I was going over P. Scholze's paper on $p$-adic Hodge Theory for rigid analytic varieties.

This question is around the "**Poincaré Lemma**" in the paper.

Throughout, let $X$ be a proper smooth rigid analytic variety over $\mathbf{Q}_p$, of pure dimension $d$.

Corollary 6.13 says that the "$B_{\rm dR}$-version" of the Poincaré Lemma for de Rham-étale cohomology follows directly from Proposition 6.10, for smooth and proper rigid analytic varieties.

Prop. 6.10 is a description of the sheaf $(\mathcal{O}\mathbb{B}_{\rm dR}^+)_X$ pro-étale locally on $X$ as $\mathbb{B}_{dR}^+[\![t_1,\ldots,t_d]\!]$ for local sections $t_1,\ldots,t_d$ 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: " pro-étale locally on $X$ the de Rham complex looks like this, hence the augmentation from $\mathbb{B}_{dR}^+[0]$ is a quasi-isomorphism".

Remark 1.Let's make an example, and call $x := (x_1,\ldots, x_d)$, $x^{\pm\infty} := (x_1^{\pm 1/p^{\infty}},\ldots, x_d^{\pm 1/p^{\infty}})$, and "restrict" the complex $\mathcal{O}\mathbb{B}^+_{\rm dR}\otimes_{\mathcal{O}_X}\Omega^{\bullet}_X$ to a $V\to X\widehat{\otimes}_{\mathbf{Q}_p}\mathbf{C}_p$ pro-étale, with $V$ small enough to admit a finite étale map to a "perfected" torus $\text{Spa}\ \mathbf{C}_p\{x^{\pm\infty}\}$, from which we pull back coordinates $x^{\pm\infty}$. Briefly, pro-étale locally on $X\widehat{\otimes}_{\mathbf{Q}_p}\mathbf{C}_p$, the complex $\mathcal{O}\mathbb{B}^+_{\rm dR}\otimes_{\mathcal{O}_X}\Omega^{\bullet}_X$ is:$$\mathbb{B}_{\rm dR}^+[\![t_1,\ldots,t_d]\!]\otimes_{\mathbf{C_p}\{x^{\infty}\}}DR^{\infty}_{\mathbf{C}_p}$$

where the $\mathbf{C}_p\{x^{\infty}\}$-algebra structure on $\mathbb{B}_{\rm dR}^+[\![t_1,\ldots,t_d]\!]$ should be spelled out in Lemmata 6.11, 6.12,

$$DR^{\infty}_{\mathbf{C}_p}: 0\to\mathbf{C}_{p}\{x^{\infty}\}\to\bigoplus_{a=1}^d\mathbf{C}_{p}\{x^{\infty}\}\text{d}x_a\to \bigoplus_{a<b}^d\mathbf{C}_{p}\{x^{\infty}\}\text{d}x_a\wedge \text{d}x_b\to\cdots\to\mathbf{C}_{p}\{x^{\infty}\}\text{d}x_1\wedge\ldots\wedge\text{d}x_d\to 0$$

and the statement of Cor. 6.13 is that the augmentation $\mathbb{B}_{\rm dR}^+[0]\to \mathbb{B}_{\rm dR}^+[\![t_1,\ldots,t_d]\!]\otimes_{\mathbf{C_p}\{x^{\infty}\}}DR^{\infty}_{\mathbf{C}_p}$ is a quasi-isomorphism.

The question is why.

There must be more to it, because locally in the topology generated by rational subsets on $X$, the de Rham complex of $X$ is:

$$DR: 0\to\mathbf{Q}_{p}\{x_1,\ldots,x_d\}\to\bigoplus_{a=1}^d\mathbf{Q}_{p}\{x_1,\ldots,x_d\}\text{d}x_a\to \bigoplus_{a<b}^d\mathbf{Q}_{p}\{x_1,\ldots,x_d\}\text{d}x_a\wedge \text{d}x_b\to\cdots\to\mathbf{Q}_{p}\{x_1,\ldots,x_d\}\text{d}x_1\wedge\ldots\wedge\text{d}x_d\to 0$$

**Question 2: if $\mathbf{Q}_p$ is the constant sheaf in the topology generated by rational subsets on $X$, is the augmentation $\mathbf{Q}_p[0]\to DR$, a quasi-isomorphism?**

Remark 2.The two augmentations $\mathbf{Q}_p[0]\to DR$ and $\mathbb{B}_{dR}^+[0]\to (\mathcal{O}\mathbb{B}_{dR}^+)_X\otimes_{\mathcal{O}_X}DR$ have really nothing to do with each other.I would expect, for

Question 2, an answer along the following lines:

the augmentation $\mathbf{Q}_p[0]\to DR$, with $$DR = 0\to\mathcal{O}_X\to\Omega^1_{X/\mathbf{Q}_p}\to\cdots\to\Omega^{d}_{X/\mathbf{Q}_p}\to 0$$ as a map of complexes of abelian sheaves on the site generated by

rational subsetson $X$,isa quasi-isomorphism. Probably because one can identify $DR$ with a Koszul complex for the ideal $(x_1,\ldots,x_d)$ in $\mathbf{Q}\{x_1,\ldots,x_d\}$, and such ideal is generated by a regular sequence?the augmentation $\mathbf{Q}_p[0]\to DR$, with $$DR = 0\to\mathcal{O}_X\to\Omega^1_{X/\mathbf{Q}_p}\to\cdots\to\Omega^{d}_{X/\mathbf{Q}_p}\to 0$$ as a map of complexes of abelian sheaves on the

étale/pro-étalesites on $X$, isnota quasi-isomorphism, 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.

**Question 2 rephrased.** Right?

Remark 3.This part of the paper contains a mistake, that has been addressed by Scholze in an erratum. Although the erratum is related, the questions asked here are not affected by this.

I would appreciate more insight/details. Thanks a lot.