On the boundedness of $k$-algebra homomorphisms between $k$-affinoid algebras

TL;DR

This paper explores conditions under which all $k$-algebra homomorphisms between $k$-affinoid algebras are bounded, revealing that this depends on the value group of $k$ or its characteristic and finiteness properties.

Contribution

It establishes a precise criterion linking the boundedness of homomorphisms to the value group and characteristic of the base field $k$ in non-archimedean analytic geometry.

Findings

Boundedness holds iff $|k^ imes|^Q = R_{>0}$ or $k$ has positive characteristic and is $F$-finite.

Provides a characterization of when algebra homomorphisms are automatically bounded.

Connects algebraic properties of $k$ with analytic boundedness conditions.

Abstract

Let $k$ be a complete non-archimedean non-trivial valued field. In this paper, we investigate whether every $k$-algebra homomorphism between $k$-affinoid algebras is automatically bounded. We show that this property holds if and only if either $|k^\times|^\mathbb{Q} = \mathbb{R}_{>0}$ holds, or $k$ has positive characteristic and is $F$-finite.