Trace Maps on Rigid Stein Spaces

TL;DR

This paper develops a relative trace map for finite tale morphisms between smooth rigid Stein spaces, linking Serre duality across spaces and base changes, enhancing understanding of duality in rigid analytic geometry.

Contribution

It introduces a relative trace map for finite tale morphisms of smooth rigid Stein spaces and explores its compatibility with Serre duality under base change.

Findings

Established a relative trace map for rigid Stein spaces.

Connected Serre duality on different spaces via the trace map.

Analyzed behavior of Stein spaces under base change and duality relations.

Abstract

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.