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On the Dimension of Modules and Algebras, I

Published online by Cambridge University Press:  22 January 2016

Samuel Eilenberg ,
Masatoshi Ikeda  and
Tadasi Nakayama
Samuel Eilenberg
Affiliation:
Columbia University
Masatoshi Ikeda
Affiliation:
Osaka University
Tadasi Nakayama
Affiliation:
Nagoya University
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In [5], Ikeda-Nagao-Nakayama gave a characterization of algebras of cohomological dimension ≦n In a subsequent paper [4] Eilenberg gave an alternative treatment of the same question. The present paper is devoted to the discussion of a number of questions suggested by the results of [4] and [5]. Among others it is shown that the conditions employed in stating the main results in [4] and [5] are equivalent, so that the main results of these two papers are in accord. Further, the cohomological dimension of a residue-algebra is studied in terms of that of the original algebra and the (module-) dimension of the associated ideal. The terminology and notation employed here are that of [3].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 8 , February 1955 , pp. 49 - 57
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1955

References

[1] Azumaya, G. and Nakayama, T., On absolutely uni-serial algebras, Jap. J. Math. 19 (1948), 263–273.Google Scholar
[2] Brauer, R. and Nesbitt, C., On the regular representations of algebras, Proc. Nat. Acad. Sci. U.S.A. 23 (1937), 236–240.Google Scholar
[3] Cartan, H. and Eilenberg, S., Homological Algebra, Princeton University Press (1954).Google Scholar
[4] Eilenberg, S., Algebras of cohomologically finite dimension, Comment. Math. Helv. (to appear).Google Scholar
[5] Ikeda, M., Nagao, H. and Nakayama, T., Algebras with vanishing n-cohomology groups, Nagoya Math. 7 (1954),..Google Scholar
[6] Nagao, H. and Nakayama, T. On the structure of (M0)- and (Mu )-modules, Math. Zeit 59 (1953), 164–170.Google Scholar