Integral refinements of rigid cohomology

Question

Disclaimer: I know absolutely nothing about p-adic cohomology, so it is possible that even the premises of this question are incorrect. But it turns out that I need to apply the theory of rigid cohomology to something in a very different field, so here I am...

Let $k$ be a perfect field of characteristic $p$. For $X$ a smooth projective variety over $k$, we can consider its crystalline cohomology $H^*_{cris}(X/W(k)),$ which is a finitely generated module over the ring $W(k)$ of Witt vectors.

Now let $X$ be a general variety over $k$. I can consider its rigid cohomology $H^*_{rig}(X)$. This is a finitely generated $W(k)\otimes\mathbb{Q}$-module, and for $X$ smooth projective, it matches up with $H^*_{cris}(X/W(k))\otimes\mathbb{Q}$.

My question is:

Is there a naturally defined cohomology theory $H^*_{?}(X/W(k)),$ giving finitely generated $W(k)$-modules for any variety $X$, so that $H^*_{?}(X/W(k))\otimes\mathbb{Q}$ recovers rigid cohomology?

For $X$ affine, I know that there is work on integral Monsky-Washnitzer cohomology (e.g. https://arxiv.org/abs/1304.7307 by Davis-Zureick-Brown) but my impression is that this is not supposed to be finitely generated in general.

Actually, I would like a version of this that would work with arithmetic $D$-module coefficients, but that is maybe too much to ask for...

Answer 1

There has been some progress on this question since the question was asked. Apparently it was "known to the experts" that there cannot be an integral $p$-adic cohomology theory which is finitely generated, agrees with rigid cohomology rationally and satisfies finite étale descent. This was mentioned by Richard Crew at least at the 2017 conference "$p$-adic cohomology and arithmetic applications" at BIRS. The complete argument (failure of descent along Artin-Schreier covers) is very nicely written up in [AC22].

If one assumes some strong form of resolution of singularities in positive characteristic, then [ESS23] shows that there is a "good" (well-defined, functorial, finitely generated, rationally agrees with rigid cohomology,...) integral $p$-adic cohomology theory which satisfies $\mathrm{cdh}$-descent. It is given by sending a variety $X$ to the logarithmic crystalline cohomology of an appropriate normal crossings compactification $(\overline{X},\overline{X}\setminus X$)$.

Without assuming resolution of singularities, there is [Mer24] which proves that a "good" integral $p$-adic cohomology theory satisfying Nisnevich descent exists. If one assumes some version of resolution of singularities, it is shown that this theory agrees with the cohomology theory in [ESS23].

As far as I know, none of this has been worked out with coefficients yet.

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[AC22] T. Abe, R. Crew, Integral $p$-adic cohomology theories, 2022, arXiv:2108.07608v2

[ESS23] V. Ertl, A. Shiho, J. Sprang, Integral $p$-adic cohomology theories for open and singular varieties, 2023, arXiv:2105.11009v2

[Mer24] A. Merici, A motivic integral $p$-adic cohomology, 2024, arXiv:2211.14303v4