On regularity of the algebra of covariants for actions of pointed Hopf algebras on regular commutative algebras (Q2762046)

The author works over an algebraically closed field \(k\). If \(H\) is a Hopf algebra with counit \(\varepsilon\) and \(A\) is a left \(H\)-module algebra, the algebra of covariants \(A_H\) is the quotient algebra \(A/I\), where \(I\) is the ideal of \(A\) generated by all the elements of the form \(ha\), where \(h\in\text{Ker}(\varepsilon)\), \(a\in A\). The aim of the paper is to prove the following theorems: (1) If \(A\) is commutative, Noetherian and local with invariant maximal ideal \(m\) such that \(A\) is a regular local ring, \(k\simeq A/m\), and \(m/m^j\) is a semisimple \(H\)-module for any \(j\geq 2\), then the local ring \(A_H\) is regular; (2) If \(H\) is pointed and \(A\) is commutative and finitely generated such that \(A_m\) is a regular local ring and \(m/m^j\) is a semisimple \(H\)-module for any invariant maximal ideal \(m\) of \(A\), and any \(j\geq 2\), then \(A_H\) is regular. A consequence of (1) is the result of \textit{B. Iversen} [Invent. Math. 16, 229-236 (1972; Zbl 0246.14010)] stating that for a linearly reductive algebraic group \(G\) acting regularly on a smooth separated \(k\)-scheme \(X\), the fixed point scheme \(X^G\) is smooth.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00024].