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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on the Nakai conjecture


Authors: Paulo Brumatti, Yves Lequain, Daniel Levcovitz and Aron Simis
Journal: Proc. Amer. Math. Soc. 130 (2002), 15-21
MSC (2000): Primary 13M05, 13M10, 13N15, 13B22; Secondary 14F10, 12H05, 13B25, 13F25
Published electronically: April 26, 2001
MathSciNet review: 1855614
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Abstract:

The conjectures of Zariski-Lipman and of Nakai are still open in general in the class of rings essentially of finite type over a field of characteristic zero. However, they have long been known to be true in dimension one. Here we give counterexamples to both conjectures in the class of one-dimensional pseudo-geometric local domains that contain a field of characteristic zero. Likewise, in connection with a recent result of Traves on the Nakai conjecture, we also show that their hypothesis of finite generation of the integral closure cannot be removed even in the class of local domains containing a field of characteristic zero.


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Additional Information

Paulo Brumatti
Affiliation: IMECC–UNICAMP, 13081-970 Campinas, São Paulo, Brazil
Email: brumatti@ime.unicamp.br

Yves Lequain
Affiliation: Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, J. Botânico, 22460-320, Rio de Janeiro, RJ, Brazil
Email: ylequain@impa.br

Daniel Levcovitz
Affiliation: Instituto de Ciências Matemáticas e de Computação, USP-SC, Av. Dr. Carlos Botelho, 1465, 13560-970 São Carlos, SP, Brazil
Email: lev@icmsc.sc.usp.br

Aron Simis
Affiliation: Departamento de Matemática, CCEN, Universidade Federal de Pernambuco, 50740-540 Recife, PE, Brazil
Email: aron@dmat.ufpe.br

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05983-4
PII: S 0002-9939(01)05983-4
Keywords: Derivations, integral closure, pseudo-geometric, ${\mathcal{D}}$-stable
Received by editor(s): January 11, 2000
Received by editor(s) in revised form: May 19, 2000
Published electronically: April 26, 2001
Additional Notes: The authors were partially supported by CNPq and by a grant from the group ALGA-PRONEX/MCT
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2001 American Mathematical Society