On Noether's Degree Bound for Finite Group Schemes

TL;DR

This paper proves a classical degree bound for finite and linearly reductive group schemes, shows unbounded bounds for infinitesimal schemes, and generalizes Molien's formula to these contexts.

Contribution

It establishes Noether's degree bound for finite and linearly reductive group schemes and extends Molien's formula to these cases, also providing examples where the bound is unbounded.

Findings

Noether's degree bound $eta(G) \

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Abstract

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