Co-stability of radicals and its applications to PI-theory (Q2822085)

\textit{M. Cohen} and \textit{S. Montgomery} proved in [Trans. Am. Math. Soc. 282, 237--258 (1984; Zbl 0533.16001)] that the Jacobson radical of an associative algebra graded by a finite group \(G\) whose order is invertible in the base field is graded. This was generalized to Hopf algebra actions in [\textit{V. Linchenko}, Math. Appl., Dordr. 555, 121--127 (2003; Zbl 1060.16038)], where it was proved that the Jacobson radical of a finite dimensional \(H\)-module associative algebra is stable under the action of the involutory Hopf algebra \(H\). In the present paper, the author provides a version of this result for coactions of infinite dimensional Hopf algebras. The main result states that if \(A\) is a finite-dimensional associative \(H\)-comodule algebra over a field \(F\) for some involutory Hopf algebra \(H\) not necessarily finite-dimensional, where either \(\mathrm{char}(F)=0\) or \(\mathrm{char}(F)>\dim(A)\), then the Jacobson radical \(J(A)\) is an \(H\)-subcomodule of \(A\). As an application, an analog of Amitsur's conjecture for graded polynomial identities of finite-dimensional associative graded algebras over a field of characteristic 0 is proved.