Tensor product theorems in positive characteristic

Abstract

In this paper we study tensor products of T -prime T -ideals over infinite fields. The behaviour of these tensor products over a field 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 tensor product theorem fails for the T -ideals of the algebras M 1,1 (E) and E ⊗ E where E is the infinite-dimensional Grassmann algebra; M 1,1 (E) consists of the 2 × 2 matrices over E having even (i.e., central) elements of E, and the other diagonal consisting of odd (anticommuting) elements of E. Note that these proofs do not depend on the structure theory of T -ideals but are "elementary" ones. All this comes to show once more that the structure theory of T -ideals is essentially about the multilinear polynomial identities.

PI (non)equivalence and Gelfand-Kirillov dimension in positive characteristic

“…It was proved that the Tensor product theorem is still valid over infinite fields of characteristic p > 2 as long as one considers multilinear polynomials only. Furthermore in [4] it was proved that the third statement of the Theorem fails, and in [5] the same was done for the first statement (when a = b = 1). In the next section we recall some of the notation and main results of these papers that we shall need.…”

“…The algebras A a,b were introduced in [4,5]. Let 0 be the set of all (i, j ) such that either 1 ≤ i, j ≤ a or a + 1 ≤ i, j ≤ a + b = n, and let 1 be the set of (i, j ) with either 1 ≤ i ≤ a, a + 1 ≤ j ≤ a + b, or 1 ≤ j ≤ a, a + 1 ≤ i ≤ a + b.…”

“…Other, elementary proofs of cases of the Tensor product theorem were given in [4,5,15]. We draw the reader's attention to the fact that in [4,5,15], the behavior of the corresponding T-ideals in positive characteristic was studied. It was proved that the Tensor product theorem is still valid over infinite fields of characteristic p > 2 as long as one considers multilinear polynomials only.…”

“…It turned out that E ⊗ E and A satisfy the same graded and hence ordinary polynomial identities. Using properties of A in [4] it was shown that T (M 1,1 (E)) T (E ⊗ E) in positive characteristic. Further on, in [5], certain subalgebras A a,b of M a+b (E) were constructed and these turned out to be quite useful in establishing the proper inclusion T (M 2 (E)) T (M 1,1 (E) ⊗ E), see [5].…”

Polynomial identities of algebras in positive characteristic

“…Other, elementary proofs of cases of the Tensor product theorem were given in [2,3,13]. We draw the reader's attention to the fact that in [2,3,13], the behavior of the corresponding T-ideals in positive characteristic was studied. It was proved that the Tensor product theorem is still valid over infinite fields of characteristic p > 2 as long as one considers multilinear polynomials only.…”