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?

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?