The cancellation problem over Noetherian one-dimensional domains. (English) Zbl 1309.14054
The general Zariski problem asks whether given a variety \(X\) over a field with \(X\times {\mathbb A}^1\) isomorphic to \({\mathbb A}^{n+1}\), is \(X\) isomorphic to \({\mathbb A}^n\). For \(n=1\), this is straight forward and for \(n=2\), this is a beautiful result of T. Fujita [Proc. Japan Acad., Ser. A 55, 106–110 (1979; Zbl 0444.14026)], M. Miyanishi and T. Sugie [J. Math. Kyoto Univ. 20, 11–42 (1980; Zbl 0445.14017)] in characteristic zero and by P. Russell [Math. Ann. 255, 287–302 (1981; Zbl 0438.14024)] in positive characteristics. For \(n>2\), the problem remains open.
In the paper under review, the authors deal with a slightly general problem, where the base field is replaced with a one dimensional Noetherian domain. The assumption on the base ring can not be made weaker and expect positive results, as shown by M. Hochster [Proc. Am. Math. Soc. 34, 81–82 (1972; Zbl 0233.13012)] for rings of dimension two. Of course, for \(n>2\) even over a field the problem is open and false in positive characteristics as shown by N. Gupta [Invent. Math. 195, No. 1, 279–288 (2014; Zbl 1309.14050)]. For \(n=1\), an affirmative answer was provided by Abhyankar, Heinzer and Eakin [S. S. Abhyankar et al., J. Algebra 23, 310–342 (1972; Zbl 0255.13008)]. The authors deal with the case of \(n=2\) and show that cancellation holds if the base ring contains the field of rational numbers.
In the paper under review, the authors deal with a slightly general problem, where the base field is replaced with a one dimensional Noetherian domain. The assumption on the base ring can not be made weaker and expect positive results, as shown by M. Hochster [Proc. Am. Math. Soc. 34, 81–82 (1972; Zbl 0233.13012)] for rings of dimension two. Of course, for \(n>2\) even over a field the problem is open and false in positive characteristics as shown by N. Gupta [Invent. Math. 195, No. 1, 279–288 (2014; Zbl 1309.14050)]. For \(n=1\), an affirmative answer was provided by Abhyankar, Heinzer and Eakin [S. S. Abhyankar et al., J. Algebra 23, 310–342 (1972; Zbl 0255.13008)]. The authors deal with the case of \(n=2\) and show that cancellation holds if the base ring contains the field of rational numbers.
Reviewer: N. Mohan Kumar (St. Louis)
MSC:
| 14R25 | Affine fibrations |
| 14R10 | Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) |
Citations:
Zbl 0444.14026; Zbl 0445.14017; Zbl 0438.14024; Zbl 0233.13012; Zbl 0255.13008; Zbl 1309.14050References:
| [1] | S. S. Abhyankar, W. Heinzer, and P. Eakin, On the uniqueness of the coefficient ring in a polynomial ring , J. Algebra 23 (1972), 310-342. · Zbl 0255.13008 · doi:10.1016/0021-8693(72)90134-2 |
| [2] | T. Asanuma, Polynomial fibre rings of algebras over Noetherian rings , Invent. Math. 87 (1987), 101-127. · Zbl 0607.13015 · doi:10.1007/BF01389155 |
| [3] | T. Asanuma and S. M. Bhatwadekar, Structure of \(\mathbf{A}^{2}\)-fibrations over one-dimensional Noetherian domains , J. Pure Appl. Algebra 115 (1997), 1-13. · Zbl 0893.13006 · doi:10.1016/S0022-4049(96)00005-9 |
| [4] | H. Bass, E. H. Connell, and D. L. Wright, Locally polynomial algebras are symmetric algebras , Invent. Math. 38 (1976/77), 279-299. · Zbl 0371.13007 · doi:10.1007/BF01403135 |
| [5] | S. M. Bhatwadekar and A. K. Dutta, “On affine fibrations” in Commutative Algebra (Trieste, 1992) , World Sci., River Edge, N. J., 1994, 1-17. · Zbl 0926.13011 |
| [6] | S. M. Bhatwadekar and A. K. Dutta, On \(\mathbf{A}^{1}\)-fibrations of subalgebras of polynomial algebras , Compositio Math. 95 (1995), 263-285. · Zbl 0840.13011 |
