PI (non)equivalence and Gelfand-Kirillov dimension in positive characteristic.
The author shows that the so called Tensor Product Theorem of Kemer for verbally prime varieties over fields of zero characteristic cannot be extended to infinite fields of positive characteristic \(p>2\). The author computes the Gelfand-Kirillov dimensions of several relatively free algebras and thus obtains a new proof of the fact that the algebras \(M_{a,a}(E)\otimes E\) and \(M_{2a}(E)\) are not PI equivalent in characteristic \(p>2\). Also, it is shown that the following algebras are not PI equivalent in positive characteristic: \(M_{a,b}(E)\otimes M_{c,d}(E)\) and \(M_{ac+bd,ad+cb}(E)\); and \(M_{a,b}(E)\otimes M_{c,d}(E)\) and \(M_{e,f}(E)\otimes M_{g,h}(E)\) when \(a\geq b\), \(c\geq d\), \(e\geq f\), \(g\geq h\), \(ac+bd=eg+fh\), \(ad+bc=eh+fg\) and \(ac\neq eg\). Here \(E\) is the infinite dimensional Grassmann algebra with 1, and \(M_{a,b}(E)\) the subalgebra of \(M_{a+b}(E)\) of the block matrices with blocks \(a\times a\) and \(b\times b\) on the main diagonal with entries from \(E_0\), and off-diagonal entries from \(E_1\), where \(E=E_0\otimes E_1\) is the natural grading.
scientific article; zbMATH DE number 5620618
scientific article