Regularity of the ring of invariants under certain actions of finite abelian Hopf algebras in characteristic \(p\) (Q1324179)

Let \(k\) be a separately closed field of positive characteristic \(p\). Let \(H\) be a commutative Hopf algebra over \(k\) with basis \(e_ 0,e_ 1,\dots,e_{p^ n - 1}\) and coalgebra structure \(\Delta e_ s = \sum_{i + j = s} e_ i \otimes e_ j\). Suppose \(A\) is a left \(H\)- module algebra which is a regular local \(k\)-algebra with maximal ideal \(m\), and that \((A,m)\) has Krull dimension \(t\) with \(e_ 1 \cdot x \notin m\) for some \(x \in m\). If \(A^ H\) denotes the ring of invariants, the authors show that \(A\) is a free \(A^ H\)-module with basis \(1,x,\dots,x^{p^ n-1}\), \(A^ H\) is a regular local ring of Krull dimension \(t\) containing \(A^{p^ n}= \{a^{p^ n}\mid a \in A\}\), and that \(A\) has a regular system of parameters \(x_ 1,\dots,x_ t\) such that \(x_ 1 = x\) and \(x^{p^ n},x_ 2,\dots, x_ t\) is a regular system of parameters in \(A^ H\).