Artin-Nagata Properties and Cohen-Macaulay Associated Graded Rings Compositio Mathematica, Tome 103, No 1 (1996), P
COMPOSITIO MATHEMATICA MARK JOHNSON BERND ULRICH Artin-Nagata properties and Cohen-Macaulay associated graded rings Compositio Mathematica, tome 103, no 1 (1996), p. 7-29 <http://www.numdam.org/item?id=CM_1996__103_1_7_0> © Foundation Compositio Mathematica, 1996, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 103 : 7-29, 1996. 7 © 1996 Kluwer Academic Publishers. Printed in the Netherlands. Artin-Nagata properties and Cohen-Macaulay associated graded rings MARK JOHNSON and BERND ULRICH* Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA Received 23 August 1994; accepted in final form 8 May 1995 1. Introduction Let R be a Noetherian local ring with infinite residue field k, and let I be an R-ideal. The Rees algebra R = R[It] , 0 Ii and the associated graded ring G = gri(R) = n @R RII -’- Q)i>o I’II’+’ are two graded algebras that reflect various algebraic and geometric properties of the ideal I. For instance, Proj(R) is the blow-up of Spec(R) along V(I), and Proj(G) corresponds to the exceptional fibre of the blow-up. One is particularly interested in when the ’blow-up algebras’ R and G are Cohen-Macaulay or Gorenstein: Besides being important in its own right, either property greatly facilitates computing various numerical invariants of these algebras, such as the Castelnuovo-Mumford regularity, or the number and degrees of their defining equations (see, for instance, [29], [5], or Section 4 of this paper).
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