Equations and algebraic geometry over profinite groups |
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Authors: | S G Melesheva |
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Institution: | 1.Novosibirsk State Univ.,Novosibirsk,Russia |
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Abstract: | The notion of an equation over a profinite group is defined, as well as the concepts of an algebraic set and of a coordinate
group. We show how to represent the coordinate group as a projective limit of coordinate groups of finite groups. It is proved
that if the set π(G) of prime divisors of the profinite period of a group G is infinite, then such a group is not Noetherian, even with respect to one-variable equations. For the case of Abelian groups,
the finiteness of a set π(G) gives rise to equational Noetherianness. The concept of a standard linear pro-p-group is introduced, and we prove that such is always equationally Noetherian. As a consequence, it is stated that free nilpotent
pro-p-groups and free metabelian pro-p-groups are equationally Noetherian. In addition, two examples of equationally non-Noetherian pro-p-groups are constructed. The concepts of a universal formula and of a universal theory over a profinite group are defined.
For equationally Noetherian profinite groups, coordinate groups of irreducible algebraic sets are described using the language
of universal theories and the notion of discriminability. |
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Keywords: | |
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