replace fs

2021_Algebra_I.tex

@@ -2664,7 +2664,7 @@ If $X$ is a set, then $\cB \se \cP(X)$ is a base for some topology on $X$ iff $X
         \end{tikzcd}
         \hspace{50pt}
         \begin{tikzcd}
-            &Y \arrow[bend right, swap]{ld}{\pi_0} \arrow[bend right, swap]{d}{\pi}&\cO_Y(Y) \cong A_\lambda \arrow{d}{\fs}& \\
+            &Y \arrow[bend right, swap]{ld}{\pi_0} \arrow[bend right, swap]{d}{\pi}&\cO_Y(Y) \cong A_\lambda \arrow{d}{\mathfrak{s}}& \\
             X \arrow[hookrightarrow]{r}{}& U \arrow[swap]{u}{\sigma} & \cO_X(U)
         \end{tikzcd}
     \end{figure}
@@ -2673,8 +2673,8 @@ If $X$ is a set, then $\cB \se \cP(X)$ is a base for some topology on $X$ iff $X
     Let $Y = V(J) \se \mathfrak{k}^{n+1}$ where $J \se \mathfrak{k}[Z,X_1,\ldots,X_n]$ is generated by the elements of $I$ and $1 - Z\ell(X_1,\ldots,X_n)$.
 
     Then $\cO_Y(Y) \cong \mathfrak{k}[Z,X_1,\ldots,X_n] / J \cong A[Z] / (1 -\lambda Z) \cong A_\lambda$.
-    By the proposition about affine varieties (\ref{propaffvar}), the morphism $\fs: \cO_Y(Y) \cong A_\lambda \to \cO_X(U)$ corresponds to a morphism $U \xrightarrow{\sigma} Y$.
-    We have $\fs(Z \mod J) = \lambda\inv$ and $\fs(X_i \mod J) = X_i \mod I$.
+    By the proposition about affine varieties (\ref{propaffvar}), the morphism $\mathfrak{s}: \cO_Y(Y) \cong A_\lambda \to \cO_X(U)$ corresponds to a morphism $U \xrightarrow{\sigma} Y$.
+    We have $\mathfrak{s}(Z \mod J) = \lambda\inv$ and $\mathfrak{s}(X_i \mod J) = X_i \mod I$.
     Thus $\sigma(x) = (\lambda(x)\inv, x)$ for $x \in U$.
     Moreover, the projection $Y \xrightarrow{\pi_0} X$ dropping the $Z$-coordinate has image contained in $U$, as for $(z,x) \in Y$ the equation
     \[

algebra.sty

@@ -36,8 +36,6 @@
 \DeclareMathOperator{\hght}{ht}
 \newcommand{\Wlog}{W.l.o.g. }
 
-\newcommand{\fl}{\ensuremath\mathfrak{l}}
-\newcommand{\fs}{\ensuremath\mathfrak{s}}
 \newcommand{\fri}{\ensuremath\mathfrak{i}}
 \newcommand{\fm}{\ensuremath\mathfrak{m}}
 \newcommand{\Vspec}{\ensuremath V_{\mathbb{S}}}%\Spec}}