By Hiroaki Hikikata
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Extra resources for Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 2
2). Let U = C - [fi=1 Qi and Vi be the valuation associated l to the D V R C~ . We may assume Q = Qi. Since C\j (= U~ C) is a semi-local Dedekind domain, one can find dx = ci/u G Cu with ci G C and u e U such 460 J. N i S H i M U R A and T. NISHIMURA that v i ( c i ) = ei - 1 and Vj(C[) > e, (2 < j < t). Taking C 2 G H Q ' - ( j ! =i U with Q' G V ( x ) - A s s ( C / x C ) , let c = C\ -C2 for sufficiently large v. Then, being i = x/c, we have £CnC = Π(ζΒΊηΒΊ) D Q (cf. 5)). Thus, £CnC = Q. Therefore, our proof of Lemma in  brings the conclusion.
On Weierstrass Models 431 Let D be the image by τ' of the proper transform o(FxB" by φ. Then e(D) > 0. 4). This is a contradiction. d. 1) continued. Take λ : Υ' —• Y so that degA is as small as possible. 10], λ is étale outside of the closed subset Ζ of Y with codimy Ζ > 2. 1). d. References [Κ] Y . Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. für die reine und angew. Math. 363 (1985), 1-46. [V] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fiber spaces, in Algebraic and Analytic Varieties, (S.
Then A is also quasi-excellent. Ideal-adic Completion of Noetherian Rings II 455 Notation. In this note, all rings are commutative with 1. Standard notation and terminology are those of Bourbaki, EGA, Matsumura and Nagata. We also use the same terminology as in  except: a universally Japanese ring is called here a nagata ring. , — For a local ring (A, m), we say P ( A ) is true ("P(A) holds", or simply P(>1)), when A have the property P . 1. Let Ay Β be noetherian rings. 2) P((B ®A k')P>) is true for any p G Spec(^l) and any P' G Spec(ß <8U k') of any finite extension field k' over κ(ρ).