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Nishimura, Note on Krull domains, J. Math. Kyoto Univ. 15 (1975), 397-400. J. Nishimura, On ideal-adic completion of noetherian rings, J. Math. Kyoto Univ. 21 (1981), 153-169. L. J. , On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals ( I ) , Amer. J. Math. 91 (1969), 508-528. C. Rotthaus, Komplettierung semilokaler quasiausgezeichneter Ringe, Nagoya Math. J. 76 (1979), 173-180. C. Rotthaus, Zur Komplettierung ausgezeichneter Ringe, Math. Ann. 253 (1980), 213-226.

We assume that the canonical bundle Κ of S has a meromorphic section with poles and no zeros along the curves, namely Κ is written as Ν K = J2(-ai)Cl (JV>1), where C^s are distinct curves and a;'s are positive integers. 1. Let S be as above. Then S satisfies one of the following conditions: (1) All ai 's are equal to one. (ii) S is a Hopf surface. (iii) S is a (CB)-surface. Moreover, in the case (iii), R = Supp(—K) is connected and the curves in R constitute a ( C B ) . R e m a r k . (1) Surfaces of the case ( i ) are completely classified in §3.

2. Let A be a noetherian normal ring and I be an ideal of A. Suppose I = Pi Π · · · Π Pr (= the intersection of prime ideals Pi, Pr). s u c ^ that (A/xA)a is Then, there exists a non-zero χ g J and a € A — Ui=i -P* reduced. 3. As A is known to be a nagata ring (cf. Marot's Theorem), it suffices to show that A is a P-ring. 2) A / a is a P-ring for any non-zero ideal a. Let L be a finite algebraic extension field of Κ and Β be a finite A-algebra s with Q(B) — L. We are to show that L B &n * normal for any η g M a x ( B ) .

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