Completed cohomology and variants

Question

I am interested in the following set up:

I have an ind-sequence of curves $\dots X_2\to X_1$ defined over a finite field of characteristic $p$ such that $X_n/X_{n-1}$ is a Galois degree $\ell$ cover and the galois group of the tower is $\mathbb Z_\ell$ for $\ell \neq p$.

I would like to study the variation in the cohomology $H^1_{et}(X_n,\mathbb Z_{\ell'})$ with $n$ for $\ell' \neq p$ (but possibly equal to $\ell$). This seems like exactly the set up of completed cohomology (for instance from here) except that continuous cohomology seems to be defined using singular cohomology.

I expect there should be a straightforward variant of completed cohomology using etale cohomology and the major theorems should hold in this case too. Is this true? Is this stuff written down anywhere?

And just in general, what would be the best place to start learning about completed cohomology?

Answer 1

There is no reason why you shouldn't consider completed etale cohomology, instead of completed singular cohomology. If you look at Emerton's 2006 Inventiones paper which started the whole theory, he allows etale cohomology as well, in order to get a Galois action on completed cohomology; and there is a huge industry of studying completed cohomology spaces as Galois modules (this is how one formulates and proves "local-global compatibility" in the p-adic Langlands programme).

Emerton doesn't consider towers of function fields, but it's totally clear that the definitions go over to that case. I suspect that shouldn't be too hard to establish the key finiteness property, that $\widetilde{H}^*$ is an admissible Banach space representation, in the function field case as well. (I don't know if this is written down in the literature explicitly, though.)

PS: As for where to start reading about completed cohomology: Emerton's 2006 paper is actually amazingly readable, miraculously so considering the density of completely new ideas in it. I got a lot out of reading it myself as a PhD student and I would strongly recommend starting with that as your reading material.