Residual intersections and linear powers

Abstract

If I I is an ideal in a Gorenstein ring S S , and S / I S/I is Cohen-Macaulay, then the same is true for any linked ideal I ′ I’ ; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal L n L_{n} of minors of a generic 2 × n 2 \times n matrix when n > 3 n>3 . In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of I I . For example, suppose that K K is the residual intersection of L n L_{n} by 2 n − 4 2n-4 general quadratic forms in L n L_{n} . In this situation we analyze S / K S/K and show that I n − 3 ( S / K ) I^{n-3}(S/K) is a self-dual maximal Cohen-Macaulay S / K S/K -module with linear free resolution over S S . The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented.