn't aware of that trick. $\endgroup$
    – Corvin Paul
    Commented Jun 21, 2018 at 13:27

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