An equivalence between two approaches to limits of local fields

TL;DR

This paper explores the relationship between two different theoretical frameworks—valued hyperfields and triples—for understanding limits of local fields, revealing their equivalence and deepening the conceptual connections in local field theory.

Contribution

It establishes an equivalence between Krasner's valued hyperfields and Deligne's triples, unifying two approaches to limits of local fields.

Findings

Proves the equivalence between hyperfields and triples

Clarifies the conceptual relationship between two theories

Enhances understanding of local field limits

Abstract

Marc Krasner proposed a theory of limits of local fields in which one relates the extensions of a local field to the extensions of a sequence of related local fields. The key ingredient in his approach was the notion of valued hyperfields, which occur as quotients of local fields. Pierre Deligne developed a different approach to the theory of limits of local fields which replaced the use of hyperfields by the use of what he termed triples, which consist of truncated discrete valuation rings plus some extra data. We study the relationship between Krasner's valued hyperfields and Deligne's triples.