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**Sample text**

O^s) 2 — • # ( S , Z ) —+ 2 H (S,Q) 442 K . NISHIGUCHI 2 and the image of L in H (S, Q ) is called the numerical class of L. , the objects on both hand sides 2 provide the same element in H (S, Q ) . A main purpose of this section is to prove the following result concerning the numerical class of the canonical bundle Κ of the surface 5. 1. Let S and Ci 's be as above. Express the numerical class of the canonical bundle Κ of S as Ν Κ = ^ a{Ci with ai G Q. Then we have ai < 0 (1 < i < N). Furthermore, no curve Ci with ai = 0 meets curves Cj with aj < 0.

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.

Let Β χ be the xBn-adic completion of Bn. *), q* Π Β φ ( 0 ) . Proof. , there exist a finite algebraic extension field L of Q(A) and the integral closure Β of A such that CL —/ q* ~ \ " a 9n — minimal prime in Sing(#*) Ί for some η G M a x ( £ ) such that q*n Π Β = (0) J is non-empty. a n Thus, one can find LQ, BO, n 0 G Max(2? 0) d q£0 G &L0 such that ht q*o = ho is minimal. 1) Ap is excellent if ht p < h0. 1). Being L a finite algebraic extension field of Q(A), let Β be the integral closure of A in L.