On the structure of certain valued fields

TL;DR

This paper investigates the structure of finitely ramified mixed characteristic valued fields, establishing isomorphisms based on residue rings and introducing a functorial correspondence that generalizes unramified valuation rings.

Contribution

It proves that isomorphisms of residue rings imply isometries of the fields and introduces a functor from certain Artinian rings to valuation rings, extending previous theorems.

Findings

Isomorphism of residue rings implies field isometry.

Existence of liftings from residue ring homomorphisms to valuation rings.

A new functorial relationship generalizing unramified valuation rings.

Abstract

In this article, we study the structure of finitely ramified mixed characteristic valued fields. 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 residue rings are isomorphic for each $n\ge 1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1\ge 1$, there is $n_2$ depending only on the ramification indices of $K_1$ and $K_2$ such that any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuation rings. Moreover, we get a functor from the category of certain principal Artinian local rings of length $n$ to the category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. Our lifting result improves Basarab's relative completeness theorem for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields.