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A Note on Gorenstein Rings of Embedding Codimension Three

Published online by Cambridge University Press:  22 January 2016

Junzo Watanabe
Junzo Watanabe*
Affiliation:
Nagoya University
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Let A = R/, where R is a regular local ring of arbitrary dimension and is an ideal of R. If A is a Gorenstein ring and if height = 2, it is easily proved that A is a complete intersection, i.e., is generated by two elements (Serre [5], Proposition 3). Hence Gorenstein rings which are not complete intersections are of embedding codimension at least three. An example of these rings is found in Bass’ paper [1] (p. 29). This is obtained as a quotient of a three dimensional regular local ring by an ideal which is generated by five elements, i.e., generated by a regular sequence plus two more elements. In this paper, suggested by this example, we prove that if A is a Gorenstein ring and if height = 3, then is minimally generated by an odd number of elements. If A has a greater codimension, presumably there is no such restriction on the minimal number of generators for , as will be conceived from the proof.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 50 , June 1973 , pp. 227 - 232
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1973

References

[1] Bass, H., Injective dimension in Noetherian rings, Trans. A.M.S., 102 (1962).Google Scholar
[2] Bass, H., On the ubiquity of Gorenstein rings, Math. Zeit., 82 (1963).Google Scholar
[3] Herzog, H. and Kunz, E., Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture notes in Math., 238, Springer (1971).Google Scholar
[4] Matlis, E., Injective modules over Noetherian rings, Pacific J. of Math., 8 (1958).CrossRefGoogle Scholar
[5] Serre, J.-P., Sur les modules projectifs, Seminaire Dubreil, 1960/61.Google Scholar