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Algebras of Finite Cohomological Dimension

Published online by Cambridge University Press:  22 January 2016

Joseph A. Wehlen
Joseph A. Wehlen*
Affiliation:
Michigan State University, East Lansing, Michigan 48823, USA and Ohio University, Athens, Ohio 45701, USA
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The cohomology theory of an associative algebra has been shown to be valuable in the study of the structure of algebras of finite cohomological dimension, especially those of dimension less than or equal to one over a field. M. Harada [9] has shown that every semi-primary hereditary algebra A (for example, A is finitely generated over a field R and has dimension < 1) is isomorphic to a generalized triangular matrix algebra. The concept of a central separable algebra over a commutative ring has been shown to be a useful generalization of the concept of a central simple algebra over a field, where a separable algebra is defined to be an algebra having cohomological dimension zero.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 43 , September 1971 , pp. 127 - 135
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

References

[1] Auslander, M., On the dimension of modules and algebras III, Global dimension, Nagoya Math. J. 9 (1955), 67–77.CrossRefGoogle Scholar
[2] Auslander, M. and Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367–409.CrossRefGoogle Scholar
[3] Azumaya, G., On maximally central algebras, Nagoya Math. J. 2 (1951) 119–150.Google Scholar
[4] Bourbaki, N., Algèbre commutative, Chapters I-II, Actualités Sci. Ind. No. 1290, Hermann, Paris, 1962.Google Scholar
[5] Eilenberg, S., Algebras of cohomologically finite dimension, Comment. Math. Helv. 28 (1954), 310–319.Google Scholar
[6] Eilenberg, S., Nagao, H., and Nakayama, T.; On the dimension of modules and algebras IV, Dimension of residue rings oj hereditary rings, Nagoya Math. J. 10 (1956), 87–95.Google Scholar
[7] Eilenberg, S., Rosenberg, A., and Zelinsky, D.; On the dimension of modules and algebras VIII Dimension of tensor product, Nagoya Math. J. 12 (1957), 71–93.Google Scholar
[8] Endo, S. and Watanabe, Y., On separable algebras over a commutative ring, Osaka J. Math. 4 (1967), 233–242.Google Scholar
[9] Harada, M., Hereditary semiprimary rings and tri-angular matrix rings, Nagoya Math. J. 27 (1966), 463–484.Google Scholar
[10] Maclane, S., Homology, Sprniger-Verlag New York, Inc., 1963.Google Scholar
[11] Northcott, D.G., An introduction to homological algebra, Cambridge University Press, 1962.Google Scholar
[12] Rosenberg, A. and Zelinksy, D., Cohomology of infinite algebras, Trans. Amer. Math. Soc. 82 (1956), 85–98.Google Scholar