Rigid rings and Makar-Limanov techniques

TL;DR

This paper explores the concept of rigidity in algebraic rings, providing examples, strategies, and tools to determine rigidity, and discusses unresolved cases in hypersurfaces.

Contribution

It introduces two general strategies for assessing ring rigidity and offers practical tools and lemmas to aid in this determination.

Findings

Provided several examples of rigid rings

Outlined parametrization and filtration techniques for rigidity analysis

Highlighted potential pitfalls and unresolved cases in hypersurfaces

Abstract

A ring is rigid if there is no nonzero locally nilpotent derivation on it. In terms of algebraic geometry, a rigid coordinate ring corresponds to an algebraic affine variety which does not allow any nontrivial algebraic additive group action. Even though it is thought that "generic" rings are rigid, it is far from trivial to show that a ring is rigid. In this paper we provide several examples of rigid rings and we outline two general strategies to help determine if a ring is rigid, which we call "parametrization techniques" and "filtration techniques". We provide many little tools and lemmas which may be useful in other situations. Also, we point out some pitfalls to beware when using these techniques. Finally, we give some reasonably simple hypersurfaces for which the question of rigidity remains unsettled.