Irreducible algebraic sets over divisible decomposed rigid groups

N. S. Romanovskii

Algebra i logika, Tome 48 (2009) no. 6, pp. 793-818.

Résumé

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, 48:2 (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.

Détail

Keywords: algebraic geometry, irreducible algebraic set, universally equivalent groups.
Mots-clés : rigid group

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