On the existence of the graded exponent for finite dimensional \(\mathbb Z_p\)-graded algebras. (Q2888780)

More than ten years ago \textit{A. Giambruno} and \textit{M. Zaicev} [Adv. Math. 140, No. 2, 145-155 (1998; Zbl 0920.16012); ibid. 142, No. 2, 221-243 (1999; Zbl 0920.16013)] proved that the codimension sequence of an associative PI-algebra \(A\) over a field \(F\) of characteristic 0 satisfies the condition that \(\exp(A)=\lim_{n\to\infty}\root n\of{c_n(A)}\) exists and is an integer. Later, similar theorems were established for other classes of algebras, including algebras with additional structure. For nonassociative algebras, counterexamples were also constructed. In particular, the papers by \textit{A. Giambruno} and \textit{M. Zaicev} [J. Algebra 222, No. 2, 471-484 (1999; Zbl 0944.16023)] and \textit{F. Benanti, A. Giambruno} and \textit{M. Pipitone} [J. Algebra 269, No. 2, 422-438 (2003; Zbl 1039.16021)] established the existence and the integrality of the corresponding version of the exponent for finite dimensional associative algebras with involution and finitely generated \(\mathbb Z_2\)-graded associative algebras, respectively.NEWLINENEWLINE In the paper under review the authors consider a finite dimensional unitary algebra \(A\) over an algebraically closed field \(F\) of characteristic zero and assume that \(A\) is graded by a group of prime order. Then the main result gives that the graded exponent of \(A\) exists and is an integer.