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Slices, Embeddings and Cancellation

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Algebraic Theory of Locally Nilpotent Derivations

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 136.3))

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Abstract

The Zariski Cancellation Problem can be viewed as a descendant of Zariski’s cancellation question for fields; see Sect.  1.1.2 . It can be stated as follows.

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Notes

  1. 1.

    Zariski’s original question for fields is nowadays referred to as the Birational Cancellation Problem.

  2. 2.

    For any affine variety, the Makar-Limanov invariant can be defined as the intersection of all rings of \(\mathbb{G}_{a}\)-invariants. This generalization was introduced by Crachiola and Makar-Limanov; see [61].

References

  1. S. Abhyankar, P. Eakin, and W. Heinzer, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), 310–342.

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Abhyankar and T.T. Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148–166.

    MathSciNet  MATH  Google Scholar 

  3. T. Asanuma, Polynomial fibre rings of algebras over noetherian rings, Invent. Math. 87 (1987), 101–127.

    Article  MathSciNet  MATH  Google Scholar 

  4. I. V. Arzhantsev, Non-linearizable algebraick ∗ -actions on affine spaces, Invent. Math. 138 (1999), 281–306.

    Article  MathSciNet  Google Scholar 

  5. S. Bhatwadekar and D. Daigle, On finite generation of kernels of locally nilpotentR-derivations ofR[X, Y, Z], J. Algebra 322 (2009), 2915–2926.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Bhatwadekar and A. K. Dutta, On affine fibrations, Commutative Algebra (Trieste, 1992) (River Edge, New Jersey), World Sci. Publ., 1994, pp. 1–17.

    Google Scholar 

  7. S. Bhatwadekar and A. Roy, Some results on embedding of a line in 3-space, J. Algebra 142 (1991), 101–109.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Crachiola, On the AK-Invariant of Certain Domains, Ph.D. thesis, Wayne State University, Detroit, Michigan, 2004.

    Google Scholar 

  9. G. V. Choodnovsky, Cancellation for two-dimensional unique factorization domains, J. Pure Appl. Algebra 213 (2009), 1735–1738.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Crachiola and L. Makar-Limanov, On the rigidity of small domains, J. Algebra 284 (2005), 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. V. Choodnovsky, An algebraic proof of a cancellation theorem for surfaces, J. Algebra 320 (2008), 3113–3119.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. C. Craighero, A result on m-flats in \(\mathbb{A}_{k}^{n}\), Rend. Sem. Mat. Univ. Padova 75 (1986), 39–46.

    MathSciNet  MATH  Google Scholar 

  13. P. C. Craighero, Families of affine fibrations, Progr. Math., vol. 278, pp. 35–43, Birkhäuser Boston, Boston, MA, 2010.

    Google Scholar 

  14. W. Danielewski, On the cancellation problem and automorphism groups of affine algebraic varieties, Preprint, Warsaw, 1989.

    Google Scholar 

  15. H. Derksen, A. van den Essen, and P. van Rossum, The cancellation problem in dimension four, Tech. Report 0022, Dept. of Mathematics, Univ. Nijmegen, The Netherlands, 2000.

    Google Scholar 

  16. H. Derksen and F. Kutzschebauch, Nonlinearizable holomorphic group actions, Math. Ann. 311 (1998), 41–53.

    Article  MathSciNet  MATH  Google Scholar 

  17. I. Dolgachev and B. Ju. Weisfeiler, Unipotent group schemes over integral rings, Math. USSR Izv. 38 (1975), 761–800.

    MATH  Google Scholar 

  18. A. Dubouloz, Rigid affine surfaces with isomorphic \(\mathbb{A}^{2}\) -cylinders, arXiv:1507.05802, 2015.

    Google Scholar 

  19. V. I. Danilov and M. H. Gizatullin, Affine open subsets in \(\mathbb{A}^{3}\) without the cancellation property, Ramanujan Math. Soc. Lect. Notes Ser. 17 (2013), 63–67.

    MathSciNet  Google Scholar 

  20. A. Dubouloz, L. Moser-Jauslin, and P.-M. Poloni, Noncancellation for contractible affine threefolds, Proc. Amer. Math. Soc. 139 (2014), 4273–4284.

    Article  MathSciNet  MATH  Google Scholar 

  21. D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, 1995.

    Google Scholar 

  22. E. B. Elliott, Polynomial Automorphisms and the Jacobian Conjecture, Birkhauser, Boston, 2000.

    Google Scholar 

  23. A. van den Essen and P. van Rossum, A class of counterexamples to the cancellation problem for arbitrary rings, Ann. Polon. Math. 76 (2001), 89–93.

