JOURNAL OF ALGEBRA 74, 466-493 (1982) Approximation Complexes of Blowing-Up Rings J. HERZOG Fachbereieh Mathematik a’er Gesamthochschule Essen, Universitritsstrasse 3, D-4300 Essen, West Germany A. SIMS Instituto de Matemdtica Pura e Aplicada, Estrada D. Castorina 110, Rio de Janeiro 20.460, Brazil AND W. V. VASCONCELOS Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903 Communicated by D. Buchsbaum Received January 22, 1981 0. INTRODUCTION The blowing-up rings we consider here are the symmetric and Rees algebras of ideals, and some of their fibers. To recall the meaning of these rings, let R be a commutative Noetherian ring and let Z be an ideal. Then the Rees algebra of Z is the form ring R(Z) = @ Z’ I>0 that occurs in the process of blowing-up the variety associated to A along the subvariety defined by I. The symmetric algebra of I, Sym(Z), also represents a blowing-up, but of a much looser structure; it has advantages over R(Z), however, in that properties of the ideal Z are more readily reflected on its arithmetic properties. There is a canonical surjection a: Sym(Z) + R(Z) which we, heuristically, view as an approximation, whenever much is known 466 OOZI-8693/82/020466-28$02.00/O Copyright @I 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.