Irreducible algebraic sets over divisible decomposed rigid groups |
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Authors: | N S Romanovskii |
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Institution: | 1.Sobolev Institute of Mathematics, Siberian Branch,Russian Academy of Sciences,Novosibirsk,Russia;2.Novosibirsk State University,Novosibirsk,Russia |
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Abstract: | A soluble group G is said to be rigid if it contains a normal series of the form G = G
1 > G
2 > …> G
p
> G
p+1 = 1, whose quotients G
i
/G
i+1 are Abelian and are torsion-free when treated as right ℤG/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 ℤG/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
1
A
2…A
p
of Abelian groups A
i
≅ G
i
/G
i+1. A decomposed divisible rigid group is uniquely defined by cardinalities α
i
of bases of suitable vector spaces A
i
, and we denote it by M(α1,…, α
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 11], 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(α1,…, α
p
). Our present goal is to derive important information directly about algebraic geometry over M(α1,… α
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(α1,…, α
p
) using the language of equations. |
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Keywords: | |
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