| [7] | D. Daigle and G. Freudenburg, “Families of affine fibrations” in Symmetry and Spaces , Progr. Math. 278 , Birkhäuser, Boston, 2010, 35-43. · Zbl 1234.14042 · doi:10.1007/978-0-8176-4875-6_3 |
| [8] | H. Derksen, A. van den Essen, and P. van Rossum, An extension of the Miyanishi-Sugie cancellation theorem to Dedekind rings , technical report, University of Nijmegen, 2002. |
| [9] | A. K. Dutta, On \(\mathbf{A}^{1}\)-bundles of affine morphisms , J. Math. Kyoto Univ. 35 (1995), 377-385. · Zbl 0861.14015 |
| [10] | A. K. Dutta and N. Onoda, Some results on codimension-one \(\mathbf{A}^{1}\)-fibrations J. Algebra 313 (2007), 905-921. · Zbl 1126.14072 · doi:10.1016/j.jalgebra.2006.06.040 |
| [11] | M. El Kahoui and M. Ouali, Fixed point free locally nilpotent derivations of \(\mathbb{A}^{2}\)-fibrations , J. Algebra 372 (2012), 480-487. · Zbl 1276.14089 · doi:10.1016/j.jalgebra.2012.09.025 |
| [12] | A. van den Essen, “Around the cancellation problem” in Affine Algebraic Geometry , Osaka Univ. Press, Osaka, 2007, 463-481. · Zbl 1129.14086 |
| [13] | A. van den Essen, S. Maubach, and S. Vénéreau, The special automorphism group of \(R[t]/(t^{m})[x_{1},\ldots,x_{n}]\) and coordinates of a subring of \(R[t][x_{1},\ldots,x_{n}]\) , J. Pure Appl. Algebra 210 (2007), 141-146. · Zbl 1112.14068 · doi:10.1016/j.jpaa.2006.09.013 |
| [14] | G. Freudenburg, Derivations of \(R[X,Y,Z]\) with a slice , J. Algebra 322 (2009), 3078-3087. · Zbl 1220.13018 · doi:10.1016/j.jalgebra.2008.05.007 |
| [15] | G. Freudenburg and P. Russell, “Open problems in affine algebraic geometry” in Affine Algebraic Geometry , Contemp. Math. 369 , Amer. Math. Soc., Providence, 2005, 1-30. · Zbl 1070.14528 · doi:10.1090/conm/369/06801 |
| [16] | T. Fujita, On Zariski problem , Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), 106-110. · Zbl 0444.14026 · doi:10.3792/pjaa.55.106 |
| [17] | E. Hamann, On the \(R\)-invariance of \(R[x]\) , J. Algebra 35 , (1975), 1-16. · Zbl 0318.13023 · doi:10.1016/0021-8693(75)90031-9 |
| [18] | M. Hochster, Nonuniqueness of coefficient rings in a polynomial ring , Proc. Amer. Math. Soc. 34 (1972), 81-82. · Zbl 0233.13012 · doi:10.2307/2037901 |
| [19] | T. Kambayashi, On the absence of nontrivial separable forms of the affine plane , J. Algebra 35 (1975), 449-456. · Zbl 0309.14029 · doi:10.1016/0021-8693(75)90058-7 |
| [20] | T. Kambayashi and D. Wright, Flat families of affine lines are affine-line bundles , Illinois J. Math. 29 (1985), 672-681. · Zbl 0599.14010 |
| [21] | H. Kraft, Challenging problems on affine \(n\)-space , Astérisque 237 (1996), 295-317, Séminaire Bourbaki 1994/95, no 802. · Zbl 0892.14003 |
| [22] | D. Lewis, Vénéreau-type polynomials as potential counterexamples , J. Pure Appl. Algebra 217 (2013), 946-957. · Zbl 1304.13013 · doi:10.1016/j.jpaa.2012.09.018 |
| [23] | M. Miyanishi, Curves on Rational and Unirational Surfaces , Tata Inst. Fund. Res. Lectures Math. Phys. 60 , Tata Inst. Fund. Res., Bombay; Narosa, New Delhi, 1978. · Zbl 0425.14008 |
| [24] | M. Miyanishi and T. Sugie, Affine surfaces containing cylinderlike open sets , J. Math. Kyoto Univ. 20 (1980), 11-42. · Zbl 0445.14017 |
| [25] | P. Russell, On affine-ruled rational surfaces , Math. Ann. 255 (1981), 287-302. · Zbl 0438.14024 · doi:10.1007/BF01450704 |
| [26] | A. Sathaye, Polynomial ring in two variables over a DVR: a criterion , Invent. Math. 74 (1983), 159-168. · Zbl 0538.13006 · doi:10.1007/BF01388536 |
| [27] | S. Vénéreau, Automorphismes et variables de l’anneau de polynomes \(A[y_{1},\ldots,y_{n}]\) , Ph.D. dissertation, Université de Grenoble I, Institut Fourier, Grenoble, 2001. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
