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I am reading stuff about Fontaine's periods rings. Let $K$ be a $p$-adic field and $C_K = \widehat{\overline{K}}$. Then $\mathcal{O}_{C_K}/p\mathcal{O}_{C_K}$ isn't perfect, otherwise $R = \varprojlim_{x \mapsto x^p} \mathcal{O}_{C_K}/p\mathcal{O}_{C_K}$ would be $\mathcal{O}_{C_K}/p\mathcal{O}_{C_K}$ itself which does not seem to be very interesting.
So my question is: what is $\mathcal{O}_{C_K}/p\mathcal{O}_{C_K}$?

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    $\begingroup$ The valuation ring $\mathcal O_{C_K}$ has value group $\mathbb Q$, so the valuation is not discrete. The maximal ideal is $\mathfrak m = (p,p^{1/2},p^{1/3}, \ldots)$, which is not finitely generated. Modding out by $p$ gives some sort of 'truncated' ring with a lot of nilpotents (e.g. $p^{1/k}$ is nilpotent for every $k \in \mathbb Z_{>0}$). This is why it's not perfect: the Frobenius is not even injective (although I think it should be surjective, because this is true on $\mathcal O_{C_K}/\mathfrak m_{\mathcal O_{C_K}}$, and the value group is $p$-divisible). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 31, 2018 at 14:14
  • $\begingroup$ Thank you for the answer @R.vanDobbendeBruyn, now all the constructions make more sense to me! $\endgroup$
    – Asdrubale Barca
    Commented Jan 31, 2018 at 15:41

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