Tensor product theorems in positive characteristic

In this paper we study tensor products of T-prime T-ideals over infinite fields. The behaviour of these tensor products over afield of characteristic 0 was described by Kemer. First we show, using methods due to Regev, that such a description holds if one restricts oneself to multilinear polynomials only. Second, applying graded polynomial identities, we prove that the tensorproduct theorem fails for the T-ideals of the algebras $M_{1,1}(E)$ and $E\otimes E$ where $E$ is the infinite dimensional Grassmann algebra; $M_{1,1}(E)$ consists of the $2\times 2$ matrices over $E$ having even (i.e. central) elements of $E$, and the other diagonal consisting of odd (anticommuting) elements of $E$. Then we pass to other tensor products and studythe respective graded identities. We obtain new proofs of some cases of Kemer's tensor product theorem. Note that these proofs do not depend on the structure theory of T-ideals but are ``elementary'' ones. Finally, using graded identities once gain,we show that the tensor product theorem fails in one more case when the base field is of positive characteristic. All this comesto show once more that the structure theory of T-ideals is essentially about the multilinear polynomial identities.