Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-23T13:33:50.022Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Brauer Group of Algebras Having a Grading and an Action

Published online by Cambridge University Press:  20 November 2018

Morris Orzech
Morris Orzech*
Affiliation:
The Institute for Advanced Study, Princeton, New Jersey
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Beginning with Wall's introduction [19] of Z2-graded central simple algebras over a field K, a number of related generalizations of the Brauer group have been proposed. In [16] the field K was replaced by a commutative ring R, building upon the theory developed in [1]. The concept of a G-graded central simple K-algebra (G an abelian group) was first defined in [12]; this work and that of [16] was subsequently unified in [6] and [7] via the construction and computation of the graded Brauer group Bφ﹛R, G) (φ a bilinear form from G X G to U(R), the units of R).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 3 , 01 June 1976 , pp. 533 - 552
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Auslander, M. and Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367–409.Google Scholar
2. Bass, H., Lectures on topics in algebraic K-theory, Tata Institute for Fundamental Research, Bombay, 1967.Google Scholar
3. Bass, H., Algebraic K-theory (Benjamin, W. A., New York, 1968).Google Scholar
4. Cartan, H. and Eilenberg, S., Homologuai algebra (Princeton University Press, 1956).Google Scholar
5. Chase, S. U., Harrison, D. K. and Rosenberg, A., Galois theory and Galois cohomology of commutative rings, Memoirs Amer. Math. Soc. 52 (1965), 1–19.Google Scholar
6. Childs, L. N., The Brauer group of graded algebras II: graded Galois extensions, Trans. Amer. Math. Soc. 201+ (1975), 137–160.Google Scholar
7. Childs, L. N., Garfinkel, G. and Orzech, M., The Brauer group of graded Azumaya algebras, Trans. Amer. Math. Soc. 175 (1973), 299–326.Google Scholar
8. Frohlich, A. and Wall, C. T. C., Generalizations of the Brauer group, to appear.Google Scholar
9. Harada, M., Some criteria for hereditarily of crossed products, Osaka J. Math. 1 (1964), 69–80.Google Scholar
10. Harrison, D. K., Abelian extensions of commutative rings, Memoirs Amer. Math. Soc. 52 (1965), 66–79.Google Scholar
11. Janusz, G. J., Separable algebras over commutative rings, Trans. Amer. Math. Soc. 122 (1966), 461–479.Google Scholar
12. Knus, M.-A., Algebras graded by a group, Category Theory, Homology Theory and their Applications, Battelle Inst. Conf. Seattle, Wash., 1968, vol. II (Springer, Berlin, 1969).Google Scholar
13. Long, F. W., A generalization of the Brauer group of graded algebras, Proc. London Math. Soc. (3) 29 (1974), 237–256.Google Scholar
14. Long, F. W., The Brauer group of dimodule algebras, J. Algebra 30 (1974), 559–601.Google Scholar
15. Orzech, M. and Small, C., The Brauer group of commutative rings, Lecture Notes in Pure and Applied Mathematics, No. 11 (Marcel Dekker, New York, 1975).Google Scholar
16. Small, C., The Brauer-Wall group of a commutative ring, Trans. Amer. Math. Soc. 156 (1971), 455–491.Google Scholar
17. Sweedler, M. E., Hopf algebras (Benjamin, W. A., New York, 1969).Google Scholar
18. Van, B. L. der Waerden, Modern algebra, vol. II (Ungar, New York, 1950).Google Scholar
19. Wall, C. T. C., Graded Brauer groups, J. Reine Angew. Math. 213 (1964), 187–199.Google Scholar
20. Yamazaki, K., On projective representations and ring extensions of finite groups, J. Fac. Science, Univ. of Tokyo 10 (1964), 147–195.Google Scholar