Ideals Of Maximal Minors Research Articles

Local cohomology with support in ideals of maximal minors

Suppose that k is a field of characteristic zero, X is an r×s matrix of indeterminates, where r≤s, and R=k[X] is the polynomial ring over k in the entries of X. We study the local cohomology modules HIi(R), where I is the ideal of R generated by the maximal minors of X. We identify the indices i for which these modules vanish, compute HIi(R) at the highest nonvanishing index, i=r(s−r)+1, and characterize all nonzero ones as submodules of certain indecomposable injective modules. These results are consequences of more general theorems regarding linearly reductive groups acting on local cohomology modules of polynomial rings.

Free resolutions and sparse determinantal ideals

A sparse generic matrix is a matrix whose entries are distinct variables and zeros. Such matrices were studied by Giusti and Merle who computed some invariants of their ideals of maximal minors. In this paper we extend these results by computing a minimal free resolution for all such sparse determinantal ideals. We do so by introducing a technique for pruning minimal free resolutions when a subset of the variables is set to zero. Our technique correctly computes a minimal free resolution in two cases of interest: resolutions of monomial ideals, and ideals resolved by the Eagon-Northcott Complex. As a consequence we can show that sparse determinantal ideals have a linear resolution over the integers, and that the projective dimension depends only on the number of columns of the matrix which are identically zero. Finally, we show that all such ideals have the property that regardless of the term order chosen, the Betti numbers of the ideal and its initial ideal are the same. In particular the nonzero generators of these ideals form a universal Grobner basis.

Blow-up of Straightening-Closed Ideals in Ordinal Hodge Algebras

We study a class of ideals $I$ in graded ordinal Hodge algebras $A$. These ideals are distinguished by the fact that their powers have a canonical standard basis. This leads to Hodge algebra structures on the Rees ring and the associated graded ring. Furthermore, from a natural standard filtration one obtains a depth bound for $A/{I^n}$ which, under certain conditions, is sharp for $n$ large. Frequently one observes that ${I^n}= {I^{(n)}}$. Under suitable hypotheses it is possible to calculate the divisor class group of the Rees algebra. Our main examples are ideals of "virtual" maximal minors and ideals of maximal minors "fixing a submatrix".

Blow-up of straightening-closed ideals in ordinal Hodge algebras

We study a class of ideals I I in graded ordinal Hodge algebras A A . These ideals are distinguished by the fact that their powers have a canonical standard basis. This leads to Hodge algebra structures on the Rees ring and the associated graded ring. Furthermore, from a natural standard filtration one obtains a depth bound for A / I n A/{I^n} which, under certain conditions, is sharp for n n large. Frequently one observes that I n = I ( n ) {I^n}= {I^{(n)}} . Under suitable hypotheses it is possible to calculate the divisor class group of the Rees algebra. Our main examples are ideals of "virtual" maximal minors and ideals of maximal minors "fixing a submatrix".