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Retracts that are kernels of locally nilpotent derivations

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Abstract

Let k be a field of characteristic zero and B a k-domain. Let R be a retract of B being the kernel of a locally nilpotent derivation of B. We show that if B = R ⊕ I for some principal ideal I (in particular, if B is a UFD), then B = R[1], i.e., B is a polynomial algebra over R in one variable. It is natural to ask that, if a retract R of a k-UFD B is the kernel of two commuting locally nilpotent derivations of B, then does it follow that B ≅ R[2]? We give a negative answer to this question. The interest in retracts comes from the fact that they are closely related to Zariski’s cancellation problem and the Jacobian conjecture in affine algebraic geometry.

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Acknowledgments

An earlier version of the paper was finished in May 2019 when the authors visited Western Michigan University. They thank Prof. Gene Freudenburg and Dr. Takanori Nagamine for helpful discussions on retracts and locally nilpotent derivations during their visiting, and they are indebt to Prof. Gene Freudenburg for showing them how to compute the Derksen invariant in Proposition 2.14.

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Authors and Affiliations

  1. School of Mathematics, Jilin University, 2699 Qianjin Street, Changchun, 130012, P. R. China

    Dayan Liu & Xiaosong Sun

Authors
  1. Dayan Liu
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  2. Xiaosong Sun
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Correspondence to Xiaosong Sun.

Additional information

This work was supported by the NSF of China (11871241), and the Science and Technology Project of Jilin Provincial Education Department (JJKH20211032KJ).

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Liu, D., Sun, X. Retracts that are kernels of locally nilpotent derivations. Czech Math J 72, 191–199 (2022). https://doi.org/10.21136/CMJ.2021.0388-20

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  • DOI: https://doi.org/10.21136/CMJ.2021.0388-20

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