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On ideals of finite homoloǵical dimension in local rings

Published online by Cambridge University Press:  24 October 2008

Lindsay Burch
Lindsay Burch
Affiliation:
Department of Mathematics, University of Dundee
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Extract

In this paper I shall demonstrate certain algebraic properties of ideals of finite homological dimension in local rings.

In the first section, I show that no non-zero ideal of finite homological dimension in a local ring can be of zero grade (this is stated by Auslander and Buchsbaum in the appendix to (l), but I cannot find a proof in the literature). Using this result, together with the complex defined by a matrix, which is described by Eagon and Northcott in (2), I prove that an ideal of homological dimension one in a local ring Q may always, for some integer n, be described as the set of determinants of matrices obtained by adjoining to a certain (n–l)× n matrix with elements in the maximal ideal of Q another row with elements arbitrarily chosen in Q. (This result was established in (3), under the additional condition that Q should be a domain.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 4 , October 1968 , pp. 941 - 948
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

REFERENCES

(1)Auslander, M. and Buchsbaum, D. A.Homological dimension in local rings. Trans. Amer. Math. Soc. 85 (1957), 390–405.CrossRefGoogle Scholar
(2)Eagon, J. A. and Northcott, D. G.Ideals defined by matrices and a certain complex associated with them. Proc. Roy. Soc. Ser. A 269 (1962), 188–204.Google Scholar
(3)Burch, A. L.A note on ideals of homological dimension one in local rings. Proc. Cambridge Philos. Soc. 63 (1967), 661–2.CrossRefGoogle Scholar
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