On finite generation of kernels of locally nilpotent \(R\)-derivations of \(R[X,Y,Z]\). (English) Zbl 1234.13027
In the article under review, the authors establish that the kernel of a locally nilpotent \(R\)-derivation of a polynomial ring in \(3\) variables over a Dedekind domain \(R\) containing \(\mathbb{Q}\) is always a finitely generated \(R\)-algebra, hence generalizing a result essentially due Zariski in the case where \(R\) is a field. They establish that the hypothesis is sharp, that is, if \(R\) is neither a field nor a Dedekind domain then there exists locally nilpotent \(R\)-derivations of \(R[X,Y,Z]\) with non finitely generated kernels.
Reviewer: Adrien Dubouloz (Dijon)
References:
| [1] | Bhatwadekar, S. M.; Dutta, A. K., Kernel of locally nilpotent \(R\)-derivations of \(R [X, Y]\), Trans. Amer. Math. Soc., 349, 3303-3319 (1997) · Zbl 0883.13006 |
| [2] | Bruns, W.; Herzog, J., Cohen-Macaulay Rings, Cambridge Stud. Adv. Math., vol. 39 (1998), Cambridge University Press · Zbl 0909.13005 |
| [3] | Daigle, D.; Freudenburg, G., A counterexample to Hilbert’s Fourteenth Problem in dimension five, J. Algebra, 221, 528-535 (1999) · Zbl 0963.13024 |
| [4] | Daigle, D.; Freudenburg, G., Triangular derivations of \(k [X_1, X_2, X_3, X_4]\), J. Algebra, 241, 328-339 (2001) · Zbl 1018.13013 |
| [5] | Giral, J. M., Krull dimension, transcendence degree and subalgebras of finitely generated algebras, Arch. Math., 36, 305-312 (1981) · Zbl 0457.13008 |
| [6] | Matsumura, H., Commutative Algebra, Mathematics Lecture Note Series (1980), Benjamin/Cummings · Zbl 0211.06501 |
| [7] | Matsumura, H., Commutative Ring Theory, Cambridge Stud. Adv. Math., vol. 8 (1989), Cambridge University Press, translated from the Japanese by M. Reid · Zbl 0666.13002 |
| [8] | M. Miyanishi, Normal affine subalgebras of a polynomial ring, in: Algebraic and Topological Theories—To the Memory of Dr. Takehiko Miyata, Kinokuniya, 1985, pp. 37-51; M. Miyanishi, Normal affine subalgebras of a polynomial ring, in: Algebraic and Topological Theories—To the Memory of Dr. Takehiko Miyata, Kinokuniya, 1985, pp. 37-51 |
| [9] | Onoda, N., Subrings of finitely generated rings over a pseudogeometric ring, Japan. J. Math., 10, 29-53 (1984) · Zbl 0594.13017 |
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