Characterization of some projective subschemes by locally free resolutions

Abstract

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.