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Preprint Number 1480
1480. Junguk Lee Hyperfields, truncated DVRs, and valued fields E-mail: Submission date: 12 September 2018 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 ≥ 1, then K_1 and K_2 are isomorphic. More generally, for n_1,n_2 ≥ 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 indcies of K_1 and K_2. Moreover if K_1 is tamely ramified, the 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. 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 index e having perfect residue fields. In the tamely ramified case, we show that a subcategory of this category of valued hyperfields is equivalent to the category of complete discrete valuation rings of mixed characteristic (0,p) having perfect residue fields. Mathematics Subject Classification: 12J25, 11S15, 12L12 Keywords and phrases: valued hyperfields, valued fields, lifting problem, homomorphism over p |

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