Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let
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\begin{align*}
S := k[x_0,\dots,x_n],
\end{align*}
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and let
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\begin{align*}
S_+ := (x_0,\dots,x_n) \trianglelefteq S
\end{align*}
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be the irrelevant ideal. For a homogeneous ideal $I\trianglelefteq S$, write
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\begin{align*}
I:S_+^\infty := \{f\in S : \text{there exists } m\in\mathbb N \text{ such that } fS_+^m\subset I\}.
\end{align*}
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Then the assignments
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\begin{align*}
X &\longmapsto I_+(X), & I &\longmapsto V_+(I)
\end{align*}
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give mutually inverse inclusion-reversing correspondences between nonempty projective algebraic subsets $X\subseteq \mathbb P_k^n$ and proper radical homogeneous ideals $I\trianglelefteq S$ satisfying
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\begin{align*}
I=I:S_+^\infty.
\end{align*}
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With the convention $I_+(\varnothing)=S$, the empty projective algebraic set corresponds to the unit ideal. Equivalently, a homogeneous ideal $I\trianglelefteq S$ defines the empty projective zero set precisely when