[Paper Review] Finding groups in Zariski-like structures

Kaisa Kangas|arXiv (Cornell University)|Apr 27, 2014

Polynomial and algebraic computation15 references3 citations

TL;DR

This paper generalizes Hrushovski's Group Configuration Theorem to quasiminimal abstract elementary classes (AECs), establishing a framework for independence and canonical bases in $\mathbb{M}^{eq}$. It introduces Zariski-like structures via axioms (ZL1)–(ZL9), proving that non-trivial bounded closure implies the existence of definable groups, with the cover of the multiplicative group of an algebraically closed field serving as a key example.

ABSTRACT

We study quasiminimal classes, i.e. abstract elementary classes (AECs) that arise from a quasiminimal pregeometry structure. For these classes, we develop an independence notion, and in particular, a theory of independence in $\M^{eq}$. We then generalize Hrushovski's Group Configuration Theorem to our setting. In an attempt to generalize Zariski geometries to the context of quasiminimal classes, we give the axiomatization for Zariski-like structures, and as an application of our group configuration theorem, show that groups can be found in them assuming that the pregeometry obtained from the bounded closure operator is non-trivial. Finally, we study the cover of the multiplicative group of an algebraically closed field and show that it provides an example of a Zariski-like structure.

Motivation & Objective

• To extend Hrushovski’s Group Configuration Theorem from first-order logic to quasiminimal abstract elementary classes (AECs).
• To develop a robust independence notion in $\mathbb{M}^{eq}$ for quasiminimal classes using bounded closure instead of algebraic closure.
• To axiomatize Zariski-like structures in the context of quasiminimal AECs and show that such structures admit definable groups when the pregeometry is non-trivial.
• To demonstrate that the cover of the multiplicative group of an algebraically closed field under the PQF-topology satisfies the Zariski-like axioms and supports definable group configurations.

Proposed method

• Develop an independence calculus in quasiminimal AECs based on the bounded closure operator (bcl), generalizing non-forking independence.
• Prove that axioms AI–AVI for independence are preserved when moving from $\mathbb{M}$ to $\mathbb{M}^{eq}$ and beyond, ensuring stability and coherence.
• Generalize Hrushovski’s Group Configuration Theorem to show that a Galois-definable rank 1 group can be constructed in $(\mathbb{M}^{eq})^{eq}$ when a group configuration exists.
• Axiomatize Zariski-like structures via (ZL1)–(ZL9), ensuring compatibility with both Zariski geometries and quasiminimal AECs.
• Use dimension-theoretic arguments and compactness to lift configurations from the base structure to $\mathbb{M}^{eq}$, preserving rank and specialization relations.
• Analyze the cover of the multiplicative group of an algebraically closed field under the PQF-topology, showing it satisfies the Zariski-like axioms and supports group configurations when $\mathrm{exp}(D)$ is smooth.

Experimental results

Research questions

• RQ1 Can Hrushovski’s Group Configuration Theorem be generalized to quasiminimal abstract elementary classes where elimination of imaginaries fails?
• RQ2 Under what conditions does a non-trivial bounded closure operator in a quasiminimal AEC imply the existence of a definable group?
• RQ3 Do Zariski-like structures—axiomatized as a generalization of Zariski geometries—admit definable groups when the pregeometry is non-trivial?
• RQ4 Does the cover of the multiplicative group of an algebraically closed field under the PQF-topology satisfy the axioms of a Zariski-like structure and support group configurations?
• RQ5 Can the dimension-theoretic axioms of Zariski geometries be preserved in irreducible subvarieties of such covers, especially when the image under the exponential map is smooth?

Key findings

• A perfect independence notion is developed in quasiminimal AECs using the bounded closure operator, satisfying axioms AI–AVI and extending to $\mathbb{M}^{eq}$.
• The generalized Group Configuration Theorem ensures that a Galois-definable rank 1 group exists in $(\mathbb{M}^{eq})^{eq}$ whenever a group configuration is present.
• Zariski-like structures satisfying axioms (ZL1)–(ZL9) admit definable groups if the bounded closure pregeometry is non-trivial.
• The cover of the multiplicative group of an algebraically closed field under the PQF-topology is shown to be a Zariski-like structure, satisfying all axioms.
• When $\mathrm{exp}(D)$ is a smooth algebraic curve, the Dimension Theorem holds on the curve, enabling the transfer of group configuration results from the cover to the curve.
• The construction of the sequence $(c_i)$ in the proof ensures that $\textrm{span}(c_i)_{i<\kappa} \cap K^* \subset K$, preserving kernel structure and enabling rank control.