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On ideals of finite homoloǵical dimension in local rings

  • Lindsay Burch (a1)
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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.)

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References
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(1)Auslander, M. and Buchsbaum, D. A.Homological dimension in local rings. Trans. Amer. Math. Soc. 85 (1957), 390405.
(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), 188204.
(3)Burch, A. L.A note on ideals of homological dimension one in local rings. Proc. Cambridge Philos. Soc. 63 (1967), 661–2.
(4)Northcott, D. G. and Rees, D.Reductions of ideals in local rings. Proc. Cambridge Philos. Soc. 50 (1954), 145158.
(5)Rees, D.A theorem of homological algebra. Proc. Cambridge Philos. Soc. 52 (1956), 605616.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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