Cancellation problem via locally nilpotent derivations
César F. Venegas R., Helbert J. Venegas R
TL;DR
This survey investigates the Zariski cancellation problem across commutative, noncommutative, and skew algebras using locally nilpotent derivations and the Makar-Limanov invariant. It develops a three-step paradigm: compute $\mathrm{ML}(A)$, establish $\mathrm{ML}$-stability under polynomial extensions, and deduce cancellativity. The authors prove complete cancellation results for GK-dimension one and two algebras in the noncommutative setting and for skew extensions of dimension one, while highlighting fundamental limitations of ML-stability in the skew setting and numerous open problems in higher dimensions. The work emphasizes combining LND methods with homological, discriminant, and Poisson-geometric techniques to address higher-dimensional and skew cases, offering a roadmap for future research.
Abstract
The Zariski cancellation problem plays a central role in affine algebraic geometry and noncommutative algebra, with locally nilpotent derivations providing a fundamental invariant-theoretic approach. This article presents a unified survey of cancellation phenomena in commutative algebras, noncommutative algebras, and skew (Ore-type) extensions, emphasizing the role of rigidity and the Makar--Limanov invariant. We explain how the locally nilpotent derivation framework successfully detects cancellation in rigid settings, while also identifying its inherent limitations, particularly in the skew case where Makar--Limanov stability fails. This perspective clarifies the scope and the boundaries of the locally nilpotent derivation method in cancellation theory.