What are the uses of coefficient systems for arithmetic cohomology theories?

Question

In topology when studying a space with non-trivial fundamental group it becomes important to consider homology and cohomology with coefficients in representations of the fundamental group, i.e. local systems. The general Serre spectral sequence for a fibration with non-simply connected base provides an example. Another use of local systems is to get a version of Poincaré duality for non-orientable manifolds.

I am wondering about similar instances when it comes to e.g. algebraic de Rham cohomology, where the coefficient systems are given by quasi-coherent sheaves with integrable connections.

What does the ability to construct the de Rham complex with coefficients in a module with integrable connection buy us? Or for crystalline cohomology; what are some uses of the full theory of crystals (rather than just the obvious crystal given by the structure sheaf)?

(note the related question: de Rham cohomology and flat vector bundles was more focused on analytic and topological situation. Also, the question: Coefficients of Weil Cohomology Theories focused on conceptualizations of these coefficient systems, whereas I am more interested in applications.)

What are some of the uses of coefficient systems in algebraic geometry, especially in arithmetic contexts?

I guess one still needs these coefficient systems to prove duality statements?

I also think that Deligne's proof of the Weil conjectures relied heavily on using constructible $l$-adic sheaves. The same goes for Kedlaya's proof of the Weil conjectures using rigid cohomology, with F-isocrystals as coefficients. What are some reasons why one needs such coefficient systems in these proofs? What are other instances where this extra flexibility is needed?

Thanks!

Answer 1

Well when you say:

The general Serre spectral sequence for a fibration with non-simply connected base provides an example.

you've already pretty much got it - except in algebraic geometry we usually use the Leray spectral sequence instead of the Serre spectral sequence.

A lot of the foundational proofs of etale cohomology (notably duality, as you state) are proved or can be proved by fibering your variety over a curve and using a Leray spectral sequence argument to reduce the statement in dimension $n$ to the statement in dimension $n-1$ plus a statement for curves, with coefficients in a local system. The first hypothesis can be handled by induction and the second by doing explicit calculations on curves.

Then of course these foundational theorems are used in the proof of the Weil conjectures. But that is probably not what you're thinking of. Deligne has two proofs of the Weil conjectures, there are later simplifications, and then of course the $p$-adic version.

However, as far as I know, the core of all these arguments is Deligne's version of Rankin's squaring trick. Fundamentally, we take some very simple observation - that for a local system with some positivity properties, the size of the largest Frobenius eigenvalue at any point is at most the position of the last pole in the zeta function, which is the size of the largest Frobenius coinvariant times q - and calculate what it implies about tensor powers of a fixed local system.

I think if you read Katz's very simple version of the proof for curves and hypersurfaces you will agree that this idea is essentially all that is needed to get the bound for some varieties, though of course more ideas are needed to get the result in full generality.

So you need local coefficient systems because you need to take high tensor powers of them to enhance this bound, which is always trivial if you try to apply it directly to bound something, into something nontrivial.

But one thing that Deligne's second proof of the Weil conjecture demonstrates, and a lesson that arithmetic geometers have taken it is that the Riemann hypothesis is really better understood as a statement about local systems the whole time. The "real" version of the statement is that the $i$th cohomology of proper variety with coefficients in a sheaf mixed of weight $\leq w$ is mixed of weight $\leq i+w$, and the original Weil conjecture is just a corollary.