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A Weil Cohomology theory is a functor from the category of smooth projective varieties (over some fixed field $k$) to graded $K$-algebras (for some fixed field $K$) satisfying various axioms. For example, Betti,de Rham, $\ell$-adic and rigid cohomologies are all Weil cohomology theories.

In each of these settings one frequently encounters the concept of a category of coefficients for a given variety. For example, in the $\ell$-adic setting the category of coefficients of a variety $X$ is the category of (not necessarily lisse) $\ell$-adic sheaves on $X$. In the Betti setting it is the category of constructible sheaves on $X$. In the de Rham setting it is the category of regular, holonomic $D$-modules on $X$. In the rigid setting a category of coefficients on $X$ is given by overholonomic arithmetic $F$-$D$-modules. For each theory there is a constant object for every variety and these give the usual Weil cohomology functor.

I have two questions (the second is related to the first):

  1. What is the conceptual meaning of a category of coefficients of a given variety for a Weil cohomology theory? Is there a set of axioms such categories must obey? If so does anyone have a reference? Why, for example, do we take only regular holonomic $D$-modules and not all $D$-modules in the de Rham setting?

  2. In each of the above examples there are important subcategories of smooth objects. For the above cases these are given by: lisse $\ell$-adic sheaves, local systems, integrable connections, and overconvergent $F$-isocrystals. What is the conceptual meaning of these privileged subcategories?

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Great question! You might be interested in "Triangulated categories of mixed motives" by Déglise and Cizinski. I think one of the points of this paper is that knowing the Weil cohomology theory already determines the coefficients. (As I understand it, that something like this should be true was first suggested by Beilinson.) –  Geordie Williamson Apr 29 '12 at 8:19
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Not an expert but my comments are too long to fit in the comment box:

As far as I know, the "coefficients" are a well-behaved (e.g. triangulated) category with some rich structures, namely six operations à la Grothendieck:

pull-back, push-forward, tensor product, inner hom, upper and lower shriek.

They are expected to satisfy various functoriality and adjunctions, which are similar to the case of sheaves of abelian groups on a topological space.

These structures at least allow you to connect coefficients on different spaces, and thus do dévissage. Once the category of coefficients are understood well, some hard theorem can be reduce to the curve case, where you still need to work hard... (Hopefully this partly answers question 1.)

Also, one Weil cohomology theory you didn't mention is crystalline cohomology, which coincide with rigid cohomology for proper smooth varieties. There, as I read from Illusie's survey, a satisfactory category of coefficients is missing. I don't know if there is any further development after that.

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Hi temp; I just changed "inner him" to "inner hom". –  Daniel Litt Jun 7 '12 at 6:07
    
Oops, thanks. Auto-correction to be blamed... –  temp Jun 7 '12 at 6:33
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