 Ideal sheaf of a point on a complex manifold

This sheaf consists of holomorphic functions vanishing at $p$ on opens containing $p$, and all holomorphic functions on opens not containing $p$. Evaluation gives $0\to {\mathcal{I}}_p\to {\mathcal{O}}_M\to {\mathbb{C}}_p\to 0$, with a skyscraper sheaf as quotient. On an elliptic curve, $H^0({\mathcal{I}}_p)=0$, $H^1({\mathcal{I}}_p)\cong \mathbb{C}$, and all higher cohomology vanishes.