Kemer's theorem for affine PI algebras over a field of characteristic zero

What is it about?

In [Math. Z. 52 (1950), 557–589; MR0035274], W. Specht asked whether every PI algebra has a finite basis of identities, i.e., if $A$ is an algebra satisfying polynomial identities, does there exist a finite set of identities of $A$ such that all identities of $A$ are consequences of these? The proof of this conjecture was in the sights of an entire generation of polynomial identity theorists until A. R. Kemer finally proved it in 1986 in [Dokl. Akad. Nauk SSSR 298 (1988), no. 2, 273–277; MR0937115]. Moreover, continuing his investigations, he proved his finite representability theorem in 1987 [Algebra i Logika 27 (1988), no. 3, 274–294, 375; MR0997959], which implies the Specht conjecture as a consequence. In the case of finitely generated algebras the finite representability theorem says that every finitely generated PI algebra satisfies the same ideal of identities as a finite-dimensional algebra (all over a field of characteristic zero). Kemer's work was highly original and contained many interesting new ideas. However, his work was so difficult that a second generation of PI theorists was not able to follow it fully. I believe that the process of assimilating Kemer's ideas did not begin until about 15 years after his papers appeared, and that this process is only now being completed. Finally, at the end of this second generation, in the paper under review Aljadeff, Kanel-Belov, and Karasik rewrite the proof, filling in Kemer's leaps and making his deep insights accessible to a new generation. This is an important service to the community of researchers in polynomial identities. Reviewed by Allan Berele

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