    Article  MathSciNet  MATH  Google Scholar 

  24. E. B. Elliott, Triangular derivations related to problems on affinen-space, Proc. Amer. Math. Soc. 130 (2001), 1311–1322.

    Google Scholar 

  25. K.-H. Fieseler, On complex affine surfaces with \(\mathbb{C}^{+}\) -actions, Comment. Math. Helvetici 69 (1994), 5–27.

    Article  MathSciNet  MATH  Google Scholar 

  26. D. Finston and S. Maubach, The automorphism group of certain factorial threefolds and a cancellation problem, Israel J. Math. 163 (2008), 369–381.

    Article  MathSciNet  MATH  Google Scholar 

  27. K.-H. Fieseler, The Vénéreau polynomials relative to \(\mathbb{C}^{{\ast}}\) -fibrations and stable coordinates, Affine Algebraic Geometry (Osaka, Japan), Osaka University Press, 2007, pp. 203–215.

    Google Scholar 

  28. K.-H. Fieseler, Derivations ofR[X, Y, Z] with a slice, J. Algebra 322 (2009), 3078–3087.

    Article  MathSciNet  Google Scholar 

  29. K.-H. Fieseler, Bivariate analogues of Chebyshev polynomials with application to embeddings of affine spaces, CRM Proc. Lecture Notes 54 (2011), 39–56.

    Article  MathSciNet  Google Scholar 

  30. K.-H. Fieseler, An affine version of a theorem of Nagata, Kyoto J. Math. 55 (2015), 663–672.

    Article  MathSciNet  Google Scholar 

  31. K.-H. Fieseler, Real and rational forms of certain \(O_{2}(\mathbb{C})\) -actions, and a solution to the Weak Complexification Problem, Transform. Groups 9 (2004), 257–272.

    MathSciNet  Google Scholar 

  32. T. Fujita, On Zariski problem, Proc. Japan Acad. 55A (1979), 106–110.

    Article  MathSciNet  MATH  Google Scholar 

  33. R. Ganong, The pencil of translates of a line in the plane, RM Proc. Lecture Notes 54 (2011), 57–71.

    Article  MathSciNet  MATH  Google Scholar 

  34. N. Gupta, On the cancellation problem for the affine space \(\mathbb{A}^{3}\) in characteristicp, Invent. Math. 195 (2014), 279–288.

    Article  MathSciNet  MATH  Google Scholar 

  35. G.-M. Greuel and, On Zariski’s cancellation problem in positive characteristic, Adv. Math. 264 (2014), 296–307.

    Google Scholar 

  36. G.-M. Greuel and, A survey on Zariski cancellation problem, Indian J. Pure Appl. Math. 46 (2015), 865–877.

    Google Scholar 

  37. R. Gurjar, A topological proof of a cancellation theorem for \(\mathbb{C}^{2}\), Math. Z. 240 (2002), 83–94.

    Article  MathSciNet  MATH  Google Scholar 

  38. E. Hamann, On theR-invariance ofR[x], Ph.D. thesis, Univ. Minnesota, Minneapolis, MN, 1973.

    Google Scholar 

  39. G.-M. Greuel and, On theR-invariance ofR[X], J. Algebra 35 (1975), 1–16.

    Google Scholar 

  40. M. Hochster, Non-uniqueness of coefficient rings in a polynomial ring, Proc. Amer. Math. Soc. 34 (1972), 81–82.

    Article  MathSciNet  MATH  Google Scholar 

  41. G.-M. Greuel and, Topics in the homological theory of modules over commutative rings, CBMS Regional Conference Series in Mathematics, vol. 24, American Mathematical Society, 1975.

    Google Scholar 

  42. D. Husemoller, Fibre Bundles, third ed., Springer-Verlag, Berlin, Heidelberg, New York, 1994.

    Book  MATH  Google Scholar 

  43. Z. Jelonek, The extension of regular and rational embeddings, Math. Ann. 277 (1987), 113–120.

    Article  MathSciNet  MATH  Google Scholar 

  44. S. Kaliman, Extensions of isomorphisms between affine algebraic subvarieties ofk n to automorphisms ofk n, Proc. Amer. Math. Soc. 113 (1991), 325–334.

    MathSciNet  MATH  Google Scholar 

  45. H. W. E. Jung, Isotopic embeddings of affine algebraic varieties into \(\mathbb{C}^{n}\), Contemp. Math. 137 (1992), 291–295.

