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.$
Categories: math.AG