Abstract:
Let & be a n x m matrix with entries in a noetherian ring R and let A’ be the
submatrix of JY consisting of the first r columns (r < n - 1). Consider the ideal
J,(A) of n x n minors of J! involving the columns of J/Z’. We obtain the primary
decomposition and the homological dimension of J,(d) in the generic case. The
proofs rely heavily on the methods and the theory of weak d-sequences and
straightening laws. As a byproduct we obtain exact conditions under which Jr(d)
is generated by a d-sequence and also a complete picture of the blowing-up algebras
of J,(d) in that case. The latter proofs rely on recent methods developed by several
authors such as those of sliding-depth, approximation complexes, Cohen-Macaulay
residual intersections. To close the discussion we construct a free resolution of
J,(d) when m=n+ 1 (the case r =n- 1 had been treated before by the present
authors). A side curiosity herein obtained is an example of a nonperfect radical
3-generated ideal of codimension 2 whose associated graded ring is a
Cohen-Macaulay reduced ring that is not Gorenstein. Examples of this sort do not
seem to abound.