# Some questions from the paper by Scholze-Weinstein

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?

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.)

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).