A Counterexample to Weitzenb\"ock's Theorem in Characteristic $p$

TL;DR

This paper provides a counterexample to Weitzenb"ock's Theorem in characteristic p, demonstrating that the ring of invariants under a certain additive group action can be non-finitely generated in positive characteristic.

Contribution

It constructs an explicit linear action of a on affine 5-space over an algebraically closed field of characteristic p, where the invariant ring is not finitely generated.

Findings

Counterexample to Weitzenbbock's Theorem in characteristic p

Invariant ring under a action is not finitely generated

Explicit linear representation inducing the counterexample

Abstract

In this paper we give a counterexample to Weitzenb\"{o}ck's Theorem in positive characteristic. Namely we show that if $ k $ is an algebraically closed field of characteristic $ p $, there is an action of $ \mathbb{G}_{a} $ on $ \mathbb{A}^{5}_{k} $, induced by a linear representation of $ \mathbb{G}_{a} $, such that the ring of invariants $ k[x_{1},\dots,x_{5}]^{\mathbb{G}_{a}} $ is not finitely generated.