Hyperfields, truncated DVRs and valued fields

TL;DR

This paper establishes that isomorphisms and homomorphisms between valued hyperfields of complete discrete valued fields of mixed characteristic can be lifted to the fields themselves, providing new insights into their structure and relationships.

Contribution

It introduces effective bounds for lifting homomorphisms between valued hyperfields to valued fields and explores categorical equivalences involving hyperfields, truncated DVRs, and valued fields.

Findings

Isomorphisms of hyperfields imply isomorphisms of fields.

Homomorphisms over p can be lifted under certain conditions.

Categorical equivalences relate hyperfields, truncated DVRs, and valued fields.

Abstract

For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th valued hyperfields of $K_1$ and $K_2$ are isomorphic over $p$ for each $n\ge1$, then $K_1$ and $K_2$ are isomorphic. More generally, for $n_1,n_2\ge 1$, if $n_2$ is large enough, then any homomorphism, which is over $p$, from the $n_1$-th valued hyperfield of $K_1$ to the $n_2$-th valued hyperfield of $K_2$ can be lifted to a homomorphism from $K_1$ to $K_2$. We compute such $n_2$ effectively, which depends only on the ramification indices of $K_1$ and $K_2$. Moreover, if $K_1$ is tamely ramified, then any homomorphism over $p$ between the first valued hyperfields is induced from a unique homomorphism of valued fields. Using this lifting result, we deduce a relative completeness theorem of AKE-style in terms of valued hyperfields. We also study some relationships between valued hyperfields, truncated discrete valuation rings, and complete discrete valued fields of mixed characteristic. For a prime number $p$ and a positive integer $e$ and for large enough $n$, we show that a certain category of valued hyperfields is equivalent to the category of truncated discrete valuation rings of length $n$ and the ramification indices $e$ having perfect residue fields of characteristic $p$. Furthermore, in the tamely ramified case, we show that a subcategory of this category of valued hyperfields is equivalent to the category of complete discrete valued rings of mixed characteristic $(0,p)$ having perfect residue fields.