On the Zariski topology of Ω-groups

Abstract

A number of geometric properties of Ω-groups from a given variety of Ω-groups can be characterized using the notions of domain and equational domain. An Ω-group H of a variety Θ is an equational domain in Θ if the union of algebraic varieties over H is an algebraic variety. We give necessary and sufficient conditions for an Ω-group H in Θ to be an equational domain in this variety. Let F = F (X) be a finitely generated by X free Ω-group in a variety of Ωgroups Θ and H be an Ω-group in Θ. One of the important questions in the algebraic geometry of varieties of Ω-groups is whether it is possible to equip the space of points Hom(F,H) with the Zariski topology, whose closed sets are precisely algebraic sets. If this is so, then the Ω-group H is called the equational domain (or stable in the terminology of the authors of the papers [P], [BPP]) in the variety Θ (see [BMR]). The same problem arises for a variety Θ(G) of Ω-groups with the given Ω-group of constants G. An important role in the study of equational domains in varieties of Ω-groups is played by the notion of domain. Necessary and sufficient conditions for linear algebras over an algebra of constants and for groups over a group of constants to be equational domains in terms of domains are given in [BMR], [BPP], [DMR] and [P]. Here we continue the study of equational domains in varieties of Ω-groups (without of the Ω-group of constants). We give necessary and sufficient conditions for an Ω-group H in Θ to be an equational domain in the variety Θ. The results presented in this paper were partially announced earlier in [L].