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/?]. 1
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