On Noether's Degree Bound for Finite Group Schemes

Abstract

This paper establishes Noether's classical degree bound ${\hat{I}}^2(G)\le |G|$ for finite and linearly reductive group schemes. On the other hand, we provide examples of infinitesimal group schemes where ${\hat{I}}^2(G)$ is unbounded. We also generalize Molien's formula to finite and linearly reductive group schemes.