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

TL;DR

We analyze when every $k$-algebra homomorphism between $k$-affinoid algebras is bounded and prove an if-and-only-if criterion: boundedness holds exactly when $|k^ imes|^{\mathbb{Q}}=\mathbb{R}_{>0}$ or when $p>0$ and $[k:k^p]<\infty$. The argument blends a reduced-case analysis using compatible systems of $p$-power roots with a deformation method that reduces the general case to the reduced situation and to a derivation obstruction via $\Omega_{k\{r^{-1}T\}/k[T]}$. A concrete counterexample over $\mathbb{Q}_p$ demonstrates nonbounded maps when the criterion fails. These results sharpen the understanding of automatic boundedness for maps between $k$-affinoid algebras and have implications for Berkovich-type non-archimedean geometry.

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.

Theorems & Definitions (42)

• Theorem 1
• Example 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Claim 2.4
• proof : Proof of \ref{['cl:A-to-AK']}