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This question might be very elementary to someone who knows p-adic hodge theory/perfectoid stuff etc.

Recall that $\mathbb C_p = \hat{\overline{\mathbb Q_p}}$ and $\mathbb C_p^\flat$ is it's tilt. We define $A_{\mathrm{inf}} = W(\mathcal O_{\mathbb C_p^\flat})$ and there is a natural map: $$\theta: A_{\mathrm{inf}} \to \mathcal O_{\mathbb C_p}.$$ Does this map have a section (perhaps after localizing at $p$)? Similarly, one defines $B_{\mathrm{dR}}$, a complete, discretely valued ring with residue field $\mathbb C_p$ using $A_{\mathrm{inf}}$. Does the residue map $B_{\mathrm{dR}} \to \mathbb C_p$ split?

I suspect that the answer to both question is no but I am not sure how to see that.

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    $\begingroup$ Split in which sense? (As rings? as topological groups?) $B_{\mathrm{dR}}^+\to \mathbb C_p$ splits as abstract rings, but not as topological rings (I think this was proved by Colmez), and both maps split as topological abelian groups, but the first map doesn't split as abstract rings (not even modulo $p$, or after inverting $p$), for example because $A_{\mathrm{inf}}$ admits the quotient $W(\overline{\mathbb F}_p)$. $\endgroup$
    – Peter Scholze
    Commented Jul 16, 2021 at 7:00
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    $\begingroup$ As Peter says: splits in what sense? If $B_{dR}^+ \to C_p$ split in a reasonable sense, then $B_{dR}$ would be isomorphic to $B_{HT}$. Whether or not this is true was unknown for a long time after the construction of $B_{dR}$. It follows from Bloch-Kato's computations that there are representations that are Hodge-Tate but not de Rham, which eventually answered the question of whether Fontaine has merely given a very complicated construction of $B_{HT}$. $\endgroup$
    – Laurent Berger
    Commented Jul 16, 2021 at 8:15
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    $\begingroup$ I was thinking primarily as rings or topological rings but I would be interested in any reasonable notion of splitting. $\endgroup$
    – Asvin
    Commented Jul 16, 2021 at 13:40
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    $\begingroup$ @LaurentBerger That's very interesting! Peter mentioned that $B_{dR}^+ \to \mathbb C_p$ splits as abstract rings, is this not enough to conclude the isomorphism to the HT period ring (as abstract rings)? $\endgroup$
    – Asvin
    Commented Jul 16, 2021 at 13:46
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    $\begingroup$ @Asvin Yes, as abstract rings. But we want the isomorphism to be compatible with the Galois action! $\endgroup$
    – Laurent Berger
    Commented Jul 16, 2021 at 14:53

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