Torsion-free Sheaves and ACM Schemes

In this paper we study short exact sequences $0 \to \mathcal P \to \mathcal N \to \ii_D(k) \to 0$ with $\mathcal P, \mathcal N$ torsion--free sheaves and $D$ closed projective scheme. This is a classical way to construct and study projective schemes (e.g. see \cite{hart-1974}, \cite{hart-2}, \cite{mdp}, \cite{serre-1960}). In particular, we give homological conditions on $\mathcal P$ and $\mathcal N$ that force $D$ to be ACM, without constrains on its codimension. As last result, we prove that if $\mathcal N$ is a higher syzygy sheaf of an ACM scheme $X,$ the scheme $D$ we get contains $X.$