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By Hiroaki Hijikata

ISBN-10: 0123480310

ISBN-13: 9780123480316

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Additional info for Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 1

Example text

1) These four polynomials have no common zero λ : μ. 2) For λι : /ii λ2 : /i2 ^ P i the matrix ( Μ{λ\-2μ\) \ λ2(λ^-2μ^) μι{2\\^μ\) μ2(2\\^μ\) \\μ\ \\μ\ \\μ\ \\μ\ \ ) has rank two. 3) For all (λ, μ) φ (0,0) the matñx of derivatives {1/2)Θχ: ( 3λ' ( i m : V - 5λ/χ4 Χ^Λ-^μ^ AV 2AV^ has rank two. § 2 . T h e sextic s p a c e c u r v e 5 . 3). So 5 C P3 is a smooth rational curve of degree six. ( 2 . , three points on a trisecant L{s,t) to S. 48 W . B A R T H and R . M O O R E Proof. Ρ 3 ( λ , μ ; 5 , 0 = 0 impHes P | = d^Pi = 0.

We put U = xl^^T, S = R[l/U], Β = So and Ü = ΠΒ (D mJ5). Since S = B[U, l/U] and U is algebraically independent over B, S^s is Cohen-Macaulay if and only if so is B^. The Cohen-Macaulayness of B^ follows from [7, Satz 3] because the elements a : i , . . , Xd/xi, y/xl] is the coordinate ring of an afiine chart of Y, the Cohen-Macaulay scheme given in [7, Satz 3](cf. 190]). In the case where p ^ yx[T, the proof is the same as the above (with Β = A[x2/xi,... ,Xd/xi,Xi/y] ) . 1 not only for a local ring A with Ass(A) = Assh(A) but also for a semi-local ring which satisfies the condition (a).

3, §1]). 2 can be proved by a similar method to one given in [3, §3]. Throughout this paper a nng means a commutative noetherian ring with unit. §2. S h a r p ' s C o n j e c t u r e , Let A be a ring. For a finitely generated A-module Μ of finite dimension, we put AsshA(M) = {pe ASSA(M) \ dim A/p = dim M } . *The authors were patially supported by Grant-in-Aid for Co-operative Reserch. Received January 30, 1987. 28 Y , ΑοΥΑΜΑ and S. G O T O Let α be an ideal of A and Ν an A-module. EA{N) denotes the injective envelope of Ν and Hl{N) is the z-th local cohomology module of Ν with respect to a.

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Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 1 by Hiroaki Hijikata

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