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Rank and Independence of Imaginaries in Proper Pairs of ACF

Zixuan Zhu

Abstract

Let $T_P$ be the theory of beautiful pairs of algebraically closed fields of fixed characteristic. It is known that for real tuples in models of $T_P$, SU-rank coincides with Morley rank and can be computed effectively. Building on Pillay's geometric description (2007) of imaginaries in $T_P$, we define an additive rank on imaginaries of $T_P$, called the geometric rank. It takes values in $ω*\mathbb N + \mathbb Z$ and coincides with SU-rank on real tuples. It refines SU-rank and characterizes forking in $T_P^{\mathrm{eq}}$, from which we derive an explicit criterion for determining forking independence.

Rank and Independence of Imaginaries in Proper Pairs of ACF

Abstract

Let be the theory of beautiful pairs of algebraically closed fields of fixed characteristic. It is known that for real tuples in models of , SU-rank coincides with Morley rank and can be computed effectively. Building on Pillay's geometric description (2007) of imaginaries in , we define an additive rank on imaginaries of , called the geometric rank. It takes values in and coincides with SU-rank on real tuples. It refines SU-rank and characterizes forking in , from which we derive an explicit criterion for determining forking independence.
Paper Structure (7 sections, 14 theorems, 38 equations)

This paper contains 7 sections, 14 theorems, 38 equations.

Table of Contents

  1. Preliminaries
  2. Proper Pairs of ACF
  3. Notation
  4. Properties for $T_P$
  5. Ranks of Real Tuples
  6. Algebraicity between Imaginaries of Pillay Form
  7. Geometric Rank and Non-forking Independence

Key Result

Theorem A

Let $\alpha=\lceil G_{B_P(\alpha)}(P)*a\rceil$, $\beta=\lceil H_{B_P(\beta)}(P)*b\rceil$ be of Pillay form. If $\beta$ is algebraic over $\alpha$, then the stabilizer of $\textup{tp}_{L_P}(ab/(ab)^c)$ is a definable homogeny from $G:=G_{B_P(\alpha)}(P)$ to $H:=H_{B_P(\beta)}(P)$. Moreover, if $\alph

Theorems & Definitions (40)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Definition 2.2
  • Remark 2.4
  • Theorem 2.5: Theorem \ref{['T:Pillay-alg']}
  • Definition 2.6
  • Remark 2.7
  • proof : Proof of Theorem \ref{['T:Pillay-alg']}
  • Remark 2.8
  • ...and 30 more