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.)