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ON GORENSTEINNESS OF HOPF MODULE ALGEBRAS

Part of: Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  10 June 2016

SERGE SKRYABIN
SERGE SKRYABIN*
Affiliation:
Institute of Mathematics and Mechanics, Kazan Federal University, Kremlevskaya St. 18, 420008 Kazan, Russia e-mail: Serge.Skryabin@kpfu.ru
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Abstract

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Let H be a Hopf algebra with a bijective antipode, A an H-simple H-module algebra finitely generated as an algebra over the ground field and module-finite over its centre. The main result states that A has finite injective dimension and is, moreover, Artin–Schelter Gorenstein under the additional assumption that each H-orbit in the space of maximal ideals of A is dense with respect to the Zariski topology. Further conclusions are derived in the cases when the maximal spectrum of A is a single H-orbit or contains an open dense H-orbit.

MSC classification

Primary: 16T05: Hopf algebras and their applications
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 2 , May 2017 , pp. 299 - 321
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Anderson, F. W. and Fuller, K. R., Rings and categories of modules (Springer, Berlin, 1974).Google Scholar
2. Bass, H., On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28.Google Scholar
3. Blair, W. D., Right Noetherian rings integral over their centers, J. Algebra 27 (1973), 187–198.Google Scholar
4. Bourbaki, N., Algèbre commutative, Ch. 10 (Masson, Paris, 1978).Google Scholar
5. Bourbaki, N., Algèbre homologique (Masson, Paris, 1980).Google Scholar
6. Brown, K. A., Noetherian Hopf algebras, Turkish J. Math. Suppl. 31 (2007), 7–23.Google Scholar
7. Brown, K. A. and Goodearl, K. R., Homological aspects of Noetherian PI Hopf algebras and irreducible modules of maximal dimension, J. Algebra, 198 (1997), 240–265.Google Scholar
8. Brown, K. A. and Hajarnavis, C. R., Injectively homogeneous rings, J. Pure Appl. Algebra 51 (1988), 65–77.Google Scholar
9. Brown, K. A., Hajarnavis, C. R. and MacEacharn, A. B., Rings of finite global dimension integral over their centres, Comm. Algebra 11 (1983), 67–93.Google Scholar
10. Curtis, C. W., Noncommutative extensions of Hilbert rings, Proc. Amer. Math. Soc. 4 (1953), 945–955.Google Scholar
11. Eilenberg, S. and Nakayama, T., On the dimension of modules and algebras. II, Nagoya Math. J. 9 (1955), 1–16.Google Scholar
12. Eisenbud, D., Commutative algebra with a view toward algebraic geometry (Springer, Berlin, 1995).Google Scholar
13. Gabriel, P., Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.Google Scholar
14. Goodearl, K. R. and Warfield, R. B., An introduction to noncommutative Noetherian rings (Cambridge Univ. Press, Cambridge, 2004).Google Scholar
15. Greco, S. and Marinari, M. G., Nagata's criterion and openness of loci for Gorenstein and complete intersection, Math. Z. 160 (1978), 207–216 Google Scholar
16. Hochster, M., Non-openness of loci in Noetherian rings, Duke Math. J. 40 (1973), 215–219.CrossRefGoogle Scholar
17. Ischebeck, F., Eine Dualität zwischen den Funktoren Ext und Tor, J. Algebra 11 (1969), 510–531.CrossRefGoogle Scholar
18. Kasch, F., Moduln und ringe (Teubner, Stuttgart, 1977).Google Scholar
19. McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings (Wiley, Chichester, 1987).Google Scholar
20. Montgomery, S., Hopf algebras and their Actions on rings (Amer. Math. Soc., Providence, RI, 1993).CrossRefGoogle Scholar
21. Müller, B., Quasi-Frobenius-Erweiterungen, Math. Z. 85 (1964), 345–368.Google Scholar
22. Müller, B., Quasi-Frobenius-Erweiterungen. II, Math. Z. 88 (1965), 380–409.Google Scholar
23. Nakayama, T., On the complete cohomology theory of Frobenius algebras, Osaka J. Math. 9 (1957), 165–187.Google Scholar
24. Pareigis, B., Einige Bemerkungen über Frobenius-Erweiterungen, Math. Ann. 153 (1964), 1–13.Google Scholar
25. Rotman, J. J., An introduction to homological algebra (Academic Press, New York, 1979).Google Scholar
26. Sharp, R. Y., Acceptable rings and homomorphic images of Gorenstein rings, J. Algebra 44 (1977), 246–261.Google Scholar
27. Skryabin, S., Hopf algebra orbits on the prime spectrum of a module algebra, Algebr. Represent. Theory 13 (2010), 1–31.CrossRefGoogle Scholar
28. Skryabin, S., Structure of H-semiprime Artinian algebras, Algebr. Represent. Theory 14 (2011), 803–822.CrossRefGoogle Scholar
29. Skryabin, S., Flatness of equivariant modules, Max-Planck-Inst. für Math. Preprint Series, 109, 2007.Google Scholar
30. Skryabin, S. and Van Oystaeyen, F., The Goldie theorem for H-semiprime algebras, J. Algebra 305 (2006), 292–320.Google Scholar
31. Stafford, J. T. and Zhang, J. J., Homological properties of (graded) Noetherian PI rings, J. Algebra 168 (1994), 988–1026.Google Scholar
32. Vasconcelos, W. V., On quasi-local regular algebras, in Convegno di Algebra Commutativa, Sympos. Math., vol. XI (Academic Press, London, 1973), 11–22.Google Scholar
33. Weibel, C., An introduction to homological algebra (Cambridge Univ. Press, Cambridge, 1994).Google Scholar
34. Wu, Q.-S. and Zhang, J. J., Homological identities for noncommutative rings, J. Algebra 242 (2001), 516–535.Google Scholar
35. Wu, Q.-S. and Zhang, J. J., Noetherian PI Hopf algebras are Gorenstein, Trans. Amer. Math. Soc. 355 (2003), 1043–1066.Google Scholar