## Journal of Commutative Algebra

### On $2$-stably isomorphic four-dimensional affine domains

#### Abstract

In this paper, we exhibit examples of four-dimensional seminormal domains $A$ and $B$ which are finitely generated over the field $\mathbb{C}$ (or $\mathbb{R}$) such that $A[X,Y] \cong B[X,Y]$ but $A[X]\ncong B[X]$.

#### Article information

Source
J. Commut. Algebra, Volume 10, Number 2 (2018), 153-162.

Dates
First available in Project Euclid: 13 August 2018

https://projecteuclid.org/euclid.jca/1534125822

Digital Object Identifier
doi:10.1216/JCA-2018-10-2-153

Zentralblatt MATH identifier
06917490

#### Citation

Asanuma, Teruo; Gupta, Neena. On $2$-stably isomorphic four-dimensional affine domains. J. Commut. Algebra 10 (2018), no. 2, 153--162. doi:10.1216/JCA-2018-10-2-153. https://projecteuclid.org/euclid.jca/1534125822

#### References

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