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Let $X$ be a smooth complex algebraic variety (for simplicity). Recall that the de Rham space (or stack) of $X$ is the quotient of $X$ by the groupoid that is the formal neighborhood of the diagonally embedded copy of $X$ in $X\times X$. This groupoid degenerates naturally to the formal completion of $X$ in its normal bundle in $X\times X$, i.e. in $T^*X$. [This is the formal completion of the degeneration of $X\hookrightarrow X\times X$ to the normal cone.] There is a corresponding degeneration of the sheaf $\mathcal{D}_X$ of differential operators on $X$ to functions on $T^*X$. [More precisely, if $\pi: T^*X\rightarrow X$ is the projection, $\mathcal{D}_X$ degenerates to $\pi_*\mathcal{O}_{T^*X}$.]

Question. What is the nicest way (is there one?) to say a corresponding degeneration of the crystalline site on $X$, so that degeneration of $\mathcal{D}$-modules corresponds to quasicoherent sheaves on the degenerating site?

If this is completely standard, can you suggest a reference?

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  • $\begingroup$ what do you mean by 'degenerates'? $\endgroup$
    – bananastack
    Mar 15, 2014 at 20:36
  • $\begingroup$ In the case of the groupoid I mean there is a flat degeneration over $\mathbb{A}^1$ whose general fiber is the de Rham groupoid and whose fiber over $0$ is the formal neighborhood of the zero section. I do not know precisely what a ``degeneration of the crystalline site'' should mean, but I would guess objects should include schemes over $X\times\mathbb{A}^1$ with additional data. $\endgroup$
    – Thomas Nevins
    Mar 15, 2014 at 21:11
  • $\begingroup$ I think I'm confused by your use of the word groupoid (not that I would know how to answer your question, I'm just trying to understand as it seems interesting). To me a groupoid is just a category where every morphism is invertible. Do you mean a groupoid object in schemes? Could you be explicit about the case of the de Rham stack? $\endgroup$
    – bananastack
    Mar 15, 2014 at 21:20
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    $\begingroup$ Clearly you don't like the answer `` just take the etale site of the degenerating family $X_{dR}\leadsto X_{Dol}$"... would you prefer a description of the family of Grothendieck topologies, of the form ``take as opens etale opens, plus [a $\lambda$-deformed version of] nilpotent thickenings"? Such thickenings can be described in terms of the tangent complex with a $\lambda$-rescaled Lie bracket.. $\endgroup$
    – David Ben-Zvi
    Mar 17, 2014 at 0:23
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    $\begingroup$ Or you could take as [Koszul dual to] an answer, ``take the derived loop space modulo $S^1$ rotation, as a family over the affinization of $BS^1$"? $\endgroup$
    – David Ben-Zvi
    Mar 17, 2014 at 0:25

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