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How should we define the function $I(\cdot)$ for ${\rm Proj} S_\bullet$(the homogeneous prime ideals not containing $S_+$) for a $\mathbb{Z}^{\ge 0}$-graded ring $S_\bullet=S_0\oplus S_+$?

I know the functions $V(\cdot)$ and $I(\cdot)$ for affine schemes. I want the projective version of them. I know the projective version of $V(\cdot)$, i.e. $V(T):=\{p| p\supset T\}\subset {\rm Proj} S_\bullet$ for $T\subset S_+.$

If we follow the affine case, we might define $$I(Z):=\cap_{p\in Z}p$$ for $Z\subset {\rm Proj} S_\bullet$.

However this definition does not satisfy $$I(Z)\subset S_+$$ because if $S_0:=\mathbb{Z}, S_\bullet:=\mathbb{Z}[x]$ and $Z=\{ (2)\}$ then $I((2))=(2)=(2\mathbb{Z})[x]\supset 2\mathbb{Z} \not\subset S_+.$

$\bullet {\bf EDIT}$ (added just after my first comment to the first answer):

How about the following? $$I(Z):=\langle (\cap_{p\in Z}p)\cap \cup_{i>0}S_i\rangle$$ Here $\cup_{i>0}S_i$ means the homogeneous elements of positive degree, and the bracket means the ideal generated by the ingredients.

How do people define $I(\cdot)$ usually?

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    This is exactly the same as Vakil's notes(5.5.H(b)).2012-11-19

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