Crystalline cohomology can be computed as the hypercohomology of the de Rham-Witt complex.

If we want to compute the etale cohomology of the constant sheaves $ \mathbb{Z}_l$ or $ \mathbb{Q}_l$ (well, to consider them as sheaves you need pro-etale site or something like that but you get the idea), is there some canonical choice of an injective resolution? There can be topological obstructions to this.

The question probably splits into 3 parts: schemes are defined over a field of zero characteristic, schemes are defined over a field of characteristic $ l=p$ , schemes are defined over a field of positive characteristic $ p\neq l$ .