On the identities of subalgebras of matrices over the Grassmann algebra (Q1100255)

The paper under review studies the asymptotic behaviour of the cocharacter and codimension sequences of K-prime T-ideals of the free associative algebra over a field of characteristic 0. The K-prime and K- semiprime T-ideals have been described by \textit{A. R. Kemer} [Izv. Akad. Nauk SSSR, Ser. Mat. 48, No.5, 1042-1059 (1984); translation: Math. USSR, Izv. 25, 359-374 (1985; Zbl 0586.16010)] who has developed the structure theory of T-ideals in the spirit of commutative algebra. The K-prime ideals coincide with the ideal of all polynomial identities for some subalgebras of matrix algebras with entries either from the field or from the Grassmann algebra. In a series of papers the author has obtained the asymptotics of the cocharacters and codimensions of the matrices. Now, the main result is that the exponential growth of the codimensions \(\{c_ n(T)\}\) of a K-prime T-ideal T is explicitly captured: For any K- prime T-ideal T there exist constants a, \(c_ 1\), \(c_ 2\), \(g_ 1\), \(g_ 2\) such that \[ c_ 1(1/n)^{g_ 1}a^ n\leq c_ n(T)\leq c_ 2(1/n)^{g_ 2}a^ n. \] As a consequence of the proof, the behaviour of the cocharacters of these T-ideals is also obtained.