Specht’s problem for associative affine algebras over commutative noetherian rings

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Abstract

In a series of papers by the authors we introduced full quivers and pseudo-quivers of representations of algebras, and used them as tools in describing PI-varieties of algebras. In this paper we apply them to obtain a complete proof of Belov’s solution of Specht’s problem for affine algebras over an arbitrary Noetherian ring. The inductive step relies on a theorem that enables one to find a “q̄-characteristic coefficient-absorbing polynomial in each T-ideal Γ”, i.e., a nonidentity of the representable algebra A arising from Γ, whose ideal of evaluations in A is closed under multiplication by q̄-powers of the characteristic coefficients of matrices corresponding to the generators of A, where q̄ is a suitably large power of the order of the base field. The passage to an arbitrary Noetherian base ring C involves localizing at finitely many elements a kind of C, and reducing to the field case by a local-global principle.

Original languageEnglish
Pages (from-to)5553-5596
Number of pages44
JournalTransactions of the American Mathematical Society
Volume367
Issue number8
DOIs
StatePublished - 2015

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Publisher Copyright:
© 2015 American Mathematical Society.

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