Ideals and Higher Derivations in Commutative Rings

Author:

Brown William C.,Kuan Wei-Eihn

Abstract

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.

Publisher

Canadian Mathematical Society

Subject

General Mathematics

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