No CrossRef data available.
Article contents
ON SEPARABLE 
$\mathbb{A}^{2}$ AND 
$\mathbb{A}^{3}$-FORMS
Part of:
General commutative ring theory
Field extensions
Ring extensions and related topics
Affine geometry
Published online by Cambridge University Press: 26 December 2018
Abstract
In this paper, we will prove that any 
$\mathbb{A}^{3}$-form over a field 
$k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of Kambayashi on the triviality of separable 
$\mathbb{A}^{2}$-forms over a field 
$k$ extends to 
$\mathbb{A}^{2}$-forms over any one-dimensional Noetherian domain containing 
$\mathbb{Q}$.
MSC classification
- Type
- Article
- Information
- Copyright
- © 2018 Foundation Nagoya Mathematical Journal
References
Asanuma, T., Polynomial fibre rings of algebras over Noetherian rings, Invent. Math. 87 (1987), 101–127.10.1007/BF01389155Google Scholar
Asanuma, T. and Bhatwadekar, S. M., Structure of 𝔸2 -fibrations over one-dimensional Noetherian domains, J. Pure Appl. Algebra 115(1) (1997), 1–13.10.1016/S0022-4049(96)00005-9Google Scholar
Asanuma, T., “Purely inseparable k-forms of affine algebraic curves”, in Contemporary Mathematics, Affine Algebraic Geometry 369, American Mathematical Society, Providence, RI, 2005, 31–46.Google Scholar
Berson, J., van den Essen, A. and Maubach, S., Derivations having divergence zero on R[X, Y], Israel J. Math. 124 (2001), 115–124.10.1007/BF02772610Google Scholar
Bhatwadekar, S. M. and Dutta, A. K., On residual variables and stably polynomial algebras, Comm. Algebra 21(2) (1993), 635–645.10.1080/00927879308824585Google Scholar
Bhatwadekar, S. M. and Dutta, A. K., Kernel of locally nilpotent R-derivations of R[X, Y], Trans. Amer. Math. Soc. 349(8) (1997), 3303–3319.10.1090/S0002-9947-97-01946-6Google Scholar
Bhatwadekar, S. M., Gupta, N. and Lokhande, S. A., Some K-theoretic properties of the kernel of a locally nilpotent derivation on k[X 1, …, X 4], Trans. Amer. Math. Soc. 369(1) (2017), 341–363.10.1090/tran/6649Google Scholar
Daigle, D., A necessary and sufficient condition for triangulability of derivations of k[X, Y, Z], J. Pure Appl. Algebra 113 (1996), 297–305.10.1016/0022-4049(95)00156-5Google Scholar
Daigle, D. and Kaliman, S., A note on locally nilpotent derivations and variables of k[X, Y, Z], Canad. Math. Bull. 52(4) (2009), 535–543.10.4153/CMB-2009-054-5Google Scholar
Dutta, A. K., On separable 𝔸1 -forms, Nagoya Math. J. 159 (2000), 45–51.10.1017/S0027763000007418Google Scholar
van den Essen, A. and van Rossum, P., Coordinates in two variables over a ℚ-algebra, Trans. Amer. Math. Soc. 359 (2004), 1691–1703.10.1090/S0002-9947-04-03492-0Google Scholar
Freudenburg, G., Algebraic Theory of Locally Nilpotent Derivations, 2nd edn., Springer, Berlin, 2017.10.1007/978-3-662-55350-3Google Scholar
Itoh, S., On weak normality and symmetric algebras, J. Algebra 85(1) (1983), 40–50.10.1016/0021-8693(83)90117-5Google Scholar
Ischebeck, F. and Rao, R. A., Ideals and Reality: Projective Modules and Number of Generators of Ideals, Springer, Berlin, 2005.Google Scholar
Kahoui, M. El and Ouali, M., A triviality criterion for 𝔸2 -fibrations over a ring containing ℚ, J. Algebra 459 (2016), 272–279.10.1016/j.jalgebra.2016.03.047Google Scholar
Kaliman, S., Free ℂ+ -actions on ℂ3 are translations, Invent. Math. 156(1) (2004), 163–173.10.1007/s00222-003-0336-1Google Scholar
Kambayashi, T., On the absence of nontrivial separable forms of the affine plane, J. Algebra 35 (1975), 449–456.10.1016/0021-8693(75)90058-7Google Scholar
Koras, M. and Russell, P., Separable forms of 𝔾m-actions on 𝔸k3, Transform. Groups 18(4) (2013), 1155–1163.10.1007/s00031-013-9242-9Google Scholar