Trimming complexes and applications to resolutions of determinantal facet ideals

Abstract

We produce a family of complexes called trimming complexes and explore its applications. We demonstrate how trimming complexes can be used to deduce the Betti table for the minimal free resolution of the ideal generated by certain subsets of a generating set for an arbitrary ideal I. In particular, we compute the Betti table of the ideal obtained by removing an arbitrary generator from the ideal of submaximal pfaffians of a generic skew symmetric matrix M. We also explicitly compute the Betti table for the ideal generated by certain subsets of the generating set of the ideal of maximal minors of a generic n × m matrix. Such ideals are a subset of a class of ideals called determinantal facet ideals, whose higher degree Betti numbers had not previously been computed.

Keywords:

• Homological commutative algebra
• free resolutions
• determinantal ideals

MATHEMATICS SUBJECT CLASSIFICATION:

• 13D02
• 13C13

Acknowledgements

The author wishes to thank the anonymous referee for many helpful comments that greatly enhanced the final draft of this paper.