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Galois theory and commutators
We prove that the relative commutator with respect to a subvariety of a
variety of Omega-groups introduced by the first author can be described in
terms of categorical Galois theory. This extends the known correspondence
between the Froehlich-Lue and the Janelidze-Kelly notions of central extension.
As an example outside the context of Omega-groups we study the reflection of
the category of loops to the category of groups where we obtain an
interpretation of the associator as a relative commutator.Comment: 14 page
Twisting commutative algebraic groups
If is a commutative algebraic group over a field , is a
commutative ring that acts on , and is a finitely generated free
-module with a right action of the absolute Galois group of , then there
is a commutative algebraic group over , which is a twist of
a power of . These group varieties have applications to cryptography (in the
cases of abelian varieties and algebraic tori over finite fields) and to the
arithmetic of abelian varieties over number fields. For purposes of such
applications we devote this article to making explicit this tensor product
construction and its basic properties.Comment: To appear in Journal of Algebra. Minor changes from original versio
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