Completeness in Zariski groups
type: Item
instance of: scholarly article
title: Completeness in Zariski groups
zbMATH Open document ID: 0922.03051
DOI: 10.1007/BF02775039
author: Markus Junker
published in: Israel Journal of Mathematics
publication date: 1999-09-12T00:00:00Z
review text: The groups of finite Morley rank are the groups where a finite dimension is assigned to every definable set that behaves much as dimension in algebraic geometry. The Cherlin-Zil'ber conjecture (still open) states that any infinite simple group of finite Morley rank is an algebraic group over an algebraically closed field. The interest of the conjecture is the hope to get an abstract characterization of algebraic groups, not mentioning fields and varieties. \textit{E. Hrushovski} and \textit{B. Zil'ber} [J. Am. Math. Soc. 9, 1-56 (1996; Zbl 0843.03020)] introduced the notion of a Zariski geometry and proved that these are exactly the objects that arise from the Zariski topologies of smooth algebraic curves over algebraically closed fields. A Zariski geometry is an infinite set \(D\) together with a family of Noetherian topologies on \(D, D^2,\dots\) satisfying certain axioms which, in particular, imply that the structure, whose universe is \(D\) and whose basic relations are the closed subsets of \(D^n\) for all \(n\), admits quantifier elimination and is strongly minimal. In the paper under review the author introduces Zariski groups as groups with axiomatically given Zariski topology. A Zariski group is an infinite group equipped with a family of Noetherian topologies on \(G, G^2,\dots\), satisfying certain axioms which hold, in particular, for algebraic groups over algebraically closed fields. The axioms guarantee that any Zariski group admits quantifier elimination (in the language whose basic relations are the closed relations) and is \(\omega\)-stable of finite Morley rank, and that any structure elementarily equivalent to it is a Zariski group, too. It is shown that a large part of algebraic geometry can be developed for Zariski groups; in particular, a notion of a smooth variety can be reasonably introduced. The author investigates a version of the Cherlin-Zil'ber conjecture for Zariski groups, the positive solution of which would provide an abstract characte…
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 20A15
zbMATH DE Number: 1286709
zbMATH Keywords: Zariski topology
zbMATH Keywords: Zariski group
zbMATH Keywords: group of finite Morley rank
zbMATH Keywords: algebraic group
zbMATH Keywords: algebraic geometry
zbMATH Keywords: smooth variety
zbMATH Keywords: bad groups
zbMATH Keywords: algebraically closed field
reviewed by: Oleg Belegradek
MaRDI profile type: Publication
cites work: Q4317873
cites work: Zariski geometries
cites work: Q3924381
cites work: Q3775535
cites work: Q3707983
cites work: Q3770506
cites work: Some Basic Theorems on Algebraic Groups
cites work: Some model theory of simple algebraic groups over algebraically closed fields
full work available at URL: https://doi.org/10.1007/bf02775039
OpenAlex ID: W2081126292
description: scientific article; zbMATH DE number 1286709