TY - JOUR

T1 - Specht’s problem for associative affine algebras over commutative noetherian rings

AU - Belov-Kanel, Alexei

AU - Rowen, Louis

AU - Vishne, Uzi

N1 - Publisher Copyright:
© 2015 American Mathematical Society.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84929430260&partnerID=8YFLogxK

U2 - 10.1090/tran/5983

DO - 10.1090/tran/5983

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AN - SCOPUS:84929430260

SN - 0002-9947

VL - 367

SP - 5553

EP - 5596

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 8

ER -