Abstract
LetH be a commutative, cocommutative and faithfully projective Hopf algebra over a commutative ringR. Using cohomological methods, we obtain a description of a subgroup of Long’s Brauer group ofH-dimodule algebras:Ψ where the mapψ is induced by the smash product, and where BDs (R, H) is the subgroup of the Brauer-Long group consisting of all elements which are split by a faithfully flat extension ofR. As an example, the Brauer-Long group of a free Hopf algebra of rank 2 is computed. The results are also applied to Orzech’s subgroup of the Brauer-Long group.
Similar content being viewed by others
References
E. Abe,Hopf Algebras, Cambridge University Press, Cambridge, 1977.
M. Artin,On the joins of Hensel rings, Adv. in Math.7 (1971), 282–296.
M. Auslander and O. Goldman,The Brauer group of a commutative ring, Trans. Am. Math. Soc.97 (1960), 367–409.
M. Beattie,Brauer groups of H-module and H-dimodule algebras, thesis, Queens University, Kingston, Ontario, 1976.
M. Beattie,A direct sum decomposition for the Brauer group of H-module algebras, J. Algebra43 (1976), 686–693.
M. Beattie,The Brauer group of central separable G-Azumaya algebras, J. Algebra54 (1978), 516–525.
M. Beattie,Computing the Brauer group of graded Azumaya algebras from its subgroups, J. Algebra101 (1986), 339–349.
M. Beattie and S. Caenepeel,The Brauer-Long group of ℤ/p t ℤ-dimodule algebras, J. Pure Appl. Algebra61 (1989), 219–236.
S. Caenepeel,Computing the Brauer-Long group of a Hopf algebra II: the Skolem-Noether theory, in preparation.
S. Caenepeel and M. Beattie,A cohomological approach to the Brauer-Long group and the groups of Galois extensions and strongly graded rings. Trans. Am. Math. Soc., to appear.
S. Chase and A. Rosenberg,Amitsur cohomology and the Brauer group, Mem. Am. Math. Soc.52 (1965), 20–45.
S. Chase and M. E. Sweedler,Hopf algebras and Galois theory, Lecture Notes in Math.97, Springer-Verlag, Berlin, 1969.
L. N. Childs,The Brauer group of graded Azumaya algebras II: graded Galois extensions, Trans. Am. Math. Soc.204 (1975), 137–160.
L. N. Childs,Representing classes in the Brauer group of quadratic number rings as smash products, Pacific J. Math.129 (1987), 243–255.
L. N. Childs, G. Garfinkel and M. Orzech,The Brauer group of graded Azumaya algebras, Trans. Am. Math. Soc.175 (1973), 299–326.
F. DeMeyer and T. Ford,Computing the Brauer-Long group of ℤ/2-dimodule algebras, J. Pure Appl. Algebra54 (1988), 197–208.
F. DeMeyer and E. Ingraham,Separable algebras over commutative rings, Lecture Notes in Math.181, Springer-Verlag, Berlin, 1971.
T. E. Early and H. F. Kreimer,Galois algebras and Harrison cohomology, J. Algebra58 (1979), 136–147.
O. Gabber,Some Theorems on Azumaya algebras, inGroupe de Brauer, Lecture Notes in Math.844, Springer-Verlag, Berlin, 1981.
S. Hurley,Galois objects with normal bases for free Hopf algebras of prime degree, J. Algebra109 (1987), 292–318.
M. A. Knus and M. Ojanguren,Théorie de la descente et algèbres d’Azumaya, Lecture Notes in Math.389, Springer-Verlag, Berlin, 1974.
H. F. Kreimer,Quadratic Hopf algebras and Galois extensions, Am. Math. Soc. Contemp. Math.13 (1981), 353–361.
H. F. Kreimer and P. M. Cook II,Galois theories and normal bases, J. Algebra43 (1976), 115–121.
F. Long,A generalization of the Brauer group of graded algebras, Proc. London Math. Soc.29 (1974), 237–256.
F. Long,The Brauer group of dimodule algebras, J. Algebra30 (1974), 559–601.
J. S. Milne,Étale Cohomology, Princeton University Press, Princeton, 1980.
A. Nakajima,On generalized Harrison cohomology and Galois object, Math. J. Okayama Univ.17 (1975), 135–148.
A. Nakajima,Some results on H-Azumaya algebras, Math. J. Okayama Univ.19 (1977), 101–110.
A. Nakajima,Free algebras and Galois objects of rank 2, Math. J. Okayama Univ.23 (1981), 181–187.
M. Orzech,On the Brauer group of algebras having a grading and an action, Can. J. Math.28 (1976), 533–552.
M. Orzech,Brauer groups of graded algebras, Lecture Notes in Math.549, Springer-Verlag, Berlin, 1976, pp. 134–147.
M. Orzech and C. Small,The Brauer group of a commutative ring, Marcel Dekker, Inc., New York, 1975.
M. E. Sweedler,Cohomology of algebras over Hopf algebras, Trans. Am. Math. Soc.133 (1968), 205–239.
M. E. Sweedler,Hopf Algebras, Benjamin, New York, 1969.
J. T. Tate, F. Oort,Group Schemes of prime order, Ann. Sci. Ec. Norm. Sup. (4ième série)3 (1970), 1–21.
F. Tilborghs,The Brauer group of R-algebras which have compatible G-action and ℤ × G-grading, Comm. Algebra18 (1990), 3351–3379.
F. Tilborghs,An anti-homomorphism for the Brauer-Long group, Math. J. Okayma Univ., to appear.
O. E. Villamayor and D. Zelinsky,Brauer groups and Amitsur cohomology for general commutative ring extensions, J. Pure Appl. Algebra10 (1977), 19–55.
C. T. C. Wall,Graded Brauer groups, J. Reine Angew. Math.213 (1964), 187–199.
D. Zelinsky,Long exact sequences and the Brauer group, inBrauer Groups, Evanston 1975, Lecture Notes in Math.549, Springer-Verlag, Berlin, 1976.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Caenepeel, S. Computing the brauer-long group of a hopf algebra I: The cohomological theory. Israel J. Math. 72, 38–83 (1990). https://doi.org/10.1007/BF02764611
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02764611