Introduction.

Let 1/4x/, XfXf, and vjx\ be matrices of indeterminates over a commutative

noetherian ring Ro, and let H(f) be the ideal I\(uX) -f I\{Xv) + I\(vu — Adj X) of

the polynomial ring R = /?o [{1^,1^,0;^ | 1 i, j /}]. Vasconcelos observed that

on numerous occasions, some specialization of H(/) is the defining ideal for the

symbolic square algebra A[Pt,

P^H2}

of the prime ideal P in the commutative ring

A. He conjectured [19] that H(f) is a perfect prime Gorenstein ideal of grade 2/. In

[16], we found the minimal homogeneous resolution of R/H(f) by free R—modules;

thereby establishing Vasconcelos' conjecture. This resolution is obtained by merging

four Koszul complexes:

F(l) — F(2)

(*)

t 1

F(3) — F(4),

where F(l) and F(4) are both Koszul complexes on the entries of [u v], F(2) is the

Koszul complex on the entries of [uX v], and F(3) is the Koszul complex on the

entries of [u Xv]. The arrows in (*) represent maps given by the various minors

of X.

In the present paper, we consider the next natural question, which is, "What

happens when the matrix X is not square?" In this case, the corresponding ideal,

K, is equal to Ii(uX) + h(Xv) -f Ij(X), where X is an g x / matrix, with f g,

v is an / x 1 matrix, and u is a 1 x g matrix. In other words, K is the ideal which

defines the variety of complexes

Q-+R-Rf

-+RF-+R,

where the middle map has rank less than / . It quickly becomes clear that the best

way to resolve R/K is to produce a family of complexes which resolves "half of

the divisor class group of R/K.

Two distinct starting points give rise to a family of complexes with similar, and

very pretty, properties. The first starting point is the theory of residual intersec-

tions. Let / be a grade two perfect ideal, or a grade three Gorenstein ideal, or a

grade g complete intersection, and let

Rn - ^ R? A R — R/I _• 0

be exact. Assume that the ring R is the polynomial ring k[P, X], where A : is a field,

X is a g x / generic matrix, and P is as generic as possible. Given this data with

grade/ / , let K be the /—residual intersection Ii(aX):/, p be the map

p = [P X]:E =

Rn®Rf

-^G =

R9,

and m and s be the integers m = / + 1 -grade / and s = / . Then, there is a family

of complexes {C^} which satisfies the following properties.

(a) The complex

C(0)

resolves R/K.

(b) The divisor class group of R/K is the infinite cyclic group Z[coker/?].

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