Towers of complements to valuation rings and truncation closed embeddings of valued fields

TL;DR

This paper characterizes when valued fields can be embedded into generalized power series fields via truncation-closed embeddings, using towers of complements, and provides intrinsic constructions for certain classes of fields.

Contribution

It introduces the concept of towers of complements to characterize truncation closed embeddings and constructs such towers intrinsically for specific valued fields.

Findings

Equivalence between truncation closed embeddings and towers of complements.

Intrinsic construction of towers for Henselian and algebraically maximal Kaplansky fields.

Extension of towers and embeddings to maximal immediate extensions.

Abstract

We study necessary and sufficient conditions for a valued field $\KF$ with value group $G$ and residue field $\kf$ (with char $\KF$ = char $\kf$) to admit a truncation closed embedding in the field of generalized power series $\kf((G, f))$ (with factor set $f$). We show that this is equivalent to the existence of a family ({\it tower of complements}) of $\kf$-subspaces of $\KF$ which are complements of the (possibly fractional) ideals of the valuation ring. If $\KF$ is a Henselian field of characteristic 0 or, more generally, an algebraically maximal Kaplansky field, we give an intrinsic construction of such a family which does not rely on a given truncation closed embedding. We also show that towers of complements and truncation closed embeddings can be extended from an arbitrary field to at least one of its maximal immediate extensions.