Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $n,m \in \mathbb{N}$, and let $X \subset \mathbb{A}^n_k$ and $Y \subset \mathbb{A}^m_k$ be affine varieties. For a subset $S \subset k[z_1,\ldots,z_r]$, write
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\begin{align*}
V(S) := \{a \in \mathbb{A}^r_k : f(a)=0 \text{ for every } f \in S\}
\end{align*}
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for its vanishing set. For an [affine variety](/page/Affine%20Variety) $Z \subset \mathbb{A}^r_k$, write
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\begin{align*}
I(Z) := \{f \in k[z_1,\ldots,z_r] : f(z)=0 \text{ for every } z \in Z\}
\end{align*}
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for its vanishing ideal, and write $k[Z] := k[z_1,\ldots,z_r]/I(Z)$ for its coordinate ring. In particular,