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Literatur
Lemma 1 on page 261 in:K. Oka, “Sur les fonctions analytique de plusieurs variables VIII”, J. Math. Soc. (Japan)3, (1951). In the proof of this lemma, Oka uses H. Cartan's theorem on three annuli.
After this paper was written, it was brought to my attention thatW. Thimm has, in the following two recent papers in these Annalen, given a proof of this generalization of Oka's lemma by an entirely different method than ours: „Über Moduln und Ideale von holomorphen Funktionen mehrerer Variablen“, vol.139, pp. 1–13 (1959); and „Untersuchungen über das Spurproblem von holomorphen Funktionen auf analytischen Mengen“, vol.139, pp. 95–114 (1959).
dimdft = dimensiondefect. This is Krull's original term and is sometimes called the “rank”. We prefer to use Krull's original term partly because we wish to reserve the term “rank” for a different concept (§ 3) which is related to the usual notion of “Jacobian rank” (see proof of 9.3).
For basic properties of quotient rings which we shall tacitly use, we refer to Chapter II in:D. G. Northcott, “Ideal Theory”. Cambridge 1953.
For (4.2) see Theorem 21 on page 99 in:I. S. Cohen, “On the structure and ideal theory of complete local rings”, Trans. Amer. Math. Soc.59, (1946); and also see Remark (13.14) in our Appendix. For (4.3) see Theorem 11 in:W. Krull, „Dimensions-theorie in Stellenringe“, Crelle J.179, (1938). For (4.4) and (4.5) see respectively Theorem 7 on page 60 and Theorem 8 on page 76 in:D. G. Northcott, “Ideal Theory”. Cambridge 1953. (4.1) has been proved for an arbitrary regular local ringR byAuslander, Buchsbaum, andSerre, by using homological algebra; see Theorem 1.11 on page 396 in:M. Auslander andD. A. Buchsbaum, “Homological dimension in local rings”, Trans. Amer. Math. Soc.85 (1957). In the special case of (formal or convergent) power series rings over a (respectively: abstract or complete nondiscrete valued) perfect infinite field, we shall, in our Appendix (§ 13), give a direct elementary proof (13.13). Note that (4.1) will not be used until § 8.
Due toSerre.
See § 1 in:J. P. Serre, «Géométrie algébrique et géométrie analytique», Ann. Inst. Four., VI, (1956).
(5.1, 5.2, 5.2a, 5.2b) readily follow from the results in:R. Remmert andK. Stein, “Über die wesentlichen Singularitäten analytischer Mengen”, Math. Ann.126, 263–306 (1953). (5.2c, 5.2d, 5.4) are inH. Cartan, «Séminaire», 1951–52, 1953–54. We shall not use (5.2c, 5.2d) until § 9 where in (9.4) they will be deduced as corollaries of (9.3).
The proof can be simplified by at once asserting equality here by invoking (8.2). However, we wanted to make § 12 independent of § 8.
Theorem 1 on page 29 in:M. Nagata, “On the closedness of singular loci”, Inst. des Hautes Études Scientifiques, Publications Mathematique, No. 2 (1959).
Zariski, O.: “The concept of a simple point of an abstract algebraic variety”, Trans. Amer. Math. Soc.62, pp. 1–52 (1947).
In this connection reference should be made to Theorem 3 on page 363 in:A. Seidenberg: “The hyperplane sections of normal varieties”. Trans. Amer. Math. Soc.69, (1950).
§ 1 in:I. S. Cohen andA. Seidenberg: “Prime ideals and integral dependence”. Bull. Amer. Math. Soc.52, pp. 252–261 (1946).
HereZ denotes an indeterminate.
Lemma 2 on page 32 in:C. Chevalley: “Intersections of algebraic and algebroid varieties”. Trans. Amer. Math. Soc.57 (1945).
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This work was supported in part by an Alfred P. Sloan Foundation Research Fellowship.
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Abhyankar, S. Concepts of order and rank on a complex space, and a condition for normality. Math. Ann. 141, 171–192 (1960). https://doi.org/10.1007/BF01360171
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DOI: https://doi.org/10.1007/BF01360171