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A Morita Cancellation Problem

Part of: Rings and algebras with additional structure Chain conditions, growth conditions, and other forms of finiteness

Published online by Cambridge University Press:  29 January 2019

D.-M. Lu ,
Q.-S. Wu  and
J. J. Zhang
D.-M. Lu
Affiliation:
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China Email: [email protected]
Q.-S. Wu
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai 200433, China Email: [email protected]
J. J. Zhang
Affiliation:
Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195, USA Email: [email protected]
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Abstract

We study a Morita-equivalent version of the Zariski cancellation problem.

Keywords

Zariski cancellation problemMorita cancellation problem

MSC classification

Primary: 16P99: None of the above, but in this section
Secondary: 16W99: None of the above, but in this section
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 3 , June 2020 , pp. 708 - 731
Copyright
© Canadian Mathematical Society 2019

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Footnotes

D.-M. Lu was partially supported by the National Natural Science Foundation of China (Grant No. 11671351). Q.-S. Wu was partially supported by the National Natural Science Foundation of China (Grant No. 11771085 and Key Project No. 11331006). J.J. Zhang was partially supported by the US National Science Foundation (Nos. DMS-1402863 and DMS-1700825).

References

Abhyankar, S., Eakin, P., and Heinzer, W., On the uniqueness of the coefficient ring in a polynomial ring. J. Algebra 23(1972), 310–342. https://doi.org/10.1016/0021-8693(72)90134-2.Google Scholar
Anderson, F. W. and Fuller, K. R., Rings and categories of modules. Graduate Texts in Mathematics, 13, Springer-Verlag, New York-Heidelberg, 1974.Google Scholar
Assem, I., Simson, D., and Skowroński, A., Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory. London Mathematical Society Student Texts, 65, Cambridge University Press, Cambridge, 2006. https://doi.org/10.1017/CBO9780511614309.Google Scholar
Auslander, M. and Goldman, O., The Brauer group of a commutative ring. Trans. Amer. Math. Soc. 97(1960), 367–409. https://doi.org/10.2307/1993378.Google Scholar
Bell, J. and Zhang, J. J., Zariski cancellation problem for noncommutative algebras. Selecta Math. (N.S.) 23(2017), no. 3, 1709–1737. https://doi.org/10.1007/s00029-017-0317-7.Google Scholar
Bell, J. and Zhang, J. J., An isomorphism lemma for graded rings. Proc. Amer. Math. Soc. 145(2017), no. 3, 989–994. https://doi.org/10.1090/proc/13276.Google Scholar
Brown, K. A. and Yakimov, M. T., Azumaya loci and discriminant ideals of PI algebras. Adv. Math. 340(2018), 1219–1255. https://doi.org/10.1016/j.aim.2018.10.024.Google Scholar
Carvalho, P. A. A. B. and Lopes, S. A., Automorphisms of generalized down-up algebras. Comm. Algebra 37(2009), no. 5, 1622–1646. https://doi.org/10.1080/00927870802209987.Google Scholar
Ceken, S., Palmieri, J., Wang, Y.-H., and Zhang, J. J., The discriminant controls automorphism groups of noncommutative algebras. Adv. Math. 269(2015), 551–584. https://doi.org/10.1016/j.aim.2014.10.018.Google Scholar
Ceken, S., Palmieri, J., Wang, Y.-H., and Zhang, J. J., The discriminant criterion and the automorphism groups of quantized algebras. Adv. Math. 286(2016), 754–801. https://doi.org/10.1016/j.aim.2015.09.024.Google Scholar
