Let $\mathcal{A}$ be an affinoid algebra over a complete non-archimedean field $K$. We have two objects we can investigate, namely the algebro-geometric spectrum $X = \operatorname{spec} \mathcal{A}$ and the non-archimedean analytic space $X^{an} = \operatorname{sp} \mathcal{A}$. The former has an etale theory from the 1960's, and the latter from V. Berkovich's '93 IHES paper. In particular for an abelian etale sheaf $\mathcal{L}$ I can analytify the sheaf to get a sheaf on the etale topology of $X^{an}$, which I'll call $\mathcal{L}^{an}$.


Are there any comparison results between $H^i(X\_{\acute{e}t},\mathcal{L})$ and $H^i(X^{an}\_{\acute{e}t},\mathcal{L}^{an})$?


The comparison theorems in that paper seem to work for algebras of finite type over $K$, or of finite type over an affinoid base. It doesn't seem directly possible to set it up so that the derived pushforward $R^q\varphi\_\*$ maps to $\operatorname{sp} K$.


I am principally interested in smooth affinoids over a discretely valued field, and $\mathcal{L}$ a locally constant sheaf of finite abelian groups whose orders are prime to the characteristic of the residue field.


I have a very convoluted argument in mind using Berkovich's most recent pre-print but I imagine there is an easier way.


EDIT: I suppose I should probably indicate that I've put some thought into this: You can take a closed immersion of $X^{an}$ into some ball (e.g. higher dimensional analogues of $E(0,r) \times D(0,s)$). This closed immersion is algebraically of finite type, and so we can use the comparison theorem to compare the two push-forwards. This plus the Leray spectral sequence would reduce the problem to showing the analogous result for constructible sheaves on balls. However there doesn't really seem to be any tools that I can obviously use to attack this.



Comparison between analytic etale cohomology and algebraic etale cohomology for affinoids
Not sure if this is still relevant, but the desired comparison in the case of "principal interest" to you can be proved in two different ways. 


On one hand, you can deduce it from some [recent results of Achinger](https://arxiv.org/abs/1701.03197), which imply that both cohomologies in question are unchanged if you replace the etale sites by the *finite* etale sites. Since the finite etale sites of $X$ and $X^{an}$ are canonically equivalent, this gives what you want. This works for any affinoid $\mathcal{A}$ over a complete discretely valued $K$ and any finite locally constant sheaf of abelian groups $\mathcal{L}$ on $X\_{et}$.


Alternatively, and now for $K$ any complete nonarchimedean field, you can make a devissage to the case where $\mathcal{L}$ is a constant sheaf of finite abelian groups, which then is handled by Corollary 3.2.3 in Huber's book. To do this, pick a finite etale Galois cover $f:X'\to X$ with Galois group $G$ such that $f^{\ast}\mathcal{L}$ is constant, and use the Hochschild-Serre spectral sequences $H^i(G,H^j\_{et}(X',f^{\ast}\mathcal{L})) \Rightarrow H^{i+j}\_{et}(X,\mathcal{L})$ and $H^i(G,H^j\_{et}(X'^{an},f^{an,\ast}\mathcal{L}^{an})) \Rightarrow H^{i+j}\_{et}(X^{an},\mathcal{L}^{an})$. There is a canonical morphism from the first spec. seq. to the second, and it's an isomorphism on all terms of the $E\_2$-page (by Huber's Corollary 3.2.3 plus some nonsense about compatibilities), so it gives an isomorphism on the abutments.


