fix \Alg

2021_Algebra_I.tex

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