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Irreducible algebraic sets over divisible decomposed rigid groups
N. S. Romanovskiiab a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Novosibirsk State University, Novosibirsk, Russia
Abstract:
A soluble group
$G$ is said to be rigid if it contains a normal series of the form
$$
G=G_1>G_2>\dots>G_p>G_{p+1}=1,
$$
whose quotients
$G_i/G_{i+1}$ are Abelian and are torsion-free when treated as right
$\mathbb Z[G/G_i]$-modules. Free soluble groups are important examples of rigid groups. A rigid group
$G$ is divisible if elements of a quotient
$G_i/G_{i+1}$ are divisible by nonzero elements of a ring
$\mathbb Z[G/G_i]$, or, in other words,
$G_i/G_{i+1}$ is a vector space over a division ring
$Q(G/G_i)$ of quotients of that ring. A rigid group
$G$ is decomposed if it splits into a semidirect product
$A_1A_2\dots A_p$ of Abelian groups
$A_i\cong G_i/G_{i+1}$. A decomposed divisible rigid group is uniquely defined by cardinalities
$\alpha_i$ of bases of suitable vector spaces
$A_i$, and we denote it by
$M(\alpha_1,\dots,\alpha_ p)$.
The concept of a rigid group appeared in [
arXiv:0808.2932v1 [math.GR]], where the dimension theory is constructed for algebraic geometry over finitely generated rigid groups. In [
Algebra i Logika, <b>48</b>:2 (2009), 258–279], all rigid groups were proved to be equationally Noetherian, and it was stated that any rigid group is embedded in a suitable decomposed divisible rigid group
$M(\alpha_1,\dots,\alpha_ p)$. Our present goal is to derive important information directly about algebraic geometry over
$M(\alpha_1,\dots,\alpha_ p)$. Namely, irreducible algebraic sets are characterized in the language of coordinate groups of these sets, and we describe groups that are universally equivalent over
$M(\alpha_1,\dots,\alpha_ p)$ using the language of equations.
Keywords:
algebraic geometry, irreducible algebraic set, rigid group, universally equivalent groups.
UDC:
512.542 Received: 15.08.2009