Some questions from the paper by Scholze-Weinstein

Question

The following is from the paper by Scholze-Weinstein on moduli of $p$ divisible groups. My question is from a part of Lemma 4.1.7: If $R$ is a semiperfect ring, then the canonical map $W(R^{\flat}) \rightarrow A_{\text{cris}}(R)$ is injective.

Here, they have claimed that elements of $W_{PD,n}$, as defined can be written uniquely as a sum $\Sigma _{i \in \mathbb{Z}} [r_i]p^i$. This is not clear to me.

Secondly, from remark 4.3.9, I have two questions:

That, for the perfect case, we can assume $R$ is local, then $T_0 ^+$ defined as $W(R) \otimes O_C$, and $T_0 = T_0[1/p]$ is connected.

Secondly, we can choose $\tilde{T_0}$ a direct limit of faithfully flat finite etale algebras such that $\text{Spec} \tilde{T_0}$ is connected without any etale covers.

EDIT: I am adding some other questions that I have from the paper and two questions that I have regarding Prof. Scholze's answer:

In Lemma 4.3.4 where it is proven that there is an isomorphism of $A_{\text{cris}}(R)$ algebras : $ A_{\text{cris}}(S) \cong A_{\text{cris}}(S^{\prime}) \hat{\otimes}_{W(R^{\flat})} A_{\text{cris}}(R) $ there's a crucial step where it's been claimed that giving divided powers on $I^{\prime} + J\tilde{S^{\prime}}$ is equivalent to giving divided powers on $I^{\prime}$ and $J \tilde{S^{\prime}}$ separately. For the forward direction, why can we say that divided powers on $I^{\prime} + J\tilde{S^{\prime}}$ restrict to divided powers on $I^{\prime}$ and $J \tilde{S^{\prime}}$.

Secondly, in proposition 4.3.6, knowing that for any $x$, the sequence $0 \rightarrow \widehat{\mathcal{F}_{1,x}} \rightarrow \widehat{\mathcal{F}_{2,x}} \rightarrow \widehat{\mathcal{F}_{3,x}} \rightarrow 0$ has cohomologies killed by $p^{1/p-1} + \epsilon \forall \epsilon>0$ we want to say that $0 \rightarrow \mathcal{F}_{1}/p^n \rightarrow \mathcal{F}_{2}/p^n \rightarrow \mathcal{F}_{3}/p^n \rightarrow 0$ has cohomologies killed by $p^{2/p-1} + \epsilon$, one way to do this would be to apply UCT to the sequence w.r.t. the change of coefficients to $T^{+}/p^n$, but then we'd have to know that $\widehat{\mathcal{F}_{i,x}}/p^n = {\mathcal{F}_{i,x}}/p^n$. Can we do this over the non-noetherian ring we are working?

Regarding the answer below

Why is $R \otimes _{k} \overline{\mathbb{F}_p}$ connected?

And, finally, how can we get a countable directed system of finite etale algebras? (as countability is important in the proof of lemma 4.3.10.)

Answer 1

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 $W_{PD}$ is a subring of $W(R^\flat)[1/p]$, and the latter is exactly the set of power series $\sum_{i\gg -\infty} [r_i] p^i$ with $r_i\in R^\flat$. So elements of $W_{PD}$ have unique expressions of the given sort. To get to $W_{PD,n}$, one kills all those elements where all $r_i\in \Phi^n(J)$. Thus, one gets similar unique representations of elements of $W_{PD,n}$, where now all $r_i\in R^\flat/\Phi^n(J)$.

For the next question: There's actually a small lapsus here: One should take the tensor product not over $\mathbb Z_p$, but over $W(k)$ where $k$ is the algebraic closure of $\mathbb F_p$ in $R$; also, the tensor product was meant to be ($p$-adically) completed. Then one argues as follows: As $T_0$ is integral perfectoid, it is almost integrally closed in $T_0[1/p]$. In fact, it is integrally closed, as any almost element of $T_0$ is already in $T_0$. (This reduces to the similar assertion in $O_C$, using that $T_0$ is a completed direct sum of copies of $O_C$ (as $W(R)$ is a completed direct sum of $\mathbb Z_p$'s).) Thus, any idempotent already lies in $T_0$. By $p$-adic completeness of $T_0$, the idempotents are then the same for $T_0$ and for $T_0/p$, or its reduced quotient, which is $R\otimes_k \overline{\mathbb F}_p$. But this has no nontrivial idempotents, as $k$ is algebraically closed in $R$.

The last thing is something very general: For any ring $A$ without nontrivial idempotents, one can fix a geometric point of $\mathrm{Spec}(A)$, and take the direct limit over all finite étale $A$-algebras $A'$ with a lift of this geometric base point. This gives such an algebra $\tilde{A}$, which is a direct limit of faithfully flat finite étale $A$-algebras, and such that any further finite étale cover of $\tilde{A}$ splits (as it can be approximated over some $A'$, but then occurs itself in this direct limit).