I was going over P. Scholze’s paper on $ p$ adic Hodge Theory for rigid analytic varieties.
Corollary 6.13 says that the “$ B_{\rm dR}$ version” of the Poincaré Lemma for de Rhamétale cohomology follows directly from Proposition 6.10, for smooth and proper rigid analytic varieties.
Prop. 6.10 is a description of the sheaf $ (\mathcal{O}\mathbb{B}_{\rm dR}^+)_X$ proétale locally on $ X$ as $ \mathbb{B}_{dR}^+[\![t_1,\ldots,t_n]\!]$ for local sections $ t_1,\ldots,t_n$ of $ \mathcal{O}\mathbb{B}_{\rm dR}^+$ .
Question 1: I don’t quite get the implication Prop. 6.10 $ \Rightarrow$ Cor. 6.13.
Could anyone who’s gone over the paper please clarify it for me? It should be something trivial along the lines: ” proétale locally on $ X$ the de Rham complex looks like this, hence the augmentation from $ \mathbb{B}_{dR}^+[0]$ is a quasiisomorphism”.
I can see that by smoothness the de Rham complex is, proétale locally on $ X$
$ $ (\mathcal{O}\mathbb{B}_{dR}^+)_X\otimes_{\mathcal{O}_X}DR: 0\to\mathbb{B}_{dR}^+[\![t_1,\ldots,t_n]\!]\to\bigoplus_{a=1}^n\mathbb{B}_{dR}^+[\![t_1,\ldots,t_n]\!]\text{d}t_a\to \bigoplus_{a<b}^n\mathbb{B}_{dR}^+[\![t_1,\ldots,t_n]\!]\text{d}t_a\wedge \text{d}t_b\to\cdots\to\mathbb{B}_{dR}^+[\![t_1,\ldots,t_n]\!]\text{d}t_1\wedge\ldots\wedge\text{d}t_n\to 0$ $
but why is $ \mathbb{B}_{dR}^+[0]\to DR$ a quasiisomorphism?
There must be more, because locally in the topology generated by rational subsets on $ X$ , the de Rham complex of $ X$ (let’s say $ X$ defined over $ \mathbf{Q}_p)$ is
$ $ DR: 0\to\mathbf{Q}_{p}[\![t_1,\ldots,t_n]\!]\to\bigoplus_{a=1}^n\mathbf{Q}_{p}[\![t_1,\ldots,t_n]\!]\text{d}t_a\to \bigoplus_{a<b}^n\mathbf{Q}_{p}[\![t_1,\ldots,t_n]\!]\text{d}t_a\wedge \text{d}t_b\to\cdots\to\mathbf{Q}_{p}[\![t_1,\ldots,t_n]\!]\text{d}t_1\wedge\ldots\wedge\text{d}t_n\to 0$ $
but:
Question 2: if $ \mathbf{Q}_p$ is the constant sheaf in the topology generated by rational subsets on $ X$ , how can the augmentation $ \mathbf{Q}_p[0]\to DR$ , be/not be a quasiisomorphism?
Remark. Morally, it seems the augmentation $ \mathbf{Q}_p[0]\to DR$ becomes a quasiisomorphism “after base change” to $ \mathbb{B}_{\rm dR}^+$ . Of course, this doesn’t make sense, since $ \mathbb{B}_{dR}^+$ is a sheaf in the proétale topology on $ X$ , while $ \mathbf{Q}_p$ should is a (constant) sheaf in the topology generated by rational subsets on $ X$ , in Question 2.
I would expect, for Question 2, an answer along the following lines:

the augmentation $ \mathbf{Q}_p[0]\to DR$ , with $ $ DR = 0\to\mathcal{O}_X\to\Omega^1_{X/\mathbf{Q}_p}\to\cdots\to\Omega^{\text{dim}(X)}_{X/\mathbf{Q}_p}\to 0$ $ as a map of complexes of abelian sheaves on the site generated by rational subsets on $ X$ , is a quasiisomorphism.

the augmentation $ \mathbf{Q}_p[0]\to DR$ , with $ $ DR = 0\to\mathcal{O}_X\to\Omega^1_{X/\mathbf{Q}_p}\to\cdots\to\Omega^{\text{dim}(X)}_{X/\mathbf{Q}_p}\to 0$ $ as a map of complexes of abelian sheaves on the étale/proétale sites on $ X$ , is not a quasiisomorphism, essentially for cohomological dimension reasons.

to fix the issue at the previous point (and since we care about étale cohomology and not cohomology wrto the topology generated by rational subsets, we do want to fix this issue) we need period sheaves.