Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-29T07:02:11.888Z Has data issue: false hasContentIssue false

Constructing (Almost) Rigid Rings and a UFD Having Infinitely Generated Derksen and Makar-Limanov Invariants

Published online by Cambridge University Press:  20 November 2018

David Finston
Affiliation:
Department of Mathematics, New Mexico State University, Las Cruces, NM, USA e-mail: dfinston@nmsu.edu
Stefan Maubach
Affiliation:
Department of Mathematics, Radboud University Nijmegen, Nijmegen, The Netherlands e-mail: s.maubach@math.ru.nl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An example is given of a UFD which has an infinitely generated Derksen invariant. The ring is “almost rigid” meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's ABC-theorem.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] de Bondt, M., Another generalisation of Mason's ABC-theorem. arXiv:math.GM0707.0434v2.Google Scholar
[2] Crachiola, A., On the AK-Invariant of Certain Domains. Ph.D. thesis, Wayne State University, May 2004.Google Scholar
[3] Crachiola, A. and Makar-Limanov, L., On the rigidity of small domains. J. Algebra 284(2005), no. 1, 112. doi:10.1016/j.jalgebra.2004.09.015Google Scholar
[4] Deveney, J. K. and Finston, D. R., Ga-actions on 3 and 7 . Comm. Algebra 22(1994), 62956302. doi:10.1080/00927879408825190Google Scholar
[5] van den Essen, A., Polynomial Automorphisms and the Jacobian Conjecture. Progress in Mathematics 190. Birkhäuser-Verlag, Basel, 2000.Google Scholar
[6] Finston, D. and Maubach, S., The automorphism group of certain factorial threefolds and a cancellation problem. Israel J. Math. 163(2008), 369381. doi:10.1007/s11856-008-0016-3Google Scholar
[7] Fossum, R., The Divisor Class Group of a Krull Domain. Ergebniss der Mathematik und ihrer Grenzgebiete 74. Springer-Verlag, New York, 1973.Google Scholar
[8] Freudenburg, G., Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia of Mathematical Sciences 136. Springer-Verlag, Berlin, 2006.Google Scholar
[9] Gurjar, R V., Masuda, K., Miyanishi, M., and Russell, P., Affine lines on affine surfaces and the Makar-Limanov invariant. Canad. J. Math. 60(2008), no. 1, 109139. doi:10.4153/CJM-2008-005-8Google Scholar
[10] Hartshorne, R. and Ogus, A., On the factoriality of local rings of small embedding codimension. Comm. Algebra 1(1974), 415437. doi:10.1080/00927877408548627Google Scholar
[11] Makar-Limanov, L., Again x + x 2 y + z 2 + t 3 = 0 . In: Affine Algebraic Geometry. Contemp. Math. 369. American Mathematical Society, Providence, RI, 2005, pp. 177182.Google Scholar
[12] Maubach, S., Hilbert's 14th Problem and Related Topics. Master's thesis, University of Nijmegen, 1998.Google Scholar
[13] Masuda, K. and Miyanishi, M., Étale endomorphisms of algebraic surfaces with G m actions. Math. Ann. 319(2001) no. 3, 493516. doi:10.1007/PL00004445Google Scholar
[14] Maubach, S., Infinitely generated Derksen and Makar-Limanov invariant. Osaka J. Math. 44(2007), no. 4, 883886.Google Scholar