On ideals of minors fixing a submatrix

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.