$p$-adic Hodge Theory for rigid spaces, after P. Scholze

Question

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 subsets on $X$, is a 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-étale sites on $X$, is not a 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.

Answer 1

Let me start with the second question first:

The usual de Rham complex is not locally acyclic in positive degrees, in any of the topologies (analytic (= of rational subsets), étale, pro-étale, ...). The problem is that rigid-analytic spaces are not "locally contractible". For example, on the annulus $$ \mathbb T = \{x \mid |x|=1\} $$ (corresponding to the algebra $\mathbb C_p\langle x^{\pm 1}\rangle$), the differential $\frac 1x dx$ cannot be integrated on any open subset that contains the Gauss point (an adic or Berkovich point); in fact, the same is true for any (pro-)étale map whose image contains the Gauss point. The problem is that the logarithm series does not converge with respect to the Gauss norm (and any admissible covering of the rigid-analytic space will contain the Gauss point). More intuitively, the problem is that one cannot admissibly cover rigid-analytic varieties by balls.

(Another minor point is that the kernel of the differential on $\mathcal O_X$ is actually larger than the constant sheaf $\mathbb Q_p$; it is the integral closure of $\mathbb Q_p$ in $\mathcal O_X$, i.e. nontrivial field extensions may appear locally.)

The de Rham complex of Corollary 6.13 is isomorphic to the continuous de Rham complex for $\mathbb B_{\mathrm{dR}}^+[[t_1,\ldots,t_d]]$ over the constants $\mathbb B_{\mathrm{dR}}^+$, in other words for $d=1$ $$ 0\to \mathbb B_{\mathrm{dR}}^+[[t]]\buildrel{\nabla_t}\over\longrightarrow \mathbb B_{\mathrm{dR}}^+[[t]] dt\to 0\ . $$ Indeed, we know that the differential vanishes by definition on $\mathbb B_{\mathrm{dR}}^+$, and $t$ corresponds to the element $T\otimes 1 - 1\otimes [T^\flat]$; the second term is killed by the differential, and the first term goes to $dT$, which is the canonical basis element for $\Omega^1_X$. By the Leibniz rule and continuity, this determines the differential.

But for any $\mathbb Q$-algebra $R$, the continuous de Rham complex for $R[[t_1,\ldots,t_d]]$ over $R$ is acyclic in positive degrees, with constants equal to $R$.

What may be a bit confusing is that this Poincaré lemma is only rather indirectly related to the usual de Rham complex; one needs to extend scalars to $\mathbb B_{\mathrm{dR}}^+$ and do a funny completion. After this completion, one arrives for some reason (Proposition 6.10) at the de Rham complex for a formal power series algebra, and that is easy.

Edit: I realized that it may be helpful (or at least fun) to explain what is going on in the example of the first paragraph. In addition to the geometric variable called $x$ in the first paragraph (corresponding to $T$ later), the sheaf $\mathcal O\mathbb B_{\mathrm{dR}}^+$ also contains the 'arithmetic' variable $[x^\flat]\in \mathbb B_{\mathrm{dR}}^+$ (after the pro-étale cover of choosing all $p$-power roots of $x$). The element $\frac x{[x^\flat]}$ is close to $1$ in $\mathcal O\mathbb B_{\mathrm{dR}}^+$ (as under $\theta$, it gets mapped to $1$, as both $x$ and $[x^\flat]$ map to $x\in \hat{\mathcal O}_X$). Therefore, the function $$ \log(\tfrac x{[x^\flat]})\in \mathcal O\mathbb B_{\mathrm{dR}}^+ $$ converges. Intuitively, $\log(\frac x{[x^\flat]}) = \log(x) - \log([x^\flat])$, but none of the two terms make sense individually. Still, as $[x^\flat]\in \mathbb B_{\mathrm{dR}}^+$ is constant, the differential of $\log(\frac x{[x^\flat]})$ agrees with $d\log(x) = \frac 1x dx$, as desired.

Thus, in a vague sense, we need to extend scalars to $\mathbb B_{\mathrm{dR}}^+$ to get a second arithmetic copy $\log([x^\flat])$ of the function $\log(x)$ we care about, and then their difference $\log(\frac x{[x^\flat]})$ converges in a suitable completion. (The minor error in the paper here was about the correct definition of this completion, by the way.)