Hyperfields, truncated DVRs, and valued fields

Abstract

For any two complete discrete valued fields K1 and K2 of mixed characteristic with perfect residue fields, we show that if the n-th valued hyperfields of K1 and K2 are isomorphic over p for each n≥1, then K1 and K2 are isomorphic. More generally, for n1,n2≥1, if n2 is large enough, then any homomorphism, which is over p, from the n1-th valued hyperfield of K1 to the n2-th valued hyperfield of K2 can be lifted to a homomorphism from K1 to K2. We compute such n2 effectively, which depends only on the ramification indices of K1 and K2. Moreover, if K1 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. Specially, in the tamely ramified case, we show that a category of valued hyperfields is equivalent to the category of complete discrete valued rings of mixed characteristic (0,p) having perfect residue fields.