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If$I$is an ideal in a Gorenstein ring$S$, and$S/I$is Cohen-Macaulay, then the same is true for any linked ideal$I’$; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal$L_{n}$of minors of a generic$2 \times n$matrix when$n>3$.

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$I$. For example, suppose that$K$is the residual intersection of$L_{n}$by$2n-4$general quadratic forms in$L_{n}$. In this situation we analyze$S/K$and show that$I^{n-3}(S/K)$is a self-dual maximal Cohen-Macaulay$S/K$-module with linear free resolution over$S$.

The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented.