On the comparison theorem for étale cohomology of non-Archimedean analytic spaces (Q1905773)

Non-archimedean analytic spaces and in particular the étale cohomology of these spaces have been developed by V. G. Berkovich, R. Huber, A. J. de Jong, M. van der Put and others. This paper treats a delicate technical detail on the comparison of the analytic étale cohomology and the ordinary étale cohomology. Let \(\varphi: {\mathcal Y}\to {\mathcal X}\) be a morphism of schemes of locally finite type over a complete non-archimedean field \(k\). Let \({\mathcal F}\) be an étale abelian torsion sheaf on \({\mathcal Y}\). This étale sheaf can be made into an étale sheaf \({\mathcal F}^{an}\) for the analytic category. Moreover, the usual constructions for étale cohomology have their counterpart for the analytic étale cohomology. The theme of this paper is to prove the existence of an isomorphism \((R^q \varphi_! {\mathcal F})^{an} \to R^q \varphi^{an}_! {\mathcal F}^{an}\). In earlier papers, the author had proved this statement in two cases: either \(\varphi\) compactifiable or \({\mathcal F}\) such that the torsion is prime to the residue characteristic of \(k\). Using recent results of W. Lütkebohmert, it is shown that the isomorphism exists (no conditions on \(\varphi\) and \({\mathcal F})\) if the characteristic of \(k\) is 0.