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There is an interpretation of D-modules over "sufficiently nice" prestacks $X$ (read: various finiteness conditions apply, perhaps even smoothness) by Gaitsgory and Rozenbylum (see chapter I.4 here and this paper), in which one views D-modules as ind-coherent sheaves (i.e. filtered colimits of coherent sheaves) over so-called de Rham spaces $X_{dR}$ attached to the previously mentioned "nice" prestacks $X$. As I understand it, this is essentially using the fact that each crystal in quasi-coherent sheaves comes canonically equipped with a flat connection, and thus can be seen as a D-module; the approach by Gaitsgory-Rozenblyum is therefore a version of infinitesimal cohomology (in the sense of Grothendieck-Ogus) wherein establishing the six functors is somewhat easier, as now the six functors for D-modules can be deduced from the general theory of ind-coherent sheaves. There is, however, a caveat: our prestack $X$ has to be an object in characteristic $0$, and preferably over a field of characteristic $0$, as smooth schemes over fields are automatically reduced.

Now I am aware of the fact that there is also a theory of "arithmetic" D-modules, developed by Berthelot, wherein one replaces infinitesimal sites and all the businesses involving de Rham spaces with crystalline sites, whose objects are pd-immersions and whose coverage is the usual Zariski coverage. Given the somewhat ad hoc definition of pd-structures, is it also possible (at least in principle) to reformulate the theory of arithmetic D-modules in the style of Gaitsgory and Rozenblyum ? Have there been any attempts at this, and if this is not possible, why so ?

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    $\begingroup$ For a smooth scheme $X$, ind-coherent sheaves are a much less significant ingredient than the stack $X_{dR}$, from section 7 of Simpson's arXiv:alg-geom/9604005v1. If you try to mimic Simpson's construction to get crystalline cohomology, you need to take the DP envelope of the diagonal ideal, and for that to represent a functor, you would have to work on a category of rings with DP structures on their radicals. $\endgroup$ Sep 28, 2021 at 20:19
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    $\begingroup$ Arithmetic D-modules are different from crystals on the crystalline site, if I understand correctly, and the later does not have six functor formalism. $\endgroup$
    – Z. M
    Jan 12, 2022 at 5:27
  • $\begingroup$ @JonPridham One could simply work with the category of (animated) rings. The de Rham stack (outside characteristic 0) of $X$ is given by $R\mapsto X(\mathbb G_a^{\operatorname{dR}}(R))$, where $\mathbb G_a^{\operatorname{dR}}:=\operatorname{cofib}(\mathbb G_a^\sharp\to\mathbb G_a)$ and $\mathbb G_a^\sharp=\operatorname{Spec}(\mathbb Z[T,T^2/2!,T^3/3!,\dots])$. See Bhatt's talk youtube.com/watch?v=v2Jfk-NTjp4 $\endgroup$
    – Z. M
    Jan 12, 2022 at 5:35
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    $\begingroup$ To clarify, if the OP's sole objection to DP structures is their non-uniqueness, then yes, simplicial rings provide a solution by allowing derived quotients, so it doesn't matter that the map $\{\gamma \in I^{\mathbb{N}} ~:~ \binom{m+n}{n}\gamma_{m+n}=\gamma_m\gamma_n\}\to I$ sending $\gamma$ to $\gamma_1$ isn't injective, but I wouldn't regard that as avoiding divided powers. $\endgroup$ Jan 12, 2022 at 21:18
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    $\begingroup$ @Z.M One hopefully final comment. For a $q$-analogue of $\hat{\mathbb{G}}_a$, you work on $\lambda$-rings, starting from free (not cofree) $\lambda$-rings over $\mathbb{Z}[q]$ rather than polynomial rings. As in the second part of Remark 1.4, $\nu^k(a):= [k]_q!(q-1)^k\lambda^k(\frac{a}{q-1})$ gives a $q$-analogue of $a^k$, the recursive formula $\nu^k(a)=\sum_{i>0}(q-1)^{i-1} \lambda^i(a)\frac{[k-1]_q!}{[k-i]_q!}\nu^{k-i}(a) $ showing it has no denominators. Instead of $T^n/n!$, you then want to include $\nu^n(T)/[n]_q!$. $\endgroup$ Jan 18, 2022 at 9:20

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The following link to a set of notes seems to contain I was looking for. It's about so-called crystalline spaces associated to proper and separated smooth schemes (which are very similar to de Rham spaces of smooth schemes) and how arithmetic D-modules can be realised as quasi-coherent sheaves on these spaces, much like how D-modules in characteristic $0$ can be thought of as quasi-coherent sheaves on de Rham spaces (I'm brushing a lot of technicalities under the rug here).

In particular, propositions 7.3 and 7.5 seem to be the result that I was looking for.

R. Gregoric, "Crystalline spaces"

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