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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

1. Heerema, N., Convergent higher derivations on local rings, Trans. Amer. Math. Soc. 132 (1968), 31–44.Google Scholar
2. Heerema, N., Higher derivations and automorphisms of complete local rings, Bull. Amer. Math. Soc. 76 (1970), 1212–1225.Google Scholar
3. Seidenberg, A., Derivations and integral closure, Pacific J. Math. 16 (1966), 167–173.Google Scholar
4. Seidenberg, A., Differential ideals in rings of finitely generated type, Amer. J. Math. 89 (1967), 22–42.Google Scholar
5. Zariski, O., Studies in equisingularity. I, Amer. J. Math. 87 (1965), 507–536.Google Scholar