Chan, K., Young, A., and Zhang, J. J., Discriminant formulas and applications. Algebra Number Theory 10(2016), no. 3, 557–596. https://doi.org/10.2140/ant.2016.10.557.Google Scholar
Chan, K., Young, A., and Zhang, J. J., Discriminants and automorphism groups of Veronese subrings of skew polynomial rings. Math. Z. 288(2018), no. 3–4, 1395–1420. https://doi.org/10.1007/s00209-017-1939-3.Google Scholar
Danielewski, W., On the cancellation problem and automorphism groups of affine algebraic varieties. Preprint, 1989, 8 pages, Warsaw.Google Scholar
DeMeyer, F. and Ingraham, E., Separable algebras over commutative rings. Lecture Notes in Mathematics, 181, Springer-Verlag, Berlin-New York, 1971.Google Scholar
Dixmier, J., Quotients simples de l’algébre enveloppante de sl2. J. Algebra 24(1973), 551–564. https://doi.org/10.1016/0021-8693(73)90127-0.Google Scholar
Eakin, P. and Heinzer, W., A cancellation problem for rings. In: Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972). Lecture Notes in Mathematics, 311, Springer, Berlin, 1973, pp. 61–77.Google Scholar
Farinati, M. A., Solotar, A., and Suárez-Álvarez, M., Hochschild homology and cohomology of generalized Weyl algebras. Ann. Inst. Fourier (Grenoble) 53(2003), no. 2, 465–488.Google Scholar
Fieseler, K.-H., On complex affine surfaces with ℂ+-action. Comment. Math. Helv. 69(1994), 5–27. https://doi.org/10.1007/BF02564471.Google Scholar
Fujita, T., On Zariski problem. Proc. Japan Acad. Ser. A Math. Sci. 55(1979), 106–110.Google Scholar
Gaddis, J., The isomorphism problem for quantum affine spaces, homogenized quantized Weyl algebras, and quantum matrix algebras. J. Pure Appl. Algebra 221(2017), no. 10, 2511–2524. https://doi.org/10.1016/j.jpaa.2016.12.036.Google Scholar
Gaddis, J., Kirkman, E., and Moore, W. F., On the discriminant of twisted tensor products. J. Algebra 477(2017), 29–55. https://doi.org/10.1016/j.jalgebra.2016.12.019.Google Scholar
Gaddis, J., Won, R., and Yee, D., Discriminants of Taft algebra smash products and applications. Algebr. Represent. Theory, to appear. https://doi.org/10.1007/s10468-018-9798-0.Google Scholar
Gupta, N., On the cancellation problem for the affine space 𝔸3 in characteristic p. Inventiones Math. 195(2014), no. 1, 279–288. https://doi.org/10.1007/s00222-013-0455-2.Google Scholar
Gupta, N., On Zariski’s cancellation problem in positive characteristic. Adv. Math. 264(2014), 296–307. https://doi.org/10.1016/j.aim.2014.07.012.Google Scholar
Gupta, N., A survey on Zariski cancellation problem. Indian J. Pure Appl. Math. 46(2015), no. 6, 865–877. https://doi.org/10.1007/s13226-015-0154-3.Google Scholar
Hasse, H. and Schmidt, F. K., Noch eine Begründung der Theorie der höheren Differrentialquotienten in einem algebraischen Funktionenkorper einer Unbestimmten. J. Reine Angew. Math. 177(1937), 215–237. https://doi.org/10.1515/crll.1937.177.215.Google Scholar
Hochster, M., Non-uniqueness of the ring of coefficients in a polynomial ring. Proc. Amer. Math. Soc. 34(1972), 81–82. https://doi.org/10.2307/2037901.Google Scholar
Joseph, A., A wild automorphism of U (sl (2)). Math. Proc. Cambridge Philos. Soc. 80(1976), no. 1, 61–65. https://doi.org/10.1017/S030500410005266X.Google Scholar
Kraft, H., Challenging problems on affine n-space. Séminaire Bourbaki, 1994/95, Astérisque 237(1996), Exp. No. 802, 5, 295–317.Google Scholar
Levitt, J. and Yakimov, M., Weyl algebras at roots of unity. Israel J. Math. 225(2018), no. 2, 681–719. https://doi.org/10.1007/s11856-018-1675-3.Google Scholar
