# Basis of $\operatorname{Proj}S$

Let $S$ be an $\mathbb{N}$-graded ring. Normally, $D_+(h)=(\operatorname{Proj}S)\backslash V_+(h)$ is defined for homogeneous elements $h$ of a positive degree. Those cover $\operatorname{Proj}S$. But do they form a basis of $\operatorname{Proj}S$ ? It appears that in order to form a basis, I need homogeneous elements of zero degree. Wouldn't it be nice if there is some magical reason so that I don't need to include those elements with zero degree?

• Recall the definition of Proj. And watch out for the irrelevant ideal. – Martin Brandenburg Dec 1 '13 at 22:10
• Are you saying it is true? Could you give me more hints? I can't seem to figure it out; yes, I'm always watching out for the irrelevant ideal. – ashpool Dec 1 '13 at 23:20

Let $J\subset S$ be a graded ideal and $\mathfrak{p}\in\operatorname{Proj}S\backslash V_+(J)$. If $J\cap S_+\subset\mathfrak{p}$, then $J\subset\mathfrak{p}$ since $S_+\not\subset\mathfrak{p}$, so we may indeed use only $D_+(h)$ where $h$ is a homogeneous element of a positive degree.