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On 2-stably isomorphic four-dimensional affine domains. (English) Zbl 1448.13014

The author gives examples of four-dimensional semi-normal domains \(A\) and \(B\) which are finitely generated over the field \(\mathbb C\) (or \(\mathbb R\)) such that \(A[X,Y]=B[X,Y]\) but \(A[X]\not\cong B[X].\) The method is purely algebraic. The first examples of this type was given by Z. Jelonek [Math. Ann. 344, No. 4, 769–778 (2009; Zbl 1178.14060)], (the examples of Z. Jelonek are smooth and they have dimension ten).

MSC:

13B25 Polynomials over commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F45 Seminormal rings
14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)

Citations:

Zbl 1178.14060
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References:

[1] T. Asanuma, Non-invariant two dimensional affine domains, Math. J. Toyama Univ. 14 (1991), 167-175. · Zbl 0778.13015
[2] T. Asanuma, Non-linearizable algebraic \(k^*\)-actions on affine spaces, Invent. Math. 138 (1999), 281-306. · Zbl 0933.14027
[3] W. Danielewski, On a cancellation problem and automorphism groups of affine algebraic varieties, 1989, preprint.
[4] Neena Gupta, On the cancellation problem for the affine space \(\A^3\) in characteristic \(p\), Invent. Math. 195 (2014), 279-288. · Zbl 1309.14050
[5] —-, On Zariski’s cancellation problem in positive characteristic, Adv. Math. 264 (2014), 296-307. · Zbl 1325.14078
[6] M. Hochster, Non-uniqueness of the ring of coefficients in a polynomial ring, Proc. Amer. Math. Soc. 34 (1972), 81-82. · Zbl 0233.13012
[7] Z. Jelonek, On the cancellation problem, Math. Annal. 344 (2009), 769-778. · Zbl 1178.14060
[8] M. Krusemeyer, Fundamental groups, algebraic \(K\)-theory, and a problem of Abhyankar, Invent. Math. 19 (1973), 15-47. · Zbl 0247.14005
[9] T.Y. Lam, Serre’s problem on projective modules, Springer-Verlag, Berlin, 1996. · Zbl 1101.13001
[10] J. Milnor, Introduction to algebraic \(K\)-theory, Ann. Math. Stud. 72, Princeton University Press, Princeton, 1971. · Zbl 0237.18005
[11] M. Nagata, A theorem on finite generation of a ring, Nagoya Math. J. 27 (1966), 193-205. · Zbl 0139.26503
[12] C. Traverso, Seminormality and Picard group, Ann. Scuola Norm. Sup. 24 (1970), 585-595.
[13] C. Weibel, The \(K\)-book: An introduction to algebraic \(K\)-theory, Grad. Stud. Math. 145 (2013).
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