The local finite basis property and local representability of varieties of associative rings

Belov, A.Y.

Автор: Belov A.Ya
Журнал: Izvestiya. Mathematics
Том: 74
Номер: 1
Год издания: 2010
Издательство: American Mathematical Society
Местоположение издательства: United States
Первая страница: 1
Последняя страница: 126
DOI: 10.1070/IM2010v074n01ABEH002481
Аннотация: Two of the main results of the celebrated theory of Kemer for structure theory of T-ideals over a field of characteristic 0 state that every T-ideal satisfies the Specht property, i.e., has a finite basis of its polynomial identities, and the representability of finitely generated relatively free algebras. The latter means that such an algebra can be embedded into a matrix algebra over an extension of the base field. On the other hand, there are several counterexamples to the Specht problem in the case of positive characteristic and the first one was constructed by the author of the paper under review. These results are the main motivation of the present very interesting paper. Its goal is to transfer some of them to the case of PI-algebras over commutative Noetherian rings. One of the central results of the paper is that PI-algebras over commutative Noetherian rings satisfy the Specht property locally: Finitely generated PI-algebras satisfy the ascending chain condition for T-ideals. Additionally, finitely generated relatively free algebras are representable. In the general case representability is understood in a more general sense. Finitely generated relatively free algebras are embeddable in algebras which are finite dimensional over their centers. Hence representability may be not realized by matrices. As a consequence the author obtains that finitely generated relatively free algebras over Noetherian commutative rings are residually finite. The main results were announced in the proceedings and the abstracts of several conferences in 1998-2000. The paper contains also interesting discussions on the history of Specht-type problems and on related problems and topics. Among them are: non-commutative algebraic geometry; the programme of Kemer which is based on the idea of a progressive mutual approximation of the two natural ways to define a variety of algebras -- in terms of identities and of generating algebras; invariant theory, etc. The author has collected a list of open problems and a comprehensive list of references, not only on PI-algebras but also on related topics both in the associative and non-associative cases.Reviewer: Vesselin Drensky (Sofia)
Добавил в систему: Белов Алексей Яковлевич

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