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 -