Cancellation problem via locally nilpotent derivations
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911453007577088 |
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| author | R., César F. Venegas R, Helbert J. Venegas |
| author_facet | R., César F. Venegas R, Helbert J. Venegas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15590 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cancellation problem via locally nilpotent derivations R., César F. Venegas R, Helbert J. Venegas Rings and Algebras 16W25 (Primary), 13B25, 14R20 (Secondary) 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. |
| title | Cancellation problem via locally nilpotent derivations |
| topic | Rings and Algebras 16W25 (Primary), 13B25, 14R20 (Secondary) |
| url | https://arxiv.org/abs/2512.15590 |