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I am trying to understand section (3) of the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties" in detail. In particular, there is the following sentence on page two that I do not understand:

"But the map $$R_{i}^{+}[X_{1},\dots,X_{n}]/(X_{1},\dots,X_{n})^{r} \to \left(R_{i}^{+}\widehat{\otimes}_{W(\kappa)}R_{i}^{+}\right)/(\ker\theta_{i})^{r}$$ is injective, with cokernel killed by a power of $p$, where $\theta_{i} \colon R_{i}^{+}\widehat{\otimes}_{W(\kappa)}R_{i}^{+} \to R_{i}^{+}$ is the multiplication."

I would be grateful, if someone could clarify this for me please.

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  • $\begingroup$ Hi, I'm curious if you have the answer now, or is it still unresolved? I'm not an expert on this subject, so I can't answer your question, but have you tried to email Mr Scholze about your question? $\endgroup$
    – Ythyb
    Sep 26 at 12:12
  • $\begingroup$ @Ythyb It is in fact still unresolved. I did not ask Mr Scholze about this, but maybe I should! $\endgroup$
    – user141099
    Nov 3 at 14:53
  • $\begingroup$ Is there someone at your university who's working on this or related subject? If there is, try talking to them first. If you do email Mr Scholze, you would be lucky if he answered your question within a month or two because he should be busy with his lectures on Analytic Stacks together with Clausen at IHES, and an upcoming paper with Hamann and Hansen that resolves Harris-Viehmann’s conjecture for an arbitrary reductive group as well as various foundational results as you can see in Remark 1.7 (1) here: arxiv.org/pdf/2309.16505.pdf from Kieu Hieu Nguyen paper a month ago. $\endgroup$
    – Ythyb
    Nov 4 at 12:17

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