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Endogenies and linearization in the non-virtually connected case

Moreno Invitti

TL;DR

This work develops a linearization framework for definable abelian groups of finite dimension acted on by two invariant pre-rings of endogenies or quasi-endomorphisms. It introduces sharp commutation, invariance notions, and the global katakernel to reduce actions to a finite definable field, yielding Z-linear structure on a quotient A/A0. The main results cover base-case and extended cases for endogenies and quasi-endomorphisms, culminating in a finite bikatakernel Kat(Γ,Δ) and, in characteristic 0, a Zilber-type field theorem. The findings generalize Zilber's Field Theorem to finite-dimensional, potentially non-connected modules, with potential applications to supersimple groups and Lie rings and implications for linearization strategies beyond the connected setting.

Abstract

We prove a linearization theorem for pre-rings of endogenies acting on a definable abelian group of finite dimension. Observe that no assumptions on the connectivity of A are made. We also prove a similar result when one of the two pre-rings is of quasi-endomorphisms. A corollary of these results is a generalization of Zilber's Field Theorem for finite-dimensional theories.

Endogenies and linearization in the non-virtually connected case

TL;DR

This work develops a linearization framework for definable abelian groups of finite dimension acted on by two invariant pre-rings of endogenies or quasi-endomorphisms. It introduces sharp commutation, invariance notions, and the global katakernel to reduce actions to a finite definable field, yielding Z-linear structure on a quotient A/A0. The main results cover base-case and extended cases for endogenies and quasi-endomorphisms, culminating in a finite bikatakernel Kat(Γ,Δ) and, in characteristic 0, a Zilber-type field theorem. The findings generalize Zilber's Field Theorem to finite-dimensional, potentially non-connected modules, with potential applications to supersimple groups and Lie rings and implications for linearization strategies beyond the connected setting.

Abstract

We prove a linearization theorem for pre-rings of endogenies acting on a definable abelian group of finite dimension. Observe that no assumptions on the connectivity of A are made. We also prove a similar result when one of the two pre-rings is of quasi-endomorphisms. A corollary of these results is a generalization of Zilber's Field Theorem for finite-dimensional theories.

Paper Structure

This paper contains 27 sections, 43 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Finite-dimensional groups
  3. Endogenies
  4. Definition
  5. Sharp commutation
  6. Invariance
  7. Global katakernel
  8. Dimensionality
  9. Theorem A: base case
  10. Theorem A: second case
  11. Lines
  12. Proof of the theorem A: second case
  13. Theorem A: third case
  14. Proof of theorem B
  15. Theorem B Second version: base case
  16. ...and 12 more sections

Key Result

Lemma 1.1

Let $G$ be a definable group of finite dimension and $\{H_i\}_{i\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\omega$ such that there is no $J=\{j_1,...,j_n\}\subseteq I$ of cardinality $n$ such that $|\bigcap_{i=1}^kH_{j_i}:\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greate

Theorems & Definitions (108)

  • Definition
  • Lemma 1.1
  • proof
  • Definition
  • Definition
  • Definition
  • Definition
  • Definition
  • Lemma 2.1
  • proof
  • ...and 98 more