  A locally free resolution of a subscheme is by definition an exact sequence
consisting of locally free sheaves (except the ideal sheaf) which has
uniqueness properties like a free resolution. The purpose of this paper is to
characterize certain locally Cohen-Macaulay subschemes by means of locally free
resolutions. First we achieve this for arithmetically Buchsbaum subschemes.
This leads to the notion of an $\Omega$-resolution and extends a result of
Chang. Second we characterize quasi-Buchsbaum subschemes by means of weak
$\Omega$-resolutions. Finally, we describe the weak $\Omega$-resolutions which
belong to arithmetically Buchsbaum surfaces of codimension two. Various
applications of our results are given.
