Noetherian PI Hopf algebras are Gorenstein
Authors:
Q.S. Wu and J. J. Zhang
Journal:
Trans. Amer. Math. Soc. 355 (2003), 10431066
MSC (2000):
Primary 16E10, 16W30
Published electronically:
October 24, 2002
MathSciNet review:
1938745
Fulltext PDF Free Access
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).
 [AjSZ]
K. Ajitabh, S. P. Smith and J. J. Zhang, AuslanderGorenstein rings, Comm. Algebra, 26 1998, 21592180.MR 99g:16010
 [ArSZ]
M. Artin, L. W. Small and J. J. Zhang, Generic flatness for strongly Noetherian algebras, J. Algebra 221 (1999), 579610. MR 2001a:16006
 [Br]
K. A. Brown, Representation theory of Noetherian Hopf algebras satisfying a polynomial identity, Trends in the representation theory of finitedimensional algebras (Seattle, WA, 1997), 4979, 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), 240265.MR 99c:16036
 [DG]
M. Demazure and P. Gabriel, ``Introduction to algebraic geometry and algebraic groups'', Translated from the French by J. Bell, NorthHolland Mathematics Studies, 39. NorthHolland Publishing Co., AmsterdamNew York, 1980. MR 82e:14001
 [GL]
K. R. Goodearl and T. H. Lenagan, Catenarity in quantum algebras, J. Pure Appl. Algebra, 111 (1996), 123142.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), 510531.MR 38:5894
 [KL]
G. R. Krause and T. H. Lenagan, ``Growth of algebras and GelfandKirillov 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, Finitedimensional cosemisimple Hopf algebras in characteristic are semisimple, J. Algebra 117 (1988), no. 2, 267289.MR 89k:16016
 [LR2]
R. G. Larson and D. E. Radford, Semisimple cosemisimple Hopf algebras, Amer. J. Math. 110 (1988), no. 1, 187195. MR 89a:16011
 [LS]
R. G. Larson and M. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969) 7594.MR 39:1523
 [Le]
T. Levasseur, Some properties of noncommutative regular graded rings, Glasgow Math. J. 34 (1992), 277300. 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), 501502.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 infinitedimensional Hopf algebras. Comm. Algebra 20 (1992), no. 5, 14891492.MR 93d:16054
 [NZ]
W. D. Nichols and M. B. Zoeller, A Hopf algebra freeness theorem. Amer. J. Math. 111 (1989), no. 2, 381385. MR 90c:16008
 [OS]
U. Oberst and H.J. Schneider, Untergruppen formeller Gruppen von endlichem Index. (German) J. Algebra 31 (1974), 1044.MR 50:13057
 [Po]
N. Popescu, Abelian categories with applications to rings and modules. London Mathematical Society Monographs, No. 3. Academic Press, LondonNew 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. 13, 1528.MR 57:16344
 [Rot]
J. J. Rotman, ``An introduction to homological algebra'', Pure and Applied Mathematics, 85. Academic Press, Inc. New YorkLondon, 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), 9881026.MR 95h:16030
 [Ta1]
M. Takeuchi, A correspondence between Hopf ideals and subHopf algebras, Manuscripta Math. 7 (1972), 251270. MR 48:328
 [Ta2]
M. Takeuchi, Relative Hopf modulesequivalences and freeness criteria, J. Algebra 60 (1979), no. 2, 452471.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, 2535.MR 80j:12016
 [WZ1]
Q.S. Wu and J. J. Zhang, Homological identities for noncommutative rings, J. Algebra, 242 (2001), 516535.
 [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), 4184.MR 94a:16077
 [YZ1]
A. Yekutieli and J. J. Zhang, Rings with Auslander dualizing complexes, J. Algebra 213 (1999), 151.MR 2000f:16012
 [YZ2]
A. Yekutieli and J. J. Zhang, Residue complexes over noncommutative rings, J. Algebra, to appear. ArXiv: math.RA/0103075.
 [AjSZ]
 K. Ajitabh, S. P. Smith and J. J. Zhang, AuslanderGorenstein rings, Comm. Algebra, 26 1998, 21592180.MR 99g:16010
 [ArSZ]
 M. Artin, L. W. Small and J. J. Zhang, Generic flatness for strongly Noetherian algebras, J. Algebra 221 (1999), 579610. MR 2001a:16006
 [Br]
 K. A. Brown, Representation theory of Noetherian Hopf algebras satisfying a polynomial identity, Trends in the representation theory of finitedimensional algebras (Seattle, WA, 1997), 4979, 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), 240265.MR 99c:16036
 [DG]
 M. Demazure and P. Gabriel, ``Introduction to algebraic geometry and algebraic groups'', Translated from the French by J. Bell, NorthHolland Mathematics Studies, 39. NorthHolland Publishing Co., AmsterdamNew York, 1980. MR 82e:14001
 [GL]
 K. R. Goodearl and T. H. Lenagan, Catenarity in quantum algebras, J. Pure Appl. Algebra, 111 (1996), 123142.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), 510531.MR 38:5894
 [KL]
 G. R. Krause and T. H. Lenagan, ``Growth of algebras and GelfandKirillov 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, Finitedimensional cosemisimple Hopf algebras in characteristic are semisimple, J. Algebra 117 (1988), no. 2, 267289.MR 89k:16016
 [LR2]
 R. G. Larson and D. E. Radford, Semisimple cosemisimple Hopf algebras, Amer. J. Math. 110 (1988), no. 1, 187195. MR 89a:16011
 [LS]
 R. G. Larson and M. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969) 7594.MR 39:1523
 [Le]
 T. Levasseur, Some properties of noncommutative regular graded rings, Glasgow Math. J. 34 (1992), 277300. 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), 501502.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 infinitedimensional Hopf algebras. Comm. Algebra 20 (1992), no. 5, 14891492.MR 93d:16054
 [NZ]
 W. D. Nichols and M. B. Zoeller, A Hopf algebra freeness theorem. Amer. J. Math. 111 (1989), no. 2, 381385. MR 90c:16008
 [OS]
 U. Oberst and H.J. Schneider, Untergruppen formeller Gruppen von endlichem Index. (German) J. Algebra 31 (1974), 1044.MR 50:13057
 [Po]
 N. Popescu, Abelian categories with applications to rings and modules. London Mathematical Society Monographs, No. 3. Academic Press, LondonNew 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. 13, 1528.MR 57:16344
 [Rot]
 J. J. Rotman, ``An introduction to homological algebra'', Pure and Applied Mathematics, 85. Academic Press, Inc. New YorkLondon, 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), 9881026.MR 95h:16030
 [Ta1]
 M. Takeuchi, A correspondence between Hopf ideals and subHopf algebras, Manuscripta Math. 7 (1972), 251270. MR 48:328
 [Ta2]
 M. Takeuchi, Relative Hopf modulesequivalences and freeness criteria, J. Algebra 60 (1979), no. 2, 452471.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, 2535.MR 80j:12016
 [WZ1]
 Q.S. Wu and J. J. Zhang, Homological identities for noncommutative rings, J. Algebra, 242 (2001), 516535.
 [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), 4184.MR 94a:16077
 [YZ1]
 A. Yekutieli and J. J. Zhang, Rings with Auslander dualizing complexes, J. Algebra 213 (1999), 151.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:
http://dx.doi.org/10.1090/S0002994702031069
PII:
S 00029947(02)031069
Keywords:
Hopf algebra,
injective dimension
Received by editor(s):
May 22, 2002
Published electronically:
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
Article copyright:
© Copyright 2002
American Mathematical Society
