fix \Alg
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@ -2465,7 +2465,7 @@ The following is somewhat harder than in the affine case:
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\item The category of sets.
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\item The category of groups.
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\item The category of rings.
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\item If $R$ is a ring, the category of $R$-modules and the category $\foralllg_R$ of $R$-algebras
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\item If $R$ is a ring, the category of $R$-modules and the category $\Alg_R$ of $R$-algebras
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\item The category of topological spaces
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\item The category $\Var_\mathfrak{k}$ of varieties over $\mathfrak{k}$ (see \ref{defvariety})
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\item If $\mathcal{A}$ is a category, then the \vocab{opposite category} or \vocab{dual category} is defined by $\Ob(\mathcal{A}\op) = \Ob(\mathcal{A})$ and $\Hom_{\mathcal{A}\op}(X,Y) = \Hom_\mathcal{A}(Y,X)$.
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@ -2482,7 +2482,7 @@ The following is somewhat harder than in the affine case:
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\item The category of abelian groups is a full subcategory of the category of groups.
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It can be identified with the category of $\Z$-modules.
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\item The category of finitely generated $R$-modules as a full subcategory of the category of $R$-modules.
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\item The category of $R$-algebras of finite type as a full subcategory of $\foralllg_R$.
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\item The category of $R$-algebras of finite type as a full subcategory of $\Alg_R$.
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\item The category of affine varieties over $\mathfrak{k}$ as a full subcategory of the category of varieties over $\mathfrak{k}$.
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\end{itemize}
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\end{example}
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@ -2563,17 +2563,17 @@ The following is somewhat harder than in the affine case:
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\begin{itemize}
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\item Let $X,Y$ be varieties over $\mathfrak{k}$. Then the map
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\begin{align}
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\phi: \Hom_{\Var_\mathfrak{k}}(X,Y) &\longrightarrow \Hom_{\foralllg_\mathfrak{k}}(\mathcal{O}_Y(Y), \mathcal{O}_X(X)) \\
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\phi: \Hom_{\Var_\mathfrak{k}}(X,Y) &\longrightarrow \Hom_{\Alg_\mathfrak{k}}(\mathcal{O}_Y(Y), \mathcal{O}_X(X)) \\
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(X \xrightarrow{f} Y) &\longmapsto (\mathcal{O}_Y(Y) \xrightarrow{f\st} \mathcal{O}_X(X))
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\end{align}
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is injective when $Y$ is quasi-affine and bijective when $Y$ is affine.
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\item The contravariant functor
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\begin{align}
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F: \Var_\mathfrak{k} &\longrightarrow \foralllg_\mathfrak{k} \\
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F: \Var_\mathfrak{k} &\longrightarrow \Alg_\mathfrak{k} \\
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X &\longmapsto \mathcal{O}_X(X)\\
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(X\xrightarrow{f} Y) &\longmapsto (\mathcal{O}_X(X) \xrightarrow{f\st} \mathcal{O}_Y(Y))
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\end{align}
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restricts to an equivalence of categories between the category of affine varieties over $\mathfrak{k}$ and the full subcategory $\mathcal{A}$ of $\foralllg_\mathfrak{k}$,
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restricts to an equivalence of categories between the category of affine varieties over $\mathfrak{k}$ and the full subcategory $\mathcal{A}$ of $\Alg_\mathfrak{k}$,
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having the $\mathfrak{k}$-algebras $A$ of finite type with $\nil A = \{0\} $ as objects.
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\end{itemize}
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\end{proposition}
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@ -2630,7 +2630,7 @@ The following is somewhat harder than in the affine case:
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\end{proof}
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\begin{remark}
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Note that giving a contravariant functor $\cC \to \cD$ is equivalent to giving a functor $\cC \to \cD\op$. We have thus shown that the category of affine varieties is equivalent to $\mathcal{A}\op$, where $\mathcal{A} \subsetneq \foralllg_\mathfrak{k}$ is the full subcategory of $\mathfrak{k}$-algebras $A$ of finite type with $\nil(A) = \{0\}$.
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Note that giving a contravariant functor $\cC \to \cD$ is equivalent to giving a functor $\cC \to \cD\op$. We have thus shown that the category of affine varieties is equivalent to $\mathcal{A}\op$, where $\mathcal{A} \subsetneq \Alg_\mathfrak{k}$ is the full subcategory of $\mathfrak{k}$-algebras $A$ of finite type with $\nil(A) = \{0\}$.
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\end{remark}
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\subsubsection{Affine open subsets are a topology base}
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