On Descent and germs

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 $\ACVF$.

Second version: We were able to drop all hypothesis in the statement (also the assumption of the existence of a non-dividing sequence, required in our previous version). A third version is uploaded including details of a number of steps upon request of the referee. Current version 50 pages