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Transfer of generalized amalgamation in simple theories

Baptiste Schilling

Abstract

We give an abstract framework to transfer generalized amalgamation from a simple theory to another, and we apply it to theories of lovely pairs and of bounded PAC structures. We show in particular that bounded pseudo-algebraically closed fields have generalized amalgamation, regardless of their imperfection degree.

Transfer of generalized amalgamation in simple theories

Abstract

We give an abstract framework to transfer generalized amalgamation from a simple theory to another, and we apply it to theories of lovely pairs and of bounded PAC structures. We show in particular that bounded pseudo-algebraically closed fields have generalized amalgamation, regardless of their imperfection degree.
Paper Structure (17 sections, 21 theorems, 13 equations)

This paper contains 17 sections, 21 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Corollary \ref{['cor:nExistencePAC']}
  3. Corollary (Corollaries \ref{['SCFfinite']}, \ref{['cor:DCF0PAC']} and \ref{['cor:namalgSCFinfiniPAC']})
  4. Generalized amalgamation
  5. Definitions
  6. First properties
  7. Transfer of generalized amalgamation
  8. The general setting
  9. Verification of the hypotheses
  10. Examples
  11. Differentially closed fields of characteristic zero
  12. Generic predicate
  13. Pairs of structures
  14. Bounded PAC substructures of stable theories
  15. The general result
  16. ...and 2 more sections

Key Result

Lemma 2.6

Let $(p_w(x_w), w\in W)$ be an amalgamation system over a set of parameters $A$. If $B$ is a superset of $A$ such every $p_w(x_w)$ has a unique nonforking extension $q_w(x_w)$ to $B$, then $(q_w(x_w), w\in W)$ is an amalgamation system over $B$.

Theorems & Definitions (54)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • ...and 44 more