Let \(V\) be an infinite dimensional vector space over a field \(F\) with basis \(B=\{e_1, e_2, \dots \}\) and let \(E\) be the Grassmann algebra of \(V\). If \(G\) is a finite abelian group, it is well known that any \(G\)-grading on \(E\) such that \(B\) is homogeneous can be obtained by defining a map \(\varphi: B \longrightarrow G \), i.e., by attributing degrees to the elements of \(B\). For \(G=\mathbb{Z}_2\), up to isomorphism, such gradings are given by one of the following maps: \(\varphi_\infty\), \(\varphi_k\) or \(\varphi_{k^*}\). Such maps are defined by: \(\varphi_\infty(e_i)=1\), if \(i\) is odd or \(0\) if \(i\) is even; \(\varphi_{k}(e_i)=0\), if \(i\leq k\) and \(1\) otherwise;\(\varphi_{k^*}(e_i)=1\), if \(i\leq k\) and \(0\) otherwise. When endowed with such gradings, \(E\) is denoted, respectively, by \(E_\infty\), \(E_k\) or \(E_{k^*}\). The canonical \(\mathbb{Z}_2\)-grading on \(E\) is given by the map \(\varphi_{0}\), and in this case, it is denoted simply by \(E\). The corresponding \(\mathbb{Z}_2\)-graded identities of such algebras, have been described in by O. M. Di Vincenzo and the second author in [Linear Algebra Appl. 431, No. 1–2, 56–72 (2009; Zbl 1225.16009)] in the case of characteristic zero, and by the first author in [Linear Algebra Appl. 435, No. 12, 3297–3313 (2011; Zbl 1230.16019)] in the case of infinite fields of characteristic \(p>2\).
From a result of O. M. Di Vincenzo and V Nardozza [Commun. Algebra 31, No. 3, 1453–1474 (2003; Zbl 1039.16023)], given a set of generators for the \(T_G\)-ideal of a \(G\)-graded algebra \(A\), in characteristic zero, one can obtain a set of generators for the \(T_{G\times \mathbb{Z}_2}\)-ideal of \(A\otimes E\) with its natural \(G\times \mathbb{Z}_2\)-grading. Moreover, the elements of the last set have the same degree as the corresponding elements of the initial set.
In the paper under review, the authors investigate \(\mathbb{Z}_2\times \mathbb{Z}_2\)-graded identities of \(E_{k^*}\otimes E\) in both, zero and positive characteristic. In both cases, they exhibit a finite basis for the ideal of \(\mathbb{Z}_2\times \mathbb{Z}_2\)-graded identities. The interesting point here is that in characteristic \(p>2\), the correspondence between degrees mentioned above is no longer valid if \(p>k\). In particular, the result of Di Vincenzo and Nardozza does not hold in positive characteristic.
In the final section, the authors show that the \(\mathbb{Z}_2\times \mathbb{Z}_2\)-graded Gelfand-Kirillov dimension of \(E_{k^*}\otimes E\) in \(m\) variables is \(m\) and, based on this and previous results, they conjecture that the GK-dimension does not change with respect to tensor product by \(E\), i.e., \(GK\dim_m^G(A)=GK\dim_m^{G\times \mathbb{Z}_2}(A\otimes E)\). The proofs presented in the paper use classical methods of the theory of polynomial identities.