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Representability of relatively free affine algebras over a Noetherian ring; Dedicated To Andre Leroy On His Retirementстатья
- Авторы: Kanel-Belov Alexei, Rowen Louis, Vishne Uzi
- Журнал: Journal of Algebra and its Applications
- Год издания: 2023
- Издательство: World Scientific
- Местоположение издательства: United States
- DOI: 10.1142/s0219498825501543
- Аннотация: Over the years questions have arisen about T-ideals of (noncommutative)polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices,one often has little control in determining the specific evaluations of the polynomial.One way of overcoming this difficulty in characteristic 0, is to reduce to multilinearpolynomials and utilizing the representation theory of the symmetric group. But thistechnique is unavailable in characteristic p > 0.An alternative method, which succeeds, is the process of “hiking” a polynomial, inwhich one specializes its indeterminates in several stages, to obtain a polynomial inwhich Capelli polynomial is embedded, in order to get control on its evaluations. Thismethod was utilized on homogeneous polynomials in the proof of Specht’s conjecturefor affine algebras over fields of positive characteristic.In this paper we develop hiking further to nonhomogeneous polynomials, to applyto the “representability question.” Kemer proved in 1988 that every affine relativelyfree PI algebra over an infinite field, is representable. In 2010, the first author ofthis paper proved more generally that every affine relatively free PI algebra overany commutative Noetherian unital ring is representable, [8]. We present a different,complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. Wethen obtain the full result over a Noetherian commutative ring, using Noetherianinduction on T-ideals.The bulk of the proof is for the case of a base field of positive characteristic. Here,whereas the usage of hiking is more direct than in proving Specht’s conjecture, onemust consider nonhomogeneous polynomials when the base ring is finite, which entailscertain difficulties to be overcome.In the appendix we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems
- Добавил в систему: Белов Алексей Яковлевич
Прикрепленные файлы
| № | Имя | Описание | Имя файла | Размер | Добавлен |
|---|---|---|---|---|---|
| 1. | Полный текст | RepringsLeroyVolume.pdf | 482,0 КБ | 27 ноября 2023 [AlexeiBelov] |