The resolution of the generic residual intersection of a complete intersection

Abstract

The concept of residual intersection, introduced by Artin and Nagata [l] in 1972, is a fruitful generalization of linkage as the following two examples attest. Let I be a strongly Cohen-Macaulay ideal in a CohenMacaulay local ring R. If Z satisfies the condition (G,), then Huneke [6, Proposition 4.31 has proved that the extended Rees algebra R[It, t‘1 is defined by an ideal which is obtained from I by way of residual intersection. Since the extended Rees algebra is a deformation of both the Rees algebra R[It] and the associated graded algebra gZ’/I” ‘, it contains considerable information about the analytic properties of I. (The definitions of residual intersection, strongly Cohen-Macaulay, and (G,) may be found in Section 4.) Huneke [6, Theorem 4.11 has also shown that the ideal .I, generated by the maximal order minors of a generic n x m matrix, is a residual intersection of a generic codimension two Cohen-Macaulay ideal. Since J is rather poorly behaved with respect to linkage [S], it is promising that it is close to a well understood ideal once we weaken the tie from linkage to residual intersection.