A Relative Trace Map and its Compatibility with Serre Duality in Rigid Analytic Geometry

Given a complete non-archimedean valued field K, we discuss a relative trace map attached to any finite étale morphism of smooth rigid-analytic Stein spaces over K and prove that it is compatible with the trace maps that arise in the Serre duality theory on the respective Stein spaces. Our proof builds on the technique of inves- tigating the trace map of a rigid Stein space via a relation between algebraic local cohomology and compactly supported rigid cohomology established in the work of Beyer [Bey97a]. For this purpose we also prove a generalization of a theorem of Bosch [Bos77] which concerns the connectedness of formal fibers of a distinguished affinoid space. This closes an argumentative gap in [Bey97a]. Furthermore, we consider the behaviour of any rigid-analytic Stein space and its trace map under (completed) base change to any complete extension field K′/K and prove that there are natural base change comparison maps that yield a com- mutative diagram relating Serre duality over K with Serre duality over K′. Finally we discuss the recent work of Abe and Lazda [AL20], which constructs a trace map on proper pushforwards of analytic adic spaces, and we explain how some of their results can be related to ours (via Huber’s functor from the category of rigid analytic varieties to the category of adic spaces)