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Rings of weak dimension one and syzygetic ideals

Author(s): Francesc Planas-Vilanova
Journal: Proc. Amer. Math. Soc. 124 (1996), 3015-3017.
MSC (1991): Primary 13F05, 13A30; Secondary 13D03
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Abstract: We prove that rings of weak dimension one are the rings with all (three-generated) ideals syzygetic. This leads to a characterization of these rings in terms of the André-Quillen homology.

References:

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M. André: Homologie des algèbres commutatives. Grundlehren 206. Heidelberg: Springer 1974. MR 50:4707

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J. Barja, A. G. Rodicio: Syzygetic Ideals, Regular sequences and a Question of Simis. J. Algebra 121 (1989), 310-314. MR 90b:13017

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N. Bourbaki: Algèbre. Chapitre 10. Algèbre homologique. Masson. Paris, 1980. MR 82j:18022

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D. Costa: On the Torsion-Freeness of the Symmetric Powers of an Ideal. J. Algebra 80 (1983), 152-158. MR 85e:13019

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A. Micali, N. Roby: Algèbres symétriques et syzygies. J. Algebra 17 (1971), 460-469. MR 44:198

[P]
F. Planas Vilanova: Ideals de tipus lineal i homologia d'André-Quillen. Ph.D. Thesis. Universitat de Barcelona, 1994.

[R]
J. Rotman: An Introduction to Homological Algebra. Academic Press, 1979. MR 80k:18011

[V]
W. V. Vasconcelos: Arithmetic of Blowup Algebras. LMS Lecture Note Series 195. Cambridge University Press, 1994. MR 95g:13005

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Additional Information:

Francesc Planas-Vilanova
Affiliation: Departament de Matemàtica Aplicada I, ETSEIB, Universitat Politècnica de Catalunya, Diagonal 647, E-08028 Barcelona, Spain
Email: planas@ma1.upc.es

DOI: 10.1090/S0002-9939-96-03416-8
PII: S 0002-9939(96)03416-8
Keywords: Ideal of linear type, syzygetic ideal, weak dimension, homology of commutative rings
Received by editor(s): November 4, 1994
Received by editor(s) in revised form: April 18, 1995
Communicated by: Wolmer V. Vasconcelos
Copyright of article: Copyright 1996, American Mathematical Society

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