On Descent and germs

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 $\ACVF$.

💡 Research Summary

This paper makes three significant contributions to model theory, particularly in the areas of stable domination and NIP theories.

1. Introduction and Background: Stable domination, introduced by Haskell, Hrushovski, and Macpherson, describes how a structure is governed by its stable part (e.g., the residue field in ACVF). It has been a key bridge for extending tools from stable theory to NIP theories like ACVF. A central result is the “descent theorem”: if a global type p is stably dominated over a larger parameter set B, then it is also stably dominated over a smaller set A ⊆ B. The original proof required a strong technical assumption (EP): that the type tp(B/A) admits a global Aut(M/A)-invariant extension. The necessity of this assumption has been an open question.

2. Main Contributions: * A New, General Proof of Descent: The paper’s primary achievement is proving the descent theorem without assuming (EP) (Theorem 4.1). The proof ingeniously uses the definability and symmetry properties of generically stable types. When p is dominated via a function f defined over Ab, the authors show how to “push down” the information to find an A-definable function h that dominates p over A. A key intermediate result (Theorem 4.3) handles the simpler case where the stable set doing the domination is A-definable, leading to a much shorter proof for ACVF (Theorem 4.4). * The Bounded Stabilizing Property of Stable Sets in NIP: The authors prove that any stable set in an NIP theory has the Bounded Stabilizing (BS) property (Theorem 3.1). This means that functions defined on such sets exhibit only finite variation with respect to changing parameters. This result clarifies the interaction between stability and NIP within a structure. * Correction of a Known Error: Using the above result, the authors identify and correct an error in Proposition 6.7 of the foundational book on stable domination in ACVF