  In this paper we extend one direction of Fr\"oberg's theorem on a
combinatorial classification of quadratic monomial ideals with linear
resolutions. We do this by generalizing the notion of a chordal graph to higher
dimensions with the introduction of d-chorded and orientably-d-cycle-complete
simplicial complexes. We show that a certain class of simplicial complexes, the
d-dimensional trees, correspond to ideals having linear resolutions over fields
of characteristic 2 and also give a necessary combinatorial condition for a
monomial ideal to be componentwise linear over all fields.
