Free Resolutions and Sparse Determinantal Ideals
By Adam Boocher. 2011
61 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/5f9786c0236a71a7e2f7d3c9b41453aa3e149e6a
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In Algebra and Number Theory
In Computational Theory and Mathematics
Citations in the corpus
- Universal Groebner bases for maximal minors (2013, background): “Boocher proved in [4] that any initial ideal of the ideal Im(X) of maximal minors of X has a linear resolution (or, equivalently in this case, defines a Cohen-Macaulay ring).”