Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-10T19:30:54.977Z Has data issue: false hasContentIssue false
  • English
  • Français

Dimension of Ideals in Polynomial Rings

Published online by Cambridge University Press:  20 November 2018

Maurice Auslander  and
Alex Rosenberg
Maurice Auslander
Affiliation:
Institute for Advanced Study
Alex Rosenberg
Affiliation:
Northwestern University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A well-known theorem asserts that if K is a field, a prime ideal in the polynomial ring S = K[X1, … Xn] and d the transcendence degree of S / over K

n = rank + d.

In the first half of this paper we extend this result to the case of arbitrary commutative noetherian K, as well as giving a purely homological proof of the classical theorem. In the second half we use our first result to compute the analogue of the dimension of the product and intersection of two affine varieties when K is a Dedekind ring.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 10 , 1958 , pp. 287 - 293
Copyright
Copyright © Canadian Mathematical Society 1958

References

1. Auslander, M. and Buchsbaum, D., Homologuai dimension in local rings, Trans. Amer. Math. Soc, 85 (1957), 390-405.Google Scholar
2. Cartan, H. and Eilenberg, S., Homologuai Algebra(Princeton, 1956).Google Scholar
3. Eilenberg, S.,Rosenberg, A., and Zelinsky, D., On the dimension of modules and algebras (VIII), Nagoya Math. J., 12 (1957).Google Scholar
4. Nagata, M., A general theory of algebraic geometry over Dedekind domains I, Amer. J. Math., 78 (1956), 78–116.Google Scholar
5. Northcott, D. G., Ideal Theory (Cambridge, 1953).Google Scholar
6. Samuel, P., Methodes d'algebre abstraite en geometrie algebrique, Ergeb. d. Math. (Neue Folge), 4 (1955).Google Scholar