Geodesic
Ideals and Higher Derivations in Commutative Rings
Canadian journal of mathematics, Tome 24 (1972) no. 3, pp. 400-415

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Brown, William C.; Kuan, Wei-Eihn. Ideals and Higher Derivations in Commutative Rings. Canadian journal of mathematics, Tome 24 (1972) no. 3, pp. 400-415. doi: 10.4153/CJM-1972-033-1
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[3] 3. Seidenberg, A., Derivations and integral closure, Pacific J. Math. 16 (1966), 167–173. Google Scholar

[4] 4. Seidenberg, A., Differential ideals in rings of finitely generated type, Amer. J. Math. 89 (1967), 22–42. Google Scholar

[5] 5. Zariski, O., Studies in equisingularity. I, Amer. J. Math. 87 (1965), 507–536. Google Scholar

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