Actions of Hopf algebras on pro-semisimple Noetherian algebras and their invariants (Q2717704)

The author obtains analogues of the first fundamental theorem of classical invariant theory for actions of finitely semisimple Hopf algebras. He proves that if a Hopf algebra \(H\) over a field \(k\) is finitely semisimple, and \(A\) is a commutative, finitely generated (resp. Noetherian) locally finite \(H\)-module algebra, then the algebra \(A^H\) of invariants is also finitely generated (resp. Noetherian). Here, finitely semisimple means that any finite-dimensional left \(H\)-module is semisimple.NEWLINENEWLINENEWLINEThe main goal of the paper is to relax the condition of local finiteness on \(A\), and find counterparts of the above theorem for \(H\)-module algebras arising from locally finite \(H\)-module algebras via completion. A pro-semisimple \(H\)-module algebra is an \(H\)-module algebra equipped with a linear topology defined by a family \(\{I_i\}\) of two-sided invariant ideals of \(A\) such that \(A/I_i\) is a semisimple \(H\)-module for all \(i\), and \(A\) is isomorphic to the inverse limit of the \(A/I_i\). Then if \(A\) is a pro-semisimple right Noetherian \(H\)-module algebra, so is \(A^H\). For commutative \(A\), if \(H\) is finitely semisimple and \((A,m)\) is a local, complete Noetherian \(H\)-module algebra such that \(m\) is invariant, and \(A/m\) is a finite field extension of \(k\), then \(A^H\) is a local, complete Noetherian algebra with unique maximal ideal \(m^H\). This applies in particular when \(A\) is the quotient of a ring of power series in \(n\) variables. In that case he shows also that if \(A=k[[X_1,\dots,X_n]]\), then \(A^H\) is a Noetherian Cohen-Macaulay ring.