Download e-book for kindle: Algebraic Geometry and Commutative Algebra. In Honor of by Hiroaki Hikikata

By Hiroaki Hikikata

ISBN-10: 0123480329

ISBN-13: 9780123480323

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Additional resources for Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 2

Example text

Thus, we have w hi P ; T^ 5 = f | I GA 7 I Pi ^ = A 6 A «7 = FN R = N 1 tA 7 < C { 1 , . . , n}; I € A « 7- L Hence, | Λ 7 » | = | Λ 7 ι | - f · · · - f | A 7 m| < η with | A 7 j | > 0. Therefore, we find 7i E Γ(7ο) such that p * , A 7 = p 7 for any 7 ' D 7 ( d 7 1 ) . §2. P r o o f o f T h e o r e m A . 1. (cf. 1)]) If (A, m i , . . , rrv) is α semilocal normal domain, one can find a non-zero χ g Χϊΐχ... m r such that A/χ A is reduced. 2. Let A be a noetherian normal ring and I be an ideal of A.

6). Take a minimal prime ideal Ρ of χ A. As C is finite over A , we find Q G A s s ( C / x C ) such that Q Π A = P. This means P G Δ(χ) (cf. 1)). Hence 1 = ht Ρ > ht p 7 + 1 for any 7 G Γ ( 7 2 ) . Therefore, ht p* = 0. 6) is proved and this completes the proof of Rotthaus' Hilfssatz. Next proposition shows that, in some practical cases, a bounded prime ideal sequence automatically becomes good. 18. Let A be a noetherian ring and m G M a x ( A ) . Fixing an element x G A and a minimal prime ideal pj^ of 6p(A^n) ( = Spec(A^) Kr(A*m)), let p; = p*m Π A; for 7 e r ( m ) ( = r ( { m } ) ) .

So, we consider the case b2(S) > 0. 2. 1, we know that —A is a non-zero effective divisor. 1. d. Remark. Nakamura studied a surface of class VIIo satisfying the condition (e) (cf. Nakamura[9 bis]), and the results of this section might be derived from [9 bis]. §5. In this section, we review a classification and the properties of surfaces containing global spherical shells (GSS). Let 5 be a surface contaning a GSS. For the definition of GSS, we refer to Kato[4]. It is easy to see that S is a surface of class V I I .

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Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 2 by Hiroaki Hikikata


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