In the paper by Bhatt and Scholze on prismatic cohomology (https://arxiv.org/pdf/1905.08229.pdf), it is stated that the de Rham comparison theorem for prismatic cohomology can be lifted to an equivalence of cdga's. This confuses me, because I don't see why the de Rham complex of a smooth and proper scheme over a field k of characteristic p which is not affine is a cdga. It seem to me that it might be an $E_\infty$ algebra which is not strictly commutative. So i'm asking:

It it true that for a smooth and proper scheme X over a field k of characteristic $p$, the global sections of the de Rham complex $R\Gamma(X,DR_X)$, which is a priori an $E_\infty$ algebra in the $\infty$-category of $k$-modules, arises from a commutative differential graded algebra $k$-vector spaces? Note that this imposes many restrictions on the complex, namely the vanishing of most of Steenrod reduced powers on its cohomology.

If it is a cdga, why is it the case?, or, alternatively, if it is not the case, what do I miss in the prismatic version of de Rham comparison stated in this paper?

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    $\begingroup$ This is obvious for an affine scheme, right? - you just take $\sum_i H^0(X, \Omega^i X)$ with its natural differential and algebra structure. So your question is if this (or some other cdga structure) glues in a canonical way. $\endgroup$ – Will Sawin Dec 12 '19 at 21:44
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    $\begingroup$ @WillSawin indeed, this is the case. The point is that the essential image of cdga's in $E_\infty$ algebras is not closed under limits (hence not obviously to global sections), otherwise every space would have a strictly commutative singular chains, and that would be really sad to all the steenrod operations story. $\endgroup$ – S. carmeli Dec 12 '19 at 21:50
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    $\begingroup$ In Corollary 15.4 of that paper, it's stated as an isomorphism of $E_\infty$ algebras. It's possible that in Theorem 1.8 they were thinking about the characteristic zero case and thought that cdga's would make the statement easier for people to understand. $\endgroup$ – Will Sawin Dec 12 '19 at 22:24
  • $\begingroup$ @WillSawin Oh, cool. So the statement in theorem 1.8 in the intro as stated might be false, and the theorem actually proven is for $E_\infty$. Still, it would be cool to see some examples of cases where it is not a cdga. I think it should be the case already for $\mathbb{P}^1$, right? $\endgroup$ – S. carmeli Dec 12 '19 at 22:28

For a smooth and proper scheme $X$ over a field $k$ of characteristic $p$, the $E_\infty$-algebra $R\Gamma(X, DR_{X/k})$ over $k$ is not represented by a $k$-cdga. Indeed, if it were, then most of the Steenrod operations on its cohomology groups would have to vanish, which is simply not true. You get a counterexample as in the first Bhatt-Morrow-Scholze (BMS1) paper by approximating $B(\mathbf{Z}/p)$ by smooth proper $k$-schemes as the Steenrod operations are nonzero on $H^*_{DR}(B(\mathbf{Z}/p))$ (it is just the singular cohomology of $B(\mathbf{Z}/p)$ with $k$-coefficients).

I suspect the phrase "it can be upgraded naturally to an isomorphism of commutative differential graded algebras" in Theorem 1.8 (3) of the quoted paper really should be interpreted at the level of sheaves. For example, the proof of 15.4 in the same paper implicitly appears to suggest the following:

Theorem: Fix a prism $(A,I)$ and a smooth formal scheme $X/(A/I)$. There is a natural identification $$\varphi_A^* \Delta_{X/A} \simeq L\eta_I \Delta_{X/A}$$ of sheaves of $E_\infty$-$A$-algebras. By the Bockstein lemma (6.12 in BMS1), the sheaf of $E_\infty$-$A/I$-algebras $(\varphi_A^* \Delta_{X/A}) \otimes_A A/I$ is represented by the sheaf of cdgas $H^*(\Delta_{X/A}/I)$, which is naturally identified with the de Rham complex $\Omega^*_{X/(A/I)}$ as a sheaf of cdgas.

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  • $\begingroup$ thank you very much. That's helps I'll try to read enough to understand all you say :) $\endgroup$ – S. carmeli Dec 15 '19 at 13:45

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