Kemer's theory for \(H\)-module algebras with application to the PI exponent (Q279731)

Throughout the review, \(H\) denotes a semisimple and finite dimensional Hopf algebra over a field \(F\) containing \(\mathbb{C}\). The main part of the paper under review is the proof of the affine \(H\)-representability theorem which states that every affine \(H\)-module algebra \(W\) over \(F\) satisfying an ordinary polynomial identity has the same \(H\)-identities as (is \(H\)-PI equivalent to) some finite dimensional \(H\)-module algebra \(A\) over some field extension \(L\) of \(F\). As a consequence of the affine \(H\)-representability theorem, the author proves:NEWLINENEWLINE (1) The \(H\)-representability theorem which states that every \(H\)-module algebra \(W\) over \(F\) satisfying an ordinary polynomial identity is \(H\)-PI equivalent to the Grassmann envelope of some finite dimensional \(H\otimes (F\mathbb{Z}/2\mathbb{Z})^*\)-module algebra \(A\) over some field extension \(L\) of \(F\);NEWLINENEWLINE (2) The Specht problem for \(H\)-module algebras which states that every \(H-T\)-ideal \(\Gamma\) containing an ordinary polynomial identity is generated as an \(H-T\)-ideal by a finite number of \(H\)-polynomials \(f_1,f_2,\ldots,f_s\), or equivalently that every ascending chain \(\Gamma_1\subseteq\Gamma_2\subseteq\ldots\) of \(H-T\)-ideals containing an ordinary polynomial identity stabilizes after a finite number of steps;NEWLINENEWLINE (3) The Amitsur conjecture for \(H\)-module algebras which states that the \(H\)-exponent of the \(H\)-codimension sequence of an \(H\)-module algebra \(W\) over \(F\) satisfying an ordinary polynomial identity is an integer.NEWLINENEWLINE The paper is a continuation of [\textit{E. Aljadeff} et al., J. Pure Appl. Algebra 220, No. 8, 2771--2808 (2016; Zbl 1342.16017)].