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.