replace \fri
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@ -363,13 +363,13 @@ Equivalent\footnote{used in a vague sense here} formulation:
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\begin{proof} (HNS2 (\ref{hns2}) $\implies$ HNS1b (\ref{hns1b}))
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\begin{proof} (HNS2 (\ref{hns2}) $\implies$ HNS1b (\ref{hns1b}))
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$I \se \fm$ for some maximal ideal. $R / \fm$ is a field, since $\fm$ is maximal.
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$I \se \fm$ for some maximal ideal. $R / \fm$ is a field, since $\fm$ is maximal.
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$R / \fm$ is of finite type, since the images of the $X_i$ generate it as a $\mathfrak{l}$-algebra.
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$R / \fm$ is of finite type, since the images of the $X_i$ generate it as a $\mathfrak{l}$-algebra.
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There are thus a field extension $\fri / \mathfrak{l}$ and an isomorphism $R / \fm \xrightarrow{\iota} \fri$ of $\mathfrak{l}$-algebras.
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There are thus a field extension $\mathfrak{i} / \mathfrak{l}$ and an isomorphism $R / \fm \xrightarrow{\iota} \mathfrak{i}$ of $\mathfrak{l}$-algebras.
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By HNS2 (\ref{hns2}), $\fri / \mathfrak{l}$ is a finite field extension.
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By HNS2 (\ref{hns2}), $\mathfrak{i} / \mathfrak{l}$ is a finite field extension.
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Let $x_i \coloneqq \iota (X_i \mod \fm)$.
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Let $x_i \coloneqq \iota (X_i \mod \fm)$.
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\[
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\[
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P(x_1,\ldots,x_m) = \iota(P \mod \fm)
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P(x_1,\ldots,x_m) = \iota(P \mod \fm)
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\]
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\]
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Both sides are morphisms $R \to \fri$ of $\mathfrak{l}$-algebras. For for $P = X_i$ the equality is trivial. It follows in general, since the $X_i$ generate $R$ as a $\mathfrak{l}$-algebra.
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Both sides are morphisms $R \to \mathfrak{i}$ of $\mathfrak{l}$-algebras. For for $P = X_i$ the equality is trivial. It follows in general, since the $X_i$ generate $R$ as a $\mathfrak{l}$-algebra.
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Thus $(x_1,\ldots,x_m)$ is a zero of $I$ (since $P \mod \fm = 0$ for $P \in I \se \fm$).
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Thus $(x_1,\ldots,x_m)$ is a zero of $I$ (since $P \mod \fm = 0$ for $P \in I \se \fm$).
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HNS1 (\ref{hns1}) can easily be derived from HNS1b.
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HNS1 (\ref{hns1}) can easily be derived from HNS1b.
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