Trace Maps on Rigid Stein Spaces

We provide a relative version of the trace map from the work of Beyer, which can be associated to any finite tale morphism $X\to Y$ of smooth rigid Stein spaces and which then relates the Serre duality on $X$ with the Serre duality on $Y$. Furthermore, we consider the behaviour of any rigid Stein space under (completed) base change to any complete extension field and deduce a commutative diagram relating Serre duality over the base field with the Serre duality over the extension field.