Timeline for Comparison for cycle class maps for de Rham and etale cohomology via p-adic Hodge theory

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Jun 28, 2022 at 17:39	comment	added	Alexander Betts		Also, one slightly subtle issue here is that there are several different definitions of "the" comparison isomorphism, and whether or not the comparison iso is compatible with Chern classes could potentially depend on this choice. Fortunately, Niziol shows that most of the standard definitions (including Tsuji's) agree with one another, making this often a non-issue. But if e.g. you want to use the comparison isomorphism coming from rigid geometry a la Scholze, then you can't just appeal to Niziol.
Jun 28, 2022 at 17:34	comment	added	Alexander Betts		An alternative reference for this compatibility is the appendix of the survey paper "Semistable Conjecture of Fontaine--Janssen: a survey" by Takeshi Tsuji, which also gives a number of other useful compatibilities.
Jun 15, 2020 at 7:27	history	edited	CommunityBot		Commonmark migration
May 9, 2018 at 21:58	comment	added	SashaP		@Bonbon There certainly are proofs, for which compatibility with the cycle maps is built in by design, e.g. Niziol's proof using K-theory. But, a priori, different proofs might give different comparison isomorphism.
May 9, 2018 at 17:24	comment	added	Bonbon		@SashaP Yes, thanks, it's a good idea. Since I am also not familiar with the concrete construction, I would do this sometime later. And I guess that $a$ would be a rational number.
May 9, 2018 at 16:33	comment	added	SashaP		Finally, for any line bundle $\mathcal{L}$ on a smooth proper $X$ we can conclude that $\alpha(1\otimes c_1^{dR})(\mathcal{L})=t^{-1}\otimes a\cdot c_1^{\'et}(\mathcal{L})$ since the class of $\mathcal{L}$ in the Neron-Severi group modulo torsion is completely determined by the restriction of $\mathcal{L}$ to all curves. However, to compute $a$ it is necessary to look at the actual construction of the comparison isomorphism which you have in mind.
May 9, 2018 at 16:33	comment	added	SashaP		I think that one can deduce that $\alpha$ respects the cycle class maps up to a scalar purely from the functorial properties of the comparison isomorphism. For $X=\mathbb{P}^1$ the element $\alpha(1\otimes c_1^{dR}(O(1)))$ is stable under the Galois action. Hence, $\alpha(1\otimes c_1^{dR}(O(1)))=t\otimes a\cdot c_1^{\'et}(O(1))$ for some $a\in\mathbb{Q}_p$. For any curve $C$ considering a finite cover $C\to\mathbb{P}^1$ we get that $\alpha$ is the multiplication by $at$ on the second cohomology of $C$.
May 7, 2018 at 21:44	history	edited	Bonbon	CC BY-SA 4.0	deleted 13 characters in body
May 7, 2018 at 21:30	comment	added	R. van Dobben de Bruyn		@JesseSilliman: that paper looks a bit fancy. I feel that this shouldn't involve studying Chern classes on higher $K$-theory. My strategy would be similar, though: replace $\operatorname{CH}^*(X)$ by $K_0(X)$ (they agree after tensoring with $\mathbb Q$), and use the splitting principle to reduce to the case of $c_1(\mathscr L)$ for $\mathscr L$ a line bundle. For this case an actual argument is needed, though.
May 7, 2018 at 21:17	comment	added	Jesse Silliman		You should look at Niziol's paper "On Uniqueness of p-adic period morphisms." In particular, she shows that comparison theorems are compatible with Chern classes.
S May 7, 2018 at 20:46	history	suggested	CommunityBot	CC BY-SA 4.0	Fixed grammar and spelling. Typed some images of diagrams.
May 7, 2018 at 20:37	review	Suggested edits
S May 7, 2018 at 20:46
May 7, 2018 at 20:01	comment	added	Bonbon		@R.vanDobbendeBruyn I'm not sure if we can replace $Z^k(X)$ with $CH^k(X)$.
May 7, 2018 at 19:58	comment	added	R. van Dobben de Bruyn		Is there a reason you wrote the first cycle class map having $Z^k(X)$ instead of $\operatorname{CH}^k(X)$ as its domain?
May 7, 2018 at 19:53	history	asked	Bonbon	CC BY-SA 4.0