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This question is about 6 operations formalism for 'crystalline' cohomology theories - more specifically the infinitesimal cohomology of smooth $\mathbb{C}$-varieties, and the overconvergent cohomology of varieties in positive characteristic. I'm not really sure of a good reference for the former, for the latter I'm talking about le Stum's site theoretic approach to rigid cohomology, as developed in his paper "The Overconvergent Site".

In particular, I'm interested in whether or not one might hope for a 6 operations formalism for the cohomology of the overconvergent site. One thing that suggests to me that this is not the 'right' set-up to get 6 operations is that I think I can convince myself that for any Cartesian square of $k$-varieties ($k$ a perfect field of characteristic $p>0$) $$ \begin{matrix} X' & \overset{g'}\rightarrow &X \\ f'\downarrow &&\downarrow f \\ Y' & \overset{g}{\rightarrow} &Y\end{matrix}$$ the induced base change morphism $$ g^*_{\mathrm{An}^\dagger} \mathbb{R}f_{\mathrm{An}^\dagger*}E \rightarrow \mathbb{R}f'_{\mathrm{An}^\dagger*}g'_{\mathrm{An}^\dagger}^*E$$ is an isomorphism. This seems 'wrong' to me - one should only expect such a base change for a proper morphism $f$. My question is whether or not this should be taken seriously as a reason why 6 operations with good 'topological' properties won't exist in this context, or is there something I've missed?

I'd also be interested to know whether or not 6 operations has been worked out for the infinitesimal site of $\mathbb{C}$-varieties (or varieties over any alg. closed char $0$ field). In the introduction to another paper of his, "Constructible $\nabla$-modules on curves", le Stum says that constructible sheaves have a definition in terms of the infinitesimal site (due to Deligne, but unpublished), but what about complexes and six operations? Is there anything known in this direction?

Also, just for reference, the argument that we have base change for any Cartesian square is basically just combining the paragraph before 1.4.2 of http://perso.univ-rennes1.fr/bernard.le-stum/Publications_files/OverconvergentSite.pdf with 7.28.1 of http://stacks.math.columbia.edu/tag/04IT. One can also see it in terms of realisations.

EDIT: It may be worth making clear that I am familiar with both Caro's theory of 6 operations for overholonomic $F\text{-}\mathcal{D}^\dagger$-modules, and the theory of algebraic $\mathcal{D}$-modules in char $0$. I am more interested in whether or not, for example in char $0$, one can do all this purely in terms of the infinitesimal site.

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Dear Chris, Are you sure that your base-change statement is correct? What if $f$ is the inclusion of $X = \mathbb A^1$ in $Y = \mathbb P^1$ and $g$ is the inclusion of $Y' =$ point at infinity. Then $X' = \emptyset$, so the right hand side of the base-change morphism vanishes. Is this really what happens on the left-hand side? Also, Berthelot's theory of $\mathcal D^{\dagger}$-modules is supposed to give the theory of coefficients and six operations. Regards, –  Emerton Mar 26 '13 at 2:26
    
P.S. I should add that $\mathcal D^{\dagger}$-modules were introduced by Berthelot to provide a six operations formalism that incorporated rigid cohomology. I'm not sure how they interact with Le Stum's theory. –  Emerton Mar 26 '13 at 2:28
    
@Emerton: le Stum's site is a "big site", analogous to the big Zariski site (so for the big Zariski site the underlying category is just schemes), so base change does hold. I'm not sure what the status of Berthelot's theory of $D^{\dagger}$-modules is now, but there used to be open problems along the lines of "$f!_$ of overholonomic is overholonomic". Googling now it looks like there are some recent papers by Daniel Caro on this. –  David Zureick-Brown Mar 26 '13 at 2:51
    
Dear David, Thanks for this clarification. My understanding was that some of these open problems had been solved, at least partially, by some of the results of Kedlaya over the last several years (at least if one considers $\mathcal D^{\dagger}-F$ modules, i.e. $\mathcal D^{\dagger}$-modules with a Frobenius structure), but I haven't kept up with the details, and so don't know the current status. Best wishes, Matt –  Emerton Mar 26 '13 at 3:19
    
At least for F-isocrystals, 6 operations has now been worked out by Caro - he has a good theory of overholonomic F-D modules which are stable under all operations and contain the category of overconvergent F-isocrystals. On quasi-projective varieties he has proved stability of holonomicity (with F-strucure). I am specifically curious as to whether one might hope to make 6 operations work within le Stum's framework, because this base change business seems to suggest not (to me anyway). –  ChrisLazda Mar 26 '13 at 10:47
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1 Answer

The six operations formalism includes a good theory of support: for any closed immersion $i:Z\to X$ with complement $j:U=X-Z\to X$, there are short exact sequences of the form $$0\to j_{!}j^\ast(F)\to F\to i_\ast i^\ast(f)\to 0$$ This is not compatible with a base change formula for arbitrary morphisms, and explain why working with too big sites is not an option (at least without further work).

For the $p$-adic version, this paper of D. Caro explains how to get the six operations for overholonomic $F$-complexes of $\mathcal{D}^\dagger$-modules. In characteristic zero, the following books are standard references:

M. Kashiwara $\mathcal{D}$-modules and microlocal calculus, Translations of Mathematical Monographs, 217. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2003.

R. Hotta, K. Takeuchi, T. Tanisaki, $\mathcal{D}$-modules, perverse sheaves, and representation theory, Progress in Mathematics, 236. Birkhäuser Boston, Inc., Boston, MA, 2008.

Note also that there is a way to produce the six operations formalism out of any reasonable cohomology theory (such as algebraic de Rham cohomology, or Berthelot's rigid cohomology), using Morel-Voevodsky's homotopy theory of schemes and mixed motives; see the last chapter of this paper (in particular, Examples 17.2.21, 17.2.22, and 17.2.23).

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