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The result is false in the open case. If true, a long exact sequence would show that also $$H^i(X_{\mathrm{proet}},\hat{\mathbb Z}_p)\otimes k\to H^i(X_{\mathrm{proet}},\hat{\mathcal O}_X)$$ is an iso …

In Lemma 4.1.7, we actually assume that $R$ is f-semiperfect (i.e. a quotient of a perfect ring by a finitely generated ideal); I doubt the result is true without this assumption. Note that $W_{PD}$ i …

This specific question is probably not addressed in the literature; let's try to figure it out! Let $K$ be an algebraic extension of $\mathbb Q_p$ such that the tilt of $\widehat{K}$ is isomorphic to …

Let me add to Z. M's answer, and note that Dustin has no reason to apologize at all: What he said is literally correct. Namely, one can directly show that $L_{F/\mathbb Z}^\blacksquare$ is isomorphic …

Inductive limit topologies are always nasty to describe: Basically you can't do better than their very definition. Namely, a subset $U\subset W(R)[1/p]$ is an open neighborhood of $0$ if and only if f …

The question is local on $X$, so we may assume that $\mathcal E$ is finite free, of rank $n$, say. In that case, as also SashaP points out, the question amounts to the question whether $\mathbb A^n_X$ …

Of course, there cannot be a direct relation at the categorical level: After all, one category is $R$-linear while the other is $R^\flat$-linear (I write $R^\flat=R'$ for the tilt, as usual). On the o …

Yes, it does admit nonempty finite coproducts. If you have two prisms $(A_1,I_1)$ and $(A_2,I_2)$ with maps $R\to A_i/I_i$, you need to find the initial prism $(A,I)$ with maps from both $(A_i,I_i)$ s …

Thanks for the question! One interpretation of the conjecture is true. Let me elaborate. The following results are kind of implicit in some discussion towards the end of www.math.uni-bonn.de/people/sc …

Let me start with the second question first: The usual de Rham complex is not locally acyclic in positive degrees, in any of the topologies (analytic (= of rational subsets), étale, pro-étale, ...). …

In general, you cannot expect that for a proper smooth scheme $X$ over $\mathbb{C}_p$, you have a canonical decomposition $$ H^i_{\mathrm{\acute{e}t}}(X)\otimes \mathbb{C}_p\cong \bigoplus_j H^{i-j}( …