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A Note on the Dubois-Efroymson Dimension Theorem

Published online by Cambridge University Press:  20 November 2018

Wojciech Kucharz
Wojciech Kucharz*
Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131, USA
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Abstract

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Let X ⊂ Rnn be an irreducible nonsingular algebraic set and let Z be an algebraic subset of X with dim Z ≦ dim X — 2. In this paper it is shown that there exists an irreducible algebraic subset Y of X satisfying the following conditions: dim Y = dim X — 1, Z ⊂ Y and that the ideal of regular functions on X vanishing on Y is principal.

Keywords

14G3014J99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 1 , 01 March 1989 , pp. 24 - 29
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Akbulut, S., King, H., The topology of real algebraic sets, L'Enseignement Math. 29 (1983), pp. 221–261. Google Scholar
2. Submanifolds and homology of nonsingular real algebraic varieties, Amer. J. Math. 107 (1985), pp. 45–83. Google Scholar
3. Bochnak, J., Kucharz, W., Shiota, M., The divisor class groups of some rings of global real analytic, Nash or rational regular functions, Lect. Notes in Math. 954, Springer 1981, pp. 218–248.Google Scholar
4. Bochnak, J., Coste, M., Roy, M. F., Géométrie algébrique réelle, Ergebnisse der Math., Vol. 12, 1987, Springer.Google Scholar
5. Dubois, D., Efroymson, G., A dimension theorem for real primes, Can. J. Math. 25 (1974), pp. 108–114. Google Scholar
6. Hironaka, H., Resolution of singularities of an algebraic variety over afield of characteristic zero, Ann. of Math. 79 (1964), I: pp. 109-203, II: pp. 205–326. Google Scholar
7. Mohan Kumar, N., Pavaman Murthy, M., Algebraic cycles and vector bundles over affine three-folds, Ann. of Math. 116 (1982), pp. 579–591. Google Scholar
8. Silhol, R., Géométrie algébrique sur un corps non algébriquement close, Comm. Alg. 6 (1978), pp. 1131–1155. Google Scholar
9. Swan, R. G., A cancellation theorem for projective modules in the metastable range, Invent. Math. 27(1974), pp. 23–43.Google Scholar
10. , Topological examples of projective modules, Trans. Amer. Math. Soc. 230 (1977), pp. 201–234. Google Scholar
11. Tognoli, A., Algebraic geometry and Nash functions, Inst. Math. Vol. III , Acad. Press London and New York 1978.Google Scholar