Abstract
This paper is motivated by the following question arising in universal algebraic geometry: when do two algebras have the same geometry? This question requires considering algebras in a variety Θ and the category Θ0of all finitely generated free algebras in Θ. The key problem is to study how far the group Aut Θ0of all automorphisms of the category Θ0is from the group Inn Θ0of inner automorphisms of Θ0(see [7, 10] for details). Recall that an automorphism ϒ of a category 𝔎 is inner, if it is isomorphic as a functor to the identity automorphism of the category 𝔎.Let Θ = 𝔑dbe the variety of all nilpotent groups whose nilpotency class is ≤ d. Using the method of verbal operations developed in [8, 9], we prove that every automorphism of the category [Formula: see text], d ≥ 2 is inner.
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