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

What is it about?

The Specht problem asks whether all polynomial identities of an associative algebra are consequences of a finite set of identities. A. R. Kemer provided positive answers for algebras over characteristic zero fields in [Dokl. Akad. Nauk SSSR 298 (1988), no. 2, 273–277; MR0937115] and for finitely generated algebras over infinite fields in [Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 4, 726–753; MR1073084]. Counterexamples for non-finitely generated algebras over fields of positive characteristic were given by A. Kanel-Belov [Mat. Sb. 191 (2000), no. 3, 13–24; MR1773251] and A. V. Grishin [Fundam. Prikl. Mat. 5 (1999), no. 1, 101–118; MR1799541]. In the current work the authors prove the Specht conjecture for finitely generated algebras over Noetherian rings. This is best possible in that it is false over non-Noetherian rings. One consequence is that the Jacobson radical of a generic PI-algebra of a finitely generated algebra over a Noetherian ring is nilpotent. Reviewed by Allan Berele

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