A conjecture of Lichtenbaum expects that for a smooth proper variety $X$ over a finite field, the etale motivic cohomology groups $H^i(X_{et}, \mathbb{Z}(n))$ are finite for $i\neq 2n, 2n+2$, finitely generated for $i=2n$, and cofinitely generated for $i=2n+2$. This especially implies that the etale $K$-theory for $X$ is finitely generated.
Is the analogue of Bass's conjecture for etale $K$-theory supposed to be true without the assumptions of properness and finiteness of the base field? (The Zariski version only requires regularity and being finite type on $\text{Spec}(\mathbb{Z}))$
Is there a Borel-Moore variant of Lichtenbaum's conjecture that does not require properness and smoothness?
There is an equivalent description of the Lichtenbaum conjecture in terms of Weil-etale cohomolgy, which translates to finite generation of integral Weil-etale motivic groups for proper and smooth $X$ over a finite field. Is there a Borel-Moore version of Weil-etale motivic cohomology that gets rid of properness and smoothness? How about when the base field is of char $0$, is it supposed to be true?