Lezama, O., Wang, Y.-H., and Zhang, J. J., Zariski cancellation problem for non-domain noncommutative algebras. Math. Z., to appear. https://doi.org/10.1007/s00209-018-2153-7.Google Scholar
Lü, J.-F., Mao, X.-F., and Zhang, J. J., Nakayama automorphism and applications. Trans. Amer. Math. Soc. 369(2017), no. 4, 2425–2460. https://doi.org/10.1090/tran/6718.Google Scholar
Makar-Limanov, L., On the hypersurface x + x 2y + z 2 + t 3 = 0 in C4 or a C3-like threefold which is not C3. Israel J. Math. 96(1996), part B, 419–429. https://doi.org/10.1007/BF02937314.Google Scholar
Miyanishi, M. and Sugie, T., Affine surfaces containing cylinderlike open sets. J. Math. Kyoto Univ. 20(1980), 11–42. https://doi.org/10.1215/kjm/1250522319.Google Scholar
Negron, C., The derived Picard group of an affine Azumaya algebra. Selecta Math. (N.S.) 23(2017), no. 2, 1449–1468. https://doi.org/10.1007/s00029-016-0249-7.Google Scholar
Nguyen, B., Trampel, K., and Yakimov, M., Noncommutative discriminants via Poisson primes. Adv. Math. 322(2017), 269–307. https://doi.org/10.1016/j.aim.2017.10.018.Google Scholar
Rogalski, D., Sierra, S. J., and Stafford, J. T., Algebras in which every subalgebra is Noetherian. Proc. Amer. Math. Soc. 142(2014), no. 9, 2983–2990. https://doi.org/10.1090/S0002-9939-2014-12052-1.Google Scholar
Russell, P., On affine-ruled rational surfaces. Math. Ann. 255(1981), 287–302. https://doi.org/10.1007/BF01450704.Google Scholar
Schack, S., Bimodules, the Brauer group, Morita equivalence, and cohomology. J. Pure Appl. Algebra 80(1992), no. 3, 315–325. https://doi.org/10.1016/0022-4049(92)90149-A.Google Scholar
Small, L. W. and Warfield, R. B. Jr., Prime affine algebras of Gel’fand-Kirillov dimension one. J. Algebra 91(1984), no. 2, 386–389. https://doi.org/10.1016/0021-8693(84)90110-8.Google Scholar
Smith, S. P., The primitive factor rings of the enveloping algebra of sl (2, ℂ). J. London Math. Soc. (2) 24(1981), no. 1, 97–108. https://doi.org/10.1112/jlms/s2-24.1.97.Google Scholar
Suzuki, M., Propriétés topologiques des polynomes de deux variables complex et automorphisms algébriques de l’espace C2. J. Math. Soc. Japan 26(1974), 241–257. https://doi.org/10.2969/jmsj/02620241.Google Scholar
Tang, X., Automorphisms for some symmetric multiparameter quantized Weyl algebras and their localizations. Algebra Colloq. 24(2017), no. 3, 419–438. https://doi.org/10.1142/S100538671700027X.Google Scholar
Tang, X., The automorphism groups for a family of generalized Weyl algebras. J. Algebra Appl. 18(2018), 1850142. https://doi.org/10.1142/S0219498818501426.Google Scholar
Wang, Y.-H. and Zhang, J. J., Discriminants of noncommutative algebras and their applications (Chinese). Sci. China Math. 48(2018), 1615–1630. https://doi.org/10.1360/N012017-00263.Google Scholar
Weibel, C. A., An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994. https://doi.org/10.1017/CBO9781139644136.Google Scholar
Wilkens, J., On the cancellation problem for surfaces. C. R. Acad. Sci. Paris Sér. I Math. 326(1998), 1111–1116. https://doi.org/10.1016/S0764-4442(98)80071-2.Google Scholar
Yekutieli, A. and Zhang, J. J., Dualizing complexes and tilting complexes over simple rings. J. Algebra 256(2002), no. 2, 556–567. https://doi.org/10.1016/S0021-8693(02)00005-4.Google Scholar
Yekutieli, A. and Zhang, J. J., Dualizing complexes and perverse modules over differential algebras. Compos. Math. 141(2005), no. 3, 620–654. https://doi.org/10.1112/S0010437X04001307.Google Scholar