TY  - GEN
N2  - 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).
A1  - Malcic, Milan
UR  - https://archiv.ub.uni-heidelberg.de/volltextserver/33617/
ID  - heidok33617
AV  - public
CY  - Heidelberg
TI  - A Relative Trace Map and its Compatibility with Serre Duality in Rigid Analytic Geometry
Y1  - 2023///
ER  -