  We produce a family of complexes called trimming complexes and explore
applications. We study how trimming complexes can be used to deduce the Betti
table for the minimal free resolution of the ideal generated by subsets of a
generating set for an arbitrary ideal $I$. In particular, we compute the Betti
table for 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 $n \times m$ 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.
