Cancellation problem via locally nilpotent derivations

2025/12/17 by R., César F. Venegas, R, Helbert J. Venegas
#13B25 #14R20 (Secondary) #16W25 (Primary) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2512.15590

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.

Citations

• A survey on Zariski cancellation problems for noncommutative and Poisson algebras
• Cancellation of Morita and Skew Types
• Noncommutative analogues of a cancellation theorem of Abhyankar, Eakin,\n and Heinzer
• Cancellation problem for AS-Regular algebras of dimension three
• Morita cancellation problem
• Azumaya loci and discriminant ideals of PI algebras
• Discriminants and Automorphism Groups of Veronese subrings of skew polynomial rings
• Noncommutative discriminants via Poisson primes
• Zariski Cancellation Problem for Noncommutative Algebras
• Stable Categories of Graded Maximal Cohen-Macaulay Modules over Noncommutative Quotient Singularities
• Nakayama automorphism and applications
• Ample Group Action on AS-regular Algebras and Noncommutative Graded Isolated Singularities
• The discriminant criterion and automorphism groups of quantized algebras
• The discriminant controls automorphism groups of noncommutative algebras
• On Zariski's Cancellation Problem in positive characteristic
• Rigid rings and Makar-Limanov techniques
• Constructing (almost) rigid rings and a UFD having infinitely generated Derksen and Makar-Limanov invariant
• On the rigidity of small domains

Related

• Zariski cancellation problem for non-domain noncommutative algebras
• A survey on Zariski cancellation problems for noncommutative and Poisson algebras
• The Filtration Induced by a Locally Nilpotent Derivation
• Cancellation of Morita and Skew Types
• On double Danielewski varieties