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On the Dimension of Modules and Algebras (III): Global Dimension1)

Published online by Cambridge University Press:  22 January 2016

Maurice Auslander
Maurice Auslander*
Affiliation:
University of Michigan
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Let Λ be a ring with unit. If A is a left Λ-module, the dimension of A (notation: 1.dimΛA) is defined to be the least integer n for which there exists an exact sequence

0 → Xn → … → X0 → A → 0

where the left Λ-modules X0, …, Xn are projective. If no such sequence exists for any n, then 1. dimAA = ∞. The left global dimension of Λ is

1. gl. dim Λ = sup 1. dimAA

where A ranges over all left Λ-modules, The condition 1. dimA A < n is equivalent with (A, C) = 0 for all left Λ-modules C. The condition 1.gl. dim Λ < n is equivalent with = 0. Similar definitions and theorems hold for right Λ-modules.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 9 , October 1955 , pp. 67 - 77
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1955

Footnotes

1)

Part of the work contained in this paper was done while the author was at the University of Chicago.

References

[1] Cartan, H. and Eilenberg, S., Homological Algebra, Princeton University Press, 1955.Google Scholar
[2] Eilenberg, S., Algebras of cohomologically finite dimension, Comment. Math. Helv., 28 (1954), 310–319.Google Scholar
[3] Jacobson, N., Theory of Rings, Amer. Math. Soc., 1943.Google Scholar
[4] Nakayama, T., On Frobeniusean Algebras II, Ann. of Math., 42 (1941), 1–21.Google Scholar