Noetherian PI Hopf algebras are Gorenstein

Wu, Q.-S.; Zhang, J.

doi:10.1090/S0002-9947-02-03106-9

Noetherian PI Hopf algebras are Gorenstein
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by Q.-S. Wu and J. J. Zhang PDF

Trans. Amer. Math. Soc. 355 (2003), 1043-1066 Request permission

Abstract:

We prove that every noetherian affine PI Hopf algebra has finite injective dimension, which answers a question of Brown (1998).

References

K. Ajitabh, S. P. Smith, and J. J. Zhang, Auslander-Gorenstein rings, Comm. Algebra 26 (1998), no. 7, 2159–2180. MR 1626582, DOI 10.1080/00927879808826267
M. Artin, L. W. Small, and J. J. Zhang, Generic flatness for strongly Noetherian algebras, J. Algebra 221 (1999), no. 2, 579–610. MR 1728399, DOI 10.1006/jabr.1999.7997
Kenneth A. Brown, Representation theory of Noetherian Hopf algebras satisfying a polynomial identity, Trends in the representation theory of finite-dimensional algebras (Seattle, WA, 1997) Contemp. Math., vol. 229, Amer. Math. Soc., Providence, RI, 1998, pp. 49–79. MR 1676211, DOI 10.1090/conm/229/03310
K. A. Brown and K. R. Goodearl, Homological aspects of Noetherian PI Hopf algebras of irreducible modules and maximal dimension, J. Algebra 198 (1997), no. 1, 240–265. MR 1482982, DOI 10.1006/jabr.1997.7109
Michel Demazure and Peter Gabriel, Introduction to algebraic geometry and algebraic groups, North-Holland Mathematics Studies, vol. 39, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from the French by J. Bell. MR 563524
K. R. Goodearl and T. H. Lenagan, Catenarity in quantum algebras, J. Pure Appl. Algebra 111 (1996), no. 1-3, 123–142. MR 1394347, DOI 10.1016/0022-4049(95)00120-4
K. R. Goodearl and R. B. Warfield Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, vol. 16, Cambridge University Press, Cambridge, 1989. MR 1020298
Friedrich Ischebeck, Eine Dualität zwischen den Funktoren Ext und Tor, J. Algebra 11 (1969), 510–531 (German). MR 237613, DOI 10.1016/0021-8693(69)90090-8
Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. MR 1721834, DOI 10.1090/gsm/022
Richard G. Larson and David E. Radford, Finite-dimensional cosemisimple Hopf algebras in characteristic $0$ are semisimple, J. Algebra 117 (1988), no. 2, 267–289. MR 957441, DOI 10.1016/0021-8693(88)90107-X
Richard G. Larson and David E. Radford, Semisimple cosemisimple Hopf algebras, Amer. J. Math. 110 (1988), no. 1, 187–195. MR 926744, DOI 10.2307/2374545
Richard Gustavus Larson and Moss Eisenberg Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 75–94. MR 240169, DOI 10.2307/2373270
Thierry Levasseur, Some properties of noncommutative regular graded rings, Glasgow Math. J. 34 (1992), no. 3, 277–300. MR 1181768, DOI 10.1017/S0017089500008843
J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572
Richard K. Molnar, A commutative Noetherian Hopf algebra over a field is finitely generated, Proc. Amer. Math. Soc. 51 (1975), 501–502. MR 376740, DOI 10.1090/S0002-9939-1975-0376740-5
Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637, DOI 10.1090/cbms/082
Warren D. Nichols and M. Bettina Richmond, Freeness of infinite-dimensional Hopf algebras, Comm. Algebra 20 (1992), no. 5, 1489–1492. MR 1157919, DOI 10.1080/00927879208824415
Warren D. Nichols and M. Bettina Zoeller, A Hopf algebra freeness theorem, Amer. J. Math. 111 (1989), no. 2, 381–385. MR 987762, DOI 10.2307/2374514
Ulrich Oberst and Hans-Jürgen Schneider, Untergruppen formeller Gruppen von endlichem Index, J. Algebra 31 (1974), 10–44 (German). MR 360610, DOI 10.1016/0021-8693(74)90003-9
N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs, No. 3, Academic Press, London-New York, 1973. MR 0340375
David E. Radford, Freeness (projectivity) criteria for Hopf algebras over Hopf subalgebras, J. Pure Appl. Algebra 11 (1977/78), no. 1-3, 15–28. MR 476790, DOI 10.1016/0022-4049(77)90035-4
Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 538169
Louis H. Rowen, Ring theory, Student edition, Academic Press, Inc., Boston, MA, 1991. MR 1095047
L. W. Small, Rings satisfying a polynomial identity, Vorlesungen aus dem Fachbereich Mathematik der Universität GH Essen [Lecture Notes in Mathematics at the University of Essen], vol. 5, Universität Essen, Fachbereich Mathematik, Essen, 1980. Written from notes taken by Christine Bessenrodt. MR 601386
J. T. Stafford and J. J. Zhang, Homological properties of (graded) Noetherian $\textrm {PI}$ rings, J. Algebra 168 (1994), no. 3, 988–1026. MR 1293638, DOI 10.1006/jabr.1994.1267
Mitsuhiro Takeuchi, A correspondence between Hopf ideals and sub-Hopf algebras, Manuscripta Math. 7 (1972), 251–270. MR 321963, DOI 10.1007/BF01579722
Mitsuhiro Takeuchi, Relative Hopf modules—equivalences and freeness criteria, J. Algebra 60 (1979), no. 2, 452–471. MR 549940, DOI 10.1016/0021-8693(79)90093-0
P. Vámos, On the minimal prime ideal of a tensor product of two fields, Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 1, 25–35. MR 489566, DOI 10.1017/S0305004100054840
Q.-S. Wu and J. J. Zhang, Homological identities for noncommutative rings, J. Algebra, 242 (2001), 516-535.
Q.-S. Wu and J. J. Zhang, Gorenstein Property of Hopf Graded Algebras, Glasgow Math. J., (to appear).
Q.-S. Wu and J. J. Zhang, Regularity of Involutory PI Hopf Algebras, J. Algebra, (to appear).
Amnon Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 (1992), no. 1, 41–84. MR 1195406, DOI 10.1016/0021-8693(92)90148-F
Amnon Yekutieli and James J. Zhang, Rings with Auslander dualizing complexes, J. Algebra 213 (1999), no. 1, 1–51. MR 1674648, DOI 10.1006/jabr.1998.7657
A. Yekutieli and J. J. Zhang, Residue complexes over noncommutative rings, J. Algebra, to appear. ArXiv: math.RA/0103075.