Geodesic
A Morita Cancellation Problem
Canadian journal of mathematics, Tome 72 (2020) no. 3, pp. 708-731

Voir la notice de l'article provenant de la source Cambridge University Press

We study a Morita-equivalent version of the Zariski cancellation problem.
DOI : 10.4153/S0008414X1900004X
Mots-clés : Zariski cancellation problem, Morita cancellation problem 
Lu, D.-M.; Wu, Q.-S.; Zhang, J. J. A Morita Cancellation Problem. Canadian journal of mathematics, Tome 72 (2020) no. 3, pp. 708-731. doi: 10.4153/S0008414X1900004X
@article{10_4153_S0008414X1900004X,
     author = {Lu, D.-M. and Wu, Q.-S. and Zhang, J. J.},
     title = {A {Morita} {Cancellation} {Problem}},
     journal = {Canadian journal of mathematics},
     pages = {708--731},
     year = {2020},
     volume = {72},
     number = {3},
     doi = {10.4153/S0008414X1900004X},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X1900004X/}
}
TY  - JOUR
AU  - Lu, D.-M.
AU  - Wu, Q.-S.
AU  - Zhang, J. J.
TI  - A Morita Cancellation Problem
JO  - Canadian journal of mathematics
PY  - 2020
SP  - 708
EP  - 731
VL  - 72
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/S0008414X1900004X/
DO  - 10.4153/S0008414X1900004X
ID  - 10_4153_S0008414X1900004X
ER  - 
%0 Journal Article
%A Lu, D.-M.
%A Wu, Q.-S.
%A Zhang, J. J.
%T A Morita Cancellation Problem
%J Canadian journal of mathematics
%D 2020
%P 708-731
%V 72
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/S0008414X1900004X/
%R 10.4153/S0008414X1900004X
%F 10_4153_S0008414X1900004X

[AEH] 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

[AF] 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

[ASS] 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

[AG] 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

[BZ1] 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

[BZ2] 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

[BY] 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

[CL] 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

[CPWZ1] 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

[CPWZ2] 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

[CYZ1] 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

[CYZ2] 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

[Da] Danielewski, W., On the cancellation problem and automorphism groups of affine algebraic varieties. Preprint, 1989, 8 pages, Warsaw. Google Scholar

[DeI] Demeyer, F. and Ingraham, E., Separable algebras over commutative rings. Lecture Notes in Mathematics, 181, Springer-Verlag, Berlin-New York, 1971. Google Scholar

[Di] Dixmier, J., Quotients simples de l’algébre enveloppante de sl. J. Algebra 24(1973), 551–564. https://doi.org/10.1016/0021-8693(73)90127-0. Google Scholar

[EH] 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

[FSS] 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

[Fi] 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

[Fu] Fujita, T., On Zariski problem. Proc. Japan Acad. Ser. A Math. Sci. 55(1979), 106–110. Google Scholar

[Ga] 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

[GKM] 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

[GWY] 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

[Gu1] 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

[Gu2] 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

[Gu3] 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

[HS] 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

[Ho] 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

[Jo] 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

[Kr] Kraft, H., Challenging problems on affine n-space. Séminaire Bourbaki, 1994/95, Astérisque 237(1996), Exp. No. 802, 5, 295–317. Google Scholar

[LY] 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

[LWZ] 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

[LMZ] 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

[Mak] Makar-Limanov, L., On the hypersurface x + x 2y + z 2 + t 3 = 0 in 4 or a 3-like threefold which is not 3. Israel J. Math. 96(1996), part B, 419–429. https://doi.org/10.1007/BF02937314. Google Scholar

[MS] 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

[Ne] 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

[NTY] 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

[RSS] 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

[Ru] Russell, P., On affine-ruled rational surfaces. Math. Ann. 255(1981), 287–302. https://doi.org/10.1007/BF01450704. Google Scholar

[Sc] 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

[SW] 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

[Sm] 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

[Su] 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

[Ta1] 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

[Ta2] 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

[WZ] 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

[We] 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

[Wi] 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

[YZ1] 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

[YZ2] 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

Cité par Sources :