On Descent and germs

TL;DR

This paper introduces a simplified proof of descent for stably dominated types in various theories, removes previous assumptions, and explores properties of stable sets in NIP theories, with applications to ACVF.

Contribution

It provides a new, simpler proof of descent for stably dominated types and extends the understanding of stable sets in NIP theories, correcting prior results.

Findings

Simplified proof of descent for stably dominated types

Descent proof in ACVF without global invariant extensions

Stable sets in NIP theories have bounded stabilizing property

Abstract

We present a new proof of descent for stably dominated types in any theory, dropping the hypothesis of the existence of global invariant extensions. Additionally, we give a much simpler proof of descent for stably dominated types in $\ACVF$. Furthermore, we demonstrate that any stable set in an $\NIP$ theory has the bounded stabilizing property. This result is subsequently used to correct Proposition 6.7 from the book on stable domination and independence in $\ACVF$.