    Article  MathSciNet  Google Scholar 

  46. S. Kaliman, S. Vénéreau, and M. Zaidenberg, Extensions birationnelles simples de l’anneau de polynômes \(\mathbb{C}^{[3]}\), C.R. Acad. Sci. Paris Ser. I Math. 333 (2001), 319–322.

    Google Scholar 

  47. H. W. E. Jung, Simple birational extensions of the polynomial ring \(\mathbb{C}^{[3]}\), Trans. Amer. Math. Soc. 356 (2004), 509–555.

    Article  MathSciNet  Google Scholar 

  48. H. W. E. Jung, Vénéreau polynomials and related fiber bundles, J. Pure Applied Algebra 192 (2004), 275–286.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. C. Kang, The biregular cancellation problem, J. Pure Appl. Algebra 45 (1987), 241–253.

    Article  MathSciNet  MATH  Google Scholar 

  50. E. G. Koshevoi, Challenging problems on affine n-space, Semináire Bourbaki 802 (1995), 295–317.

    Google Scholar 

  51. D. Lewis, Vénéreau-type polynomials as potential counterexamples, J. Pure Appl. Algebra 217 (2013), 946–957.

    Article  MathSciNet  MATH  Google Scholar 

  52. E. G. Koshevoi, Strongly residual coordinates overA[x], Springer Proc. Math. Stat. 79 (2014), 407–430.

    Article  Google Scholar 

  53. E. G. Koshevoi, Locally nilpotent derivations, a new ring invariant and applications, Lecture notes, Bar-Ilan University, 1998. Avail. at http://www.math.wayne.edu/~lml/.

  54. E. G. Koshevoi, AK invariant, some conjectures, examples and counterexamples, Ann. Polon. Math. 76 (2001), 139–145.

    Article  MathSciNet  MATH  Google Scholar 

  55. E. G. Koshevoi, The commuting derivations conjecture, J. Pure Appl. Algebra 179 (2003), 159–168.

    Article  MathSciNet  Google Scholar 

  56. J. H. McKay and, Some remarks on polynomial rings, Osaka J. Math. 10 (1973), 617–624.

    Google Scholar 

  57. J. H. McKay and, Algebraic characterization of the affine plane, J. Math. Kyoto Univ. 15 (1975), 169–184.

    Google Scholar 

  58. M. Miyanishi and T. Sugie, Affine surfaces containing cylinderlike open sets, J. Math. Kyoto U. 20 (1980), 11–42.

    Article  MathSciNet  MATH  Google Scholar 

  59. V. L. Popov, Around the Abyhankar-Sathaye Conjecture, Doc. Math. (2015), 513–528, Extra vol.: Alexander S. Merkurjev’s sixtieth birthday.

    Google Scholar 

  60. M. Raynaud, Modules projectifs universels, Invent. Math. 6 (1968), 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  61. K. P. Russell, On affine-ruled rational surfaces, Math. Ann. 255 (1981), 287–302.

    Article  MathSciNet  MATH  Google Scholar 

  62. K. P. Russell, Cancellation, Springer Proc. Math. Stat. 79 (2014), 495–518.

    Article  MathSciNet  MATH  Google Scholar 

  63. B. Segre, Corrispondenze di möbius e trasformazioni cremoniane intere (Italian), Atti Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 91 (1956/1957), 3–19.

    Google Scholar 

  64. A. R. Shastri, Polynomial representations of knots, Tôhoku Math. J. 44 (1992), 11–17.

    Article  MathSciNet  MATH  Google Scholar 

  65. V. Srinivas, On the embedding dimension of an affine variety, Math. Ann. 289 (1991), 125–132.

    Article  MathSciNet  MATH  Google Scholar 

  66. A. A. Suslin, Mennicke symbols and their applications in theK-theory of fields, Algebraic K-Theory, Part I (Oberwolfach, 1980), Lecture Notes in Math., vol. 966, Springer, 1982, pp. 344–356.

    Google Scholar 

  67. M. Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace \(\mathbb{C}^{2}\), J. Math. Soc. Japan 26 (1974), 241–257.

    Article  MathSciNet  MATH  Google Scholar 

  68. S. Vénéreau, Automorphismes et variables de l’anneau de polynômesA[ y 1, …, y n ], Ph.D. thesis, Institut Fourier des mathématiques, Grenoble, 2001.

    Google Scholar 

  69. J. Wilkens, On the cancellation problem for surfaces, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 1111–1116.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Western Michigan University, Kalamazoo, Michigan, USA

    Gene Freudenburg

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Freudenburg, G. (2017). Slices, Embeddings and Cancellation. In: Algebraic Theory of Locally Nilpotent Derivations. Encyclopaedia of Mathematical Sciences, vol 136.3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55350-3_10

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