From fa11abf4cbd6c6a839c9bba5256d0f760d386087 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Maximilian=20