Affine principal neighbourhoods from ideal-sheaf vanishing
= Affine principal neighbourhoods from ideal-sheaf vanishing
Let $P$ have an affine open neighbourhood $U$, and let $Y=X\setminus U$. Evaluation gives $\mathcal I_Y\twoheadrightarrow k_P$ with <kernel> $\mathcal I_Y\cap\mathcal I_P$. If first <sheaf cohomology> of every <coherent ideal sheaf> vanishes, a <global section> $f$ of $\mathcal I_Y$ can be chosen with $f(P)=1$. Then $X_f\subseteq U$ is a <principal open subset> of an <affine variety> and is affine.