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Noetherian PI Hopf algebras are Gorenstein

Author(s): Q.-S. Wu; J. J. Zhang
Journal: Trans. Amer. Math. Soc. 355 (2003), 1043-1066.
MSC (2000): Primary 16E10, 16W30
Posted: October 24, 2002
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Abstract | References | Similar articles | Additional information

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

References:

[AjSZ]
K. Ajitabh, S. P. Smith and J. J. Zhang, Auslander-Gorenstein rings, Comm. Algebra, 26 1998, 2159-2180.MR 99g:16010
[ArSZ]
M. Artin, L. W. Small and J. J. Zhang, Generic flatness for strongly Noetherian algebras, J. Algebra 221 (1999), 579-610. MR 2001a:16006

[Br]
K. A. Brown, Representation theory of Noetherian Hopf algebras satisfying a polynomial identity, Trends in the representation theory of finite-dimensional algebras (Seattle, WA, 1997), 49-79, Contemp. Math., 229, American Mathematical Society, Providence, RI, 1998.MR 99m:16056

[BG]
K. A. Brown and K. R. Goodearl, Homological aspects of Noetherian PI Hopf algebras and irreducible modules of maximal dimension, J. Algebra 198 (1997), 240-265.MR 99c:16036

[DG]
M. Demazure and P. Gabriel, ``Introduction to algebraic geometry and algebraic groups'', Translated from the French by J. Bell, North-Holland Mathematics Studies, 39. North-Holland Publishing Co., Amsterdam-New York, 1980. MR 82e:14001

[GL]
K. R. Goodearl and T. H. Lenagan, Catenarity in quantum algebras, J. Pure Appl. Algebra, 111 (1996), 123-142.MR 97e:16054
[GW]
K. R. Goodearl and R. B. Warfield, Jr., ``An Introduction to Noncommutative Noetherian Rings,'' London Math. Soc. Student Texts, Vol. 16, Cambridge Univ. Press, Cambridge, 1989.MR 91c:16001
[Is]
F. Ischebeck, Eine Dualität zwischen den Funktoren Ext und Tor (German), J. Algebra 11 (1969), 510-531.MR 38:5894

[KL]
G. R. Krause and T. H. Lenagan, ``Growth of algebras and Gelfand-Kirillov dimension'', Revised edition, Graduate Studies in Mathematics, 22. American Mathematical Society, Providence, RI, 2000.MR 2000j:16035

[LR1]
R. G. Larson and D. E. Radford, Finite-dimensional cosemisimple Hopf algebras in characteristic $0$ are semisimple, J. Algebra 117 (1988), no. 2, 267-289.MR 89k:16016

[LR2]
R. G. Larson and D. E. Radford, Semisimple cosemisimple Hopf algebras, Amer. J. Math. 110 (1988), no. 1, 187-195. MR 89a:16011
[LS]
R. G. Larson and M. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969) 75-94.MR 39:1523

[Le]
T. Levasseur, Some properties of noncommutative regular graded rings, Glasgow Math. J. 34 (1992), 277-300. MR 93k:16045
[MR]
J. C. McConnell and J. C. Robson, ``Noncommutative Noetherian Rings,'' Wiley, Chichester, 1987.MR 89j:16023
[Mol]
R. K. Molnar, A commutative Noetherian Hopf algebra over a field is finitely generated. Proc. Amer. Math. Soc. 51 (1975), 501-502.MR 51:12915

[Mon]
S. Montgomery, Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics, 82, American Mathematical Society, Providence, RI, 1993.MR 94i:16019

[NR]
W. D. Nichols and M. B. Richmond, Freeness of infinite-dimensional Hopf algebras. Comm. Algebra 20 (1992), no. 5, 1489-1492.MR 93d:16054

[NZ]
W. D. Nichols and M. B. Zoeller, A Hopf algebra freeness theorem. Amer. J. Math. 111 (1989), no. 2, 381-385. MR 90c:16008
[OS]
U. Oberst and H.-J. Schneider, Untergruppen formeller Gruppen von endlichem Index. (German) J. Algebra 31 (1974), 10-44.MR 50:13057

[Po]
N. Popescu, Abelian categories with applications to rings and modules. London Mathematical Society Monographs, No. 3. Academic Press, London-New York, 1973.MR 49:5130

[Ra]
D. E. Radford, Freeness (projectivity) criteria for Hopf algebras over Hopf subalgebras, J. Pure Appl. Algebra 11 (1977/78), no. 1-3, 15-28.MR 57:16344

[Rot]
J. J. Rotman, ``An introduction to homological algebra'', Pure and Applied Mathematics, 85. Academic Press, Inc. New York-London, 1979.MR 80k:18001

[Row]
L. H. Rowen, ``Ring theory'', Student edition. Academic Press, Inc., Boston, MA, 1991.MR 94e:16001

[Sm]
L.W. Small, Rings satisfying a polynomial identity, Lecture Notes, Universität Essen (1980).MR 82j:16028

[SZ]
J. T. Stafford and J. J. Zhang, Homological properties of (graded) noetherian PI rings, J. Algebra 168 (1994), 988-1026.MR 95h:16030

[Ta1]
M. Takeuchi, A correspondence between Hopf ideals and sub-Hopf algebras, Manuscripta Math. 7 (1972), 251-270. MR 48:328
[Ta2]
M. Takeuchi, Relative Hopf modules--equivalences and freeness criteria, J. Algebra 60 (1979), no. 2, 452-471.MR 82m:16006

[Va]
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 80j:12016

[WZ1]
Q.-S. Wu and J. J. Zhang, Homological identities for noncommutative rings, J. Algebra, 242 (2001), 516-535.

[WZ2]
Q.-S. Wu and J. J. Zhang, Gorenstein Property of Hopf Graded Algebras, Glasgow Math. J., (to appear).

[WZ3]
Q.-S. Wu and J. J. Zhang, Regularity of Involutory PI Hopf Algebras, J. Algebra, (to appear).

[Ye]
A. Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 (1992), 41-84.MR 94a:16077
[YZ1]
A. Yekutieli and J. J. Zhang, Rings with Auslander dualizing complexes, J. Algebra 213 (1999), 1-51.MR 2000f:16012

[YZ2]
A. Yekutieli and J. J. Zhang, Residue complexes over noncommutative rings, J. Algebra, to appear. ArXiv: math.RA/0103075.

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Additional Information:

Q.-S. Wu
Affiliation: Institute of Mathematics, Fudan University, Shanghai, 200433, China
Email: qswu@fudan.edu.cn

J. J. Zhang
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: zhang@math.washington.edu

DOI: 10.1090/S0002-9947-02-03106-9
PII: S 0002-9947(02)03106-9
Keywords: Hopf algebra, injective dimension
Received by editor(s): May 22, 2002
Posted: October 24, 2002
Additional Notes: The first author was supported in part by the NSFC (project 10171016) and the second author was supported in part by the NSF
Copyright of article: Copyright 2002, American Mathematical Society

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