Ideal sheaf of a point on a complex manifold
= Ideal sheaf of a point on a complex manifold
{title2=$\mathcal I_p$}
= Point ideal sheaf
{synonym}
This sheaf consists of <holomorphic functions> vanishing at $p$ on opens containing $p$, and all <holomorphic functions> on opens not containing $p$. Evaluation gives $0\to\mathcal I_p\to\mathcal O_M\to\mathbb C_p\to0$, 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.