Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-27T22:02:42.606Z Has data issue: false hasContentIssue false
  • English
  • Français

ON SEPARABLE $\mathbb{A}^{2}$ AND $\mathbb{A}^{3}$-FORMS

Part of: General commutative ring theory Affine geometry Field extensions Ring extensions and related topics

Published online by Cambridge University Press:  26 December 2018

AMARTYA KUMAR DUTTA ,
NEENA GUPTA  and
ANIMESH LAHIRI
AMARTYA KUMAR DUTTA
Affiliation:
Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India email [email protected]
NEENA GUPTA
Affiliation:
Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India email [email protected]
ANIMESH LAHIRI
Affiliation:
Swami Vivekananda Research Centre, Ramakrishna Mission Vidyamandira, P.O. Belur Math, Howrah 711202, India email [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Primary: 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
Secondary: 13B25: Polynomials over commutative rings 12F10: Separable extensions, Galois theory 14R25: Affine fibrations 13A50: Actions of groups on commutative rings; invariant theory
Type
Article
Information
Nagoya Mathematical Journal , Volume 239 , September 2020 , pp. 346 - 354
Check for updates
Copyright
© 2018 Foundation Nagoya Mathematical Journal

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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
Gurjar, R. V., Masuda, K. and Miyanishi, M., Affine space fibration, preprint.Google 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