  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.
