Kemer's Theory for H-Module Algebras with Application to the PI Exponent

Let H be a semisimple finite dimensional Hopf algebra over a field F of zero characteristic. We prove three major theorems: 1. The Representability theorem which states that every H-module (associative) F-algebra W satisfying an ordinary PI, has the same H-identities as the Grassmann envelope of an $H\otimes\left(F\mathbb{Z}/2\mathbb{Z}\right)^{*}$-module algebra which is finite dimensional over a field extension of F. 2. The Specht problem for H-module (ordinary) PI algebras. That is, every H-T-ideal $\Gamma$ which contains an ordinary PI contains H-polynomials $f_{1},...,f_{s}$ which generates $\Gamma$ as an H-T-ideal. 3. Amitsur's conjecture for H-module algebras, saying that the exponent of the H-codimension sequence of an ordinary PI H-module algebra is an integer.