Completeness in Zariski groups
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 characterization of the Zariski topology for algebraic groups. The main result: there is no bad smooth Zariski group. (A group of finite Morley rank is said to be bad if it is non-solvable, and all proper connected definable subgroups of it are nilpotent. As is known, no algebraic group over an algebraically closed field is bad. The problem of the existence of bad groups is open.) It follows, by Zil\('\)ber's theorem, that any smooth simple Zariski group interprets an algebraically closed field.
scientific article; zbMATH DE number 1286709
scientific article