Endogenies and linearization in the non-virtually connected case

TL;DR

This paper establishes a linearization theorem for pre-rings of endogenies on definable abelian groups without connectivity assumptions, extending Zilber's Field Theorem to finite-dimensional theories.

Contribution

It introduces a new linearization theorem for pre-rings of endogenies in the non-virtually connected case, generalizing previous results.

Findings

Proves a linearization theorem for pre-rings of endogenies

Extends results to cases with quasi-endomorphisms

Generalizes Zilber's Field Theorem for finite-dimensional theories

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.