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Retracts that are kernels of locally nilpotent derivations
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 1, pp. 191-199
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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 $B=R\oplus I$ 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 $B\cong R^{[2]}$? 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.
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 $B=R\oplus I$ 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 $B\cong R^{[2]}$? 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.
DOI : 10.21136/CMJ.2021.0388-20
Classification : 13N15, 14R10
Keywords: retract; locally nilpotent derivation; kernel; Zariski's cancellation problem 
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Liu, Dayan; Sun, Xiaosong. Retracts that are kernels of locally  nilpotent derivations. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 1, pp. 191-199. doi: 10.21136/CMJ.2021.0388-20

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