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Examples of bad noetherian local rings

Published online by Cambridge University Press:  22 January 2016

Maria Grazia Marinari
Maria Grazia Marinari*
Affiliation:
Istituto Matematico-Universita di Genova
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In this paper we use a general technique, due to Larfeldt-Lech [9], to show that there exist local noetherian rings which are “bad” with respect to some properties which are obviously verified in the algebro-geometric case. In particular we show that there exist local Gorenstein rings which are not homomorphic images of regular (local) rings and that there exist local rings which do have canonical module but do not have canonical algebra.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 70 , July 1978 , pp. 105 - 110
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1978

References

[1] Bass, H.: Injective dimension in noetherian rings. Trans. A.M.S. 102, 1962.CrossRefGoogle Scholar
[2] Dieudonne, J.: Topics in local algebra. Notre Dame Math. Lect. 10, 1967.Google Scholar
[3] Fossum, R., Foxby, H. B., Griffith, P. and Reiten, I.: Minimal injective resolution with application to dualizing modules and Gorenstein modules. Publ. Math. I.H.E.S. 45, 1975.CrossRefGoogle Scholar
[4] Greco, S. and Salmon, P.: Topics in m-adic topologies. Erg. der Math. Springer Verlag 58, 1971.Google Scholar
[5] Gulliksen, T. H. and Levin, G.: Homology of local rings. Queen’s Papers 20, 1969.Google Scholar
[6] Herzog, J.: Komplexe, Auflösungen und Dualität in der lokalen Algebra. Habilitationsschrift (unpublished), Universität Regensburg, Regensburg, Germany.Google Scholar
[7] Herzog, J. and Kunz, E.: Der kanonische Modul eines C.M. Rings. Lect. Notes in Math. 238, 1971.CrossRefGoogle Scholar
[8] Kunz, E.: Almost complete intersections are not Gorenstein rings. J. of Algebra 28, 1974.CrossRefGoogle Scholar
[9] Larfeldt, L. and Lech, C.: On flat couples (to appear).Google Scholar
[10] Reiten, I.: The converse to a theorem of Sharp on Gorenstein modules. Proc. A.M.S. 32, 1972.CrossRefGoogle Scholar
[11] Scheja, G.: Über die Bettizahlen lokaler Ringe. Math. Ann. 155, 1964.CrossRefGoogle Scholar
[12] Sharp, R. Y.: Gorenstein modules. Math. Z. 115, 1970.CrossRefGoogle Scholar
[13] Watanabe, K., Ishikawa, T., Tachibana, S. and Otsuka, K.: On tensor product of Gorenstein rings. J. Math. Kyoto Univ. 9, 1969.Google Scholar