Irreducibility of an affine space in algebraic geometry over a group

Abstract

We prove a theorem which states that if G is an equationally Noetherian group that is locally approximated by finite p-groups for each prime p then an affine space G n in a respective Zariski topology is irreducible for any n. The hypothesis of the theorem is satisfied by free groups, free soluble groups, free nilpotent groups, finitely generated torsion-free nilpotent groups, and rigid soluble groups. Also corrections to a valuation lemma, which has been used in some of the author’s previous works, are introduced.