Consider a variety $X$ over a field $k$ and let $\ell$ be a prime different from the characteristic of $k$. One has the derived category $D(X, Q_{\ell})$ of $\ell$-adic sheaves. There are very important abelian subcategories (of perverse sheaves) corresponding to $t$-structures given by various perversity conditions.

Question: What is the precise crystalline analogue of $D(X, Q_{\ell})$ and the subcategories of perverse sheaves? Here I am thinking of crystalline as being some sort of "$p$-adic sheaves" where $p$ is the characteristic of $k$. Perhaps one needs to assume $k$ is perfect..

Thank you in advance for your help.

  • 2
    $\begingroup$ Berthelot's theory of $\mathcal D^{\dagger}$-modules on $X$ is one possible answer. $\endgroup$
    – Emerton
    Aug 31, 2011 at 2:59
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    $\begingroup$ Another is the theory of Arabia and Mebkhout. (See their Ann. Inst. Fourier paper; my understanding is that there is much more to come, though.) $\endgroup$
    – Emerton
    Aug 31, 2011 at 3:04

1 Answer 1


Your question seems to contain an implicit assumption that analogues are unique. It is not clear to me that for any choice of formalism you will get precise analogues of all the theorems you see in the $\ell$-adic world.

If $k$ is the complex field, you can apply the Riemann-Hilbert correspondence to get an equivalence between regular holonomic $\mathcal{D}_X$-modules and middle-perverse sheaves in the analytic topology with complex coefficients. Then, you can look for a $p$-adic analogue of regular holonomic $\mathcal{D}_X$-modules, and as Emerton commented, one possible answer is provided by Berthelot's theory of $\mathcal{D}^\dagger$-modules. Caro has some work on the ArXiv on overholonomic modules that produces the same sort of finiteness under smooth proper maps you get from the $\ell$-adic setting, but I do not know enough about this field to commment further.

  • $\begingroup$ @Carnahan: A very good point indeed. Now that you have pointed it out to me, I see that perhaps there may be more than one crystalline analogue. Very interesting! Thanks $\endgroup$
    – SGP
    Sep 1, 2011 at 10:50

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