$p$-adic comparison of cohomology with coefficients in $\mathbb{Z}_{p}$ and $\mathbb{B}_{\textrm{dR}}$ on general smooth algebraic varieties

Question

This is something which I'm sure is well known to experts which I would appreciate some information about. In his paper [1], Scholze proves (e.g. Theorem 8.4, Theorem 8.8) that on a proper adic space $X$ over $\textrm{Spa}(k,\mathcal{O}_{k})$, with $k$ an appropriate non-archimedean field, the natural map $H^{i}(X_{\textrm{pet}}, \hat{\mathbb{Z}}_{p}) \otimes B_{\textrm{dR}} \to H^{i}(X_{\textrm{pet}}, \mathbb{B}^{+}_{\textrm{dR}})$ is an isomorphism, where I use the notation "pet" for the pro-etale site. I think a similar statement should be true for $X$ the adic space associated to a smooth algebraic variety over $k$, but I cannot find the appropriate reference.

I am aware of work [2] of Diao, Lan, Liu and Zhu which works with a general smooth variety, but I am really interested in the "intermediate" comparison between cohomology wich coefficients in $\hat{\mathbb{Z}}_{p}$ and $\mathbb{B}^{+}_{\textrm{dR}}$ rather than the actual comparison isomorphism of $p$-adic Hodge theory, and it seems to me they only handle this intermediate fact in the proper case. The related work [3] seems to do something like what I want in Proposition 3.3.4, but in a compactly supported setting which isn't really what I'm interested in.

If it helps I am really just looking to understand the maps $H^{i}(X_{\textrm{pet}}, \hat{\mathbb{Z}}_{p}) \otimes B_{\textrm{dR}} \to H^{i}(X_{\textrm{pet}}, \mathbb{B}^{+}_{\textrm{dR}})$ when $X = \mathbb{G}_{m}^{w}$ is the $w$-fold product of the multiplicative group, so an explicit description of this map would also be appreciated.

Edit: I may have found what I want in Theorem 7.14 of [4]; I will take a look at the paper and report back.

1. https://www.math.uni-bonn.de/people/scholze/pAdicHodgeTheory.pdf
2. https://arxiv.org/pdf/1803.05786.pdf
3. https://arxiv.org/pdf/1912.13030.pdf
4. https://arxiv.org/pdf/1801.01779.pdf

Answer 1

The result is false in the open case.

If true, a long exact sequence would show that also

$$H^i(X_{\mathrm{proet}},\hat{\mathbb Z}_p)\otimes k\to H^i(X_{\mathrm{proet}},\hat{\mathcal O}_X)$$

is an isomorphism, where I assume $k$ is algebraically closed (which I think is also implicit in the question, or otherwise one should base change $X$ to an algebraic closure before taking cohomology). But this can't even be true for $i=0$ in general, as the right-hand side receives a map from $H^0(X,\mathcal O_X)$ -- if $X$ is affine, this is infinite-dimensional.

One key issue is that if $X$ is open, it is not quasicompact as a rigid space, and hence taking cohomology does not commute with filtered colimits. Thus, it is not even clear that the map

$$ H^i(X_{\mathrm{proet}},\hat{\mathbb Z}_p)\otimes \mathbb Q_p\to H^i(X_{\mathrm{proet}},\hat{\mathbb Q}_p)$$

is an isomorphism. In fact, it is not an isomorphism even for $X=\mathbb A^1$ and $i=1$, where the left-hand side vanishes but right-hand side is infinite-dimensional, see Corollary 3.10 of Le Bras' Thesis.