Let ti
l x
/, X/Xy, and Vfxi be matrices of indeterminates over a com-
mutative noetherian ring RQ, and let H(f) be the ideal I\(uX) + h(Xv) -f I\(vu
AdjX) of the polynomial ring R = JRO[it, X, v]. Vasconcelos conjectured that
the ideal H(f) is a perfect Gorenstein ideal of grade 2/. In [16], we found the
minimal homogeneous resolution of RfH{f) by free R—modules; thereby estab-
lishing Vasconcelos' conjecture. The present paper considers the situation when
the matrix X is not square. In this case, the corresponding ideal, K, is equal to
I\(uX)-\-I\{Xv) + If{X), where X is an # x / matrix with / g. We have resolved
the ideal K. It is perfect of grade / + g 1; the Cohen-Macaulay type of R/K is
(JlJ); and the last twist in the minimal resolution of R/K is g -h 3 / 3. We have
also resolved half of the divisor class group of R/K.
1991 Mathematics Subject Classificdticn. 13D25.
Key words and phrases. Canonical module, Divisor class group, Koszul complex, Symmetric
algebra, Variety of complexes.
The author was supported in part by the National Science Foundation grant DMS-9322556.
Received by editor July 28, 1998.
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