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Ideals and Higher Derivations in Commutative Rings

Published online by Cambridge University Press:  20 November 2018

William C. Brown  and
Wei-Eihn Kuan
William C. Brown
Affiliation:
Michigan State University, East Lansing, Michigan
Wei-Eihn Kuan
Affiliation:
Michigan State University, East Lansing, Michigan
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In this paper, we wish to generalize the following lemma first proven by O. Zariski [5, Lemma 4]. Let O be a complete local ring containing the rational numbers and let m denote the maximal ideal of O. Assume there exists a derivation δ of O such that δ(x) is a unit in O for some x in m. Then O contains a ring O1 of representatives of the (complete local) ring O/Ox having the following properties: (a) δ is zero O1; (b) x is analytically independent over O1; (c) O is the power series ring O1[[x]]. In [4], A. Seidenberg used Zariski's lemma extensively to study conditions under which an affine algebraic variety V over a base field of characteristic zero is analytically a product along a given subvariety W of V. We should like to generalize Zariski's lemma by removing the condition that O contain the rationals. We could then get some conditions under which an arbitrary affine variety V would be analytically a product along a subvariety W.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 3 , June 1972 , pp. 400 - 415
Copyright
Copyright © Canadian Mathematical Society 1972

References

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