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,
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
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 =
-^G =
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
resolves R/K.
(b) The divisor class group of R/K is the infinite cyclic group Z[coker/?].
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