Observations on crossed products and invariants of Hopf algebras (Q1337787)

Let \(B = A\#_ \sigma H\) denote a crossed product of the associative algebra \(A\) with the finite-dimensional Hopf algebra \(H\). We show that if \(H\) is pointed then the Jacobson radicals of \(B\) and \(A\) are related by \(J(B)^{\text{dim}_ kH} \subseteq J(A)B\). In the case where \(A\) is an \(H\)-module algebra and the trace map from \(A\) to the algebra of \(H\)- invariants \(A^ H\) is surjective we establish the following estimate for the right global dimension of \(A^ H\): \[ \text{r.gldim} A^ H \leq \text{r.gldim} A + \text{gldim} H + \min\{\text{fdim} _{A^ H} A, \text{pdim} A_{A^ H}\}. \] We also investigate the behavior of \(A\)- modules \(W\) under restriction to \(A^ H\) in this case. Assuming \(H\) to be pointed we show, among other things, that if \(W\) is Noetherian (Artinian) then \(W_{A^ H}\) is likewise.