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

Example text

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.

EA{N) denotes the injective envelope of Ν and Hl{N) is the z-th local cohomology module of Ν with respect to a. , Α(Α,α) = 0 α - ^ Α [ α Τ ] ς Α [ Γ ] n>0 with an indeterminate T. We put V(a) = {pe Spec(A) | ρ 3 a}, CM(A) = {p G Spec(A) | Ap is Cohen-Macaulay} and nonCM(A) = Spec(A) \ CM(A). We denote by the maximal-ideaUadic completion over a local ring. Defínition 2 . 1 . Lett be an integer. A finitely generated module Μ over a nng A is said to be [St) if depthMp > mm{t, dimMp} for every p G Supp^(M).

Hall, Fundamental dualizing complexes for commutative noetherian rings, Quart. J. Math. (2), 3 0 (1979), 21-32. R. 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.

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