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An equivalence between two approaches to limits of local fields

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.