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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
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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.

Information

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

Footnotes

Written with the support of the National Science Foundation.

References

1. Auslander, M. and Buchsbaum, D., Homologuai dimension in local rings, Trans. Amer. Math. Soc, 85 (1957), 390-405.Google Scholar
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