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Some Examples of Smooth and Regular Rings(1)

Published online by Cambridge University Press:  20 November 2018

Leslie G. Roberts
Leslie G. Roberts*
Affiliation:
Department of Mathematics Queen′s University, Kingston, Ontario, K7L 3N6
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In this note I investigate smoothness and regularity of the ring the ring of integers. These results do not seem to be well known, especially those dealing with regularity. At any rate, several of my colleagues knew that was not smooth, but thought that it might be regular. I was surprised by the number of possibilities that can occur, as well as by the fact that A can be regular but not smooth. As expected the prime 2 plays a special role. Smoothness depends on a, b, c mod 2, and regularity depends on a, b, c mod 4.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 23 , Issue 3 , 01 September 1980 , pp. 255 - 259
Copyright
Copyright © Canadian Mathematical Society 1980

Footnotes

(1)

This research was supported in part by NRC grant A-7209.

References

[1] Arf, C., Untersuchungen uber quadratische Formen in Korpem der Charakteristik 2 (Teil 2), J. fur reine und angew Mathv 183 (1940) 148-167.Google Scholar
[2] Matsumura, H., Commutative Algebra, Benjamin, New York, 1970.Google Scholar
[3] Quillen, D., Higher Algebraic K-theory I, pp. 179-198, Lecture Notes in Mathematics, No. 341, Springer-Verlag, New York, 1973.Google Scholar
[4] Swift, R., Smooth Algebras, M.Sc. Thesis, Queen's University, Kingston, Ontario, 1978.Google Scholar
[5] Zariski, O. and Samuel, P., Commutative Algebra Volume II, Van Nostrand, Princeton, 1958.Google Scholar