On residual domination and types orthogonal to the value group

TL;DR

This paper unifies the concept of residual domination in henselian valued fields, showing its equivalence to types orthogonal to the value group and linking it to stability and simplicity in various valued field theories.

Contribution

It introduces a unifying framework for residual domination, connecting it with orthogonality to the value group and stability notions in valued field types.

Findings

Residual domination coincides with orthogonality to the value group.

Types with stable residue fields relate to generic stability.

Results apply to ultraproducts of p-adic fields and the limit theory VFA_0.

Abstract

We present a unifying framework of residual domination for (expansions of) henselian valued fields of equicharacteristic zero, encompassing some valued fields with operators. We show that the class of residually dominated types coincides with the types that are orthogonal to the value group, and with the class of types whose reduct to ACVF (the theory of algebraically closed valued fields with a non-trivial valuation) are generically stable. When the residue field is stable (resp. simple) we relate these equivalent notions to generic stability (resp. simplicity). Those results apply in particular to ultraproducts of $p$-adic fields and to the limit theory VFA$_{0}$ of algebraically closed valued fields of characteristic $p$ with the Frobenius automorphism (as $p$ tends to infinity).