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By Hiroaki Hijikata

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

Example text

X d ( G Ap) are contained in a pAp-primary ideal b such that, for every system of parameters a i , . ,α»)Αρ : a¿+i = ( α ϊ , . . , ai)Ap : b holds for 0 < ¿ < d (cf. 18]). For every prime ideal p 2 H\D*)p we have =0 for ¿ > 0 as Ap is Cohen-Macaulay. Hence there is a positive integer t such that PHI^{A) = 0 for i < d. Let p and q be prime ideals such that p 2 ^ and q 2 / + p. Since Ap is Cohen-Macaulay and height q > d, depth Ap + dim Aq/pAq = dim Ap -f dim Aq/pAq = dim Aq > d. Hence there is a positive integer u such that Ι^Η}{Α) = 0 for i < d by virtue of [6].

Hartshorne, Residues and duality, Lect. Notes in Math. 20, Springer Verlag, 1966. J. Herzog, Ε. Kunz et al.. Der kanonische Modul eines Cohen-MacaulayRings, Lect. Notes in Math. 238, Springer Verlag, 1971. P. Schenzel, Standard systems of parameters and their blowing-up rings, J. Reine Angew. , 344 (1983), 201-220. R. Y. Sharp, Dualizing complexes for commutative Noetherian rings. Math. Proc. Cambridge Philos. Soc, 78 (1975), 369-386. R. Y. Sharp, Necessary conditions for the existence of dualizing complexes in commutative algebra, Sém.

Li s = t, the proof is aheady done. Let s < t. Put a = qi Π · · · Π q^ and b = q^^i Π · · · Π qt. Let p be a prime ideal with dim A / p > 1. Then Ap is Cohen-Macaulay, especially Ass(Ap) = Assh(Ap). Hence we have p 2 ^ 4- b and dim A / ( a + b) < 1. If p D a, {A/a)p ^ Ap is Cohen-Macaulay. If p D b, (A/b)p ^ Ap is CohenMacaulay. Therefore dimnonCM(A/a) < 1 and dimnonCM(A/b) < 1. Since Ass(A/a) = Assh(A/a). A / a is a homomorphic image of a Gorenstein local ring R. Since dim A / b < d, A / b is a homomorphic image of a Gorenstein local ring S by the induction h3φothesis.

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