Tensor product theorems in positive characteristic.
In his celebrated paper [Math. USSR, Izv. 25, 359-374 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 5, 1042-1059 (1984; Zbl 0586.16010)] \textit{A. R. Kemer} showed the importance of the polynomial identities of some algebras as building blocks of the polynomial identities of any PI-algebra over a field of characteristic 0. The corresponding T-ideals of these algebras are T-prime, i.e. prime in the class of T-ideals of the free algebra. The complete list of algebras with T-prime T-ideals includes matrix algebras with entries from the field and from the Grassmann algebra and some of their subalgebras. Up to PI-equivalence, this list is closed under tensor products and the theory of Kemer gives rules for the tensor multiplication. A direct combinatorial proof, using graded polynomial identities, for these rules was given by \textit{A. Regev} [J. Algebra 133, No. 2, 512-526 (1990; Zbl 0738.16007)]. In the paper under review the authors transfer the proof of Regev's to the case of an infinite field of odd characteristic and show that the tensor product of two T-prime algebras from the list of Kemer has the same multilinear polynomial identities as the T-prime algebra from the case of characteristic 0. Nevertheless, one cannot expect the complete analogue to hold in positive characteristic. The authors show that the tensor square \(E\otimes E\) of the Grassmann algebra has more polynomial identities than the algebra \(M_{1,1}(E)\). In particular, \([x_1^{p^2},x_2]=0\) is a polynomial identity for \(E\otimes E\) which does not hold for \(M_{1,1}(E)\), where \(p\) is the characteristic of the base field.
scientific article; zbMATH DE number 2116886
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