Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T17:59:22.892Z Has data issue: false hasContentIssue false
  • English
  • Français

Projectivity of Hopf algebras over subalgebras with semilocal central localizations

Published online by Cambridge University Press:  11 January 2008

Serge Skryabin
Serge Skryabin
Affiliation:
[email protected] Research Institute, Universitetskaya St. 17, 420008 Kazan, Russia
Get access

Abstract

The purpose of this paper is to extend the class of pairs A, H where H is a Hopf algebra over a field and A a right coideal subalgebra for which H is proved to be either projective or flat as an A-module. The projectivity is obtained under the assumption that H is residually finite dimensional, A has semilocal localizations with respect to a central subring, and there exists a Hopf subalgebra B of H such that the antipode of B is bijective and B is a finitely generated A-module. The flatness of H over A is shown to hold when H is a directed union of residually finite dimensional Hopf subalgebras, and there exists a Hopf subalgebra of H whose center contains A. More general projectivity and flatness results are established for (co)equivariant modules over an H-(co)module algebra under similar assumptions.

Keywords

Hopf algebrasModule algebrasComodule algebrasCoideal subalgebrasProjective modulesFlat modules
Type
Research Article
Information
Journal of K-Theory , Volume 2 , Issue 1 , August 2008 , pp. 1 - 40
Copyright
Copyright © ISOPP 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Azumaya, G., On maximally central algebras, Nagoya Math. J. 2 (1951) 119–150CrossRefGoogle Scholar
2.Bass, H., Algebraic K-theory, Benjamin, (1968)Google Scholar
3.Björk, J.-E., Conditions which imply that subrings of artinian rings are artinian, J. reine angew. Math. 247 (1971) 123–138Google Scholar
4.Björk, J.-E., Noetherian and Artinian chain conditions of associative rings, Arch. Math. 24 (1973) 366–378CrossRefGoogle Scholar
5.Bourbaki, N., Commutative Algebra, Springer, (1989)Google Scholar
6.Brown, K.A. and Goodearl, K.R., Lectures on Algebraic Quantum Groups, Birkhäuser, (2002)CrossRefGoogle Scholar
7.Camps, R. and Dicks, W., On semilocal rings, Isr. J. Math. 81 (1993) 203–211CrossRefGoogle Scholar
8.Chin, W., Spectra of smash products, Isr. J. Math. 72 (1990) 84–98CrossRefGoogle Scholar
9.Cortzen, B. and Small, L.W., Finite extensions of rings, Proc. Amer. Math. Soc. 103 (1988) 1058–1062CrossRefGoogle Scholar
10.Curtis, C.W., Noncommutative extensions of Hilbert rings, Proc. Amer. Math. Soc. 4 (1953) 945–955CrossRefGoogle Scholar
11.De Concini, C. and Lyubashenko, V., Quantum function algebra at roots of 1, Adv. Math. 108 (1994) 205–262CrossRefGoogle Scholar
12.Doi, Y., On the structure of relative Hopf modules, Comm. Algebra 11 (1983) 243–255CrossRefGoogle Scholar
13.Etingof, P. and Ostrik, V., Finite tensor categories, Moscow Math. J. 4 (2004) 627–654CrossRefGoogle Scholar
14.Hoffmann, K., Coidealunteralgebren in endlich dimensionalen Hopfalgebren, Dissertation, Univ. München, (1991)Google Scholar
15.Kharchenko, V.K., PBW-bases of coideal subalgebras and a freeness theorem, Preprint.Google Scholar
16.Lam, T.-Y., A First Course in Noncommutative Rings, Graduate Texts in Mathematics, Vol. 131, Springer, (1991)CrossRefGoogle Scholar
17.Masuoka, A., On Hopf algebras with cocommutative coradicals, J. Algebra 144 (1991) 451–466CrossRefGoogle Scholar
18.Masuoka, A., Freeness of Hopf algebras over coideal subalgebras, Comm. Algebra 20 (1992) 1353–1373CrossRefGoogle Scholar
19.Masuoka, A. and Wigner, D., Faithful flatness of Hopf algebras, J. Algebra 170 (1994) 156–164CrossRefGoogle Scholar
20.Montgomery, S., Hopf algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics, Vol. 82, American Mathematical Society, (1993)Google Scholar
21.Montgomery, S. and Schneider, H.-J., Hopf crossed products, rings of quotients, and prime ideals, Adv. Math. 112 (1995) 1–55CrossRefGoogle Scholar
22.Montgomery, S. and Schneider, H.-J., Prime ideals in Hopf Galois extensions, Isr. J. Math. 112 (1999) 187–235CrossRefGoogle Scholar
23.Nichols, W.D. and Zoeller, M.B., A Hopf algebra freeness theorem, Amer. J. Math. 111 (1989) 381–385CrossRefGoogle Scholar
24.Procesi, C., Non commutative Jacobson-rings, Ann. Scuola Norm. Sup. Pisa 21 (1967) 281–290Google Scholar
25.Radford, D.E., Pointed Hopf algebras are free over Hopf subalgebras, J. Algebra 45 (1977) 266–273CrossRefGoogle Scholar
26.Radford, D.E., Freeness (projectivity) criteria for Hopf algebras over Hopf subalgebras, J. Pure Appl. Algebra 11 (1977) 15–28CrossRefGoogle Scholar
27.Robson, J.C. and Small, L.W., Liberal extensions, Proc. London Math. Soc. 42 (1981) 87–103CrossRefGoogle Scholar
28.Schauenburg, P., Faithful flatness over Hopf subalgebras: counterexamples, in “Interactions between ring theory and representations of algebras”, Lect. Notes Pure Appl. Math., Vol. 210, Marcel Dekker, (2000), pp. 331–344Google Scholar
29.Schauenburg, P. and Schneider, H.-J., Galois type extensions of noncommutative algebras, to appear in Banach Center PublGoogle Scholar
30.Schneider, H.-J., Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra 152 (1992) 289–312CrossRefGoogle Scholar
31.Schneider, H.-J., Some remarks on exact sequences of quantum groups, Comm. Algebra 21 (1993) 3337–3357CrossRefGoogle Scholar
32.Skryabin, S., Projectivity and freeness over comodule algebras, Trans. Amer. Math. Soc. 359 (2007) 2597–2623CrossRefGoogle Scholar
33.Skryabin, S. and Van Oystaeyen, F., The Goldie Theorem for H-semiprime algebras, J. Algebra 305 (2006) 292–320CrossRefGoogle Scholar
34.Sweedler, M.E., Hopf Algebras, Benjamin, (1969)Google Scholar
35.Takeuchi, M., Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971) 561–582Google Scholar
36.Takeuchi, M., Relative Hopf modules—equivalences and freeness criteria, J. Algebra 60 (1979) 452–471CrossRefGoogle Scholar
37.Wu, Q.-S. and Zhang, J.J., Noetherian PI Hopf algebras are Gorenstein, Trans. Amer. Math. Soc. 355 (2003) 1043–1066CrossRefGoogle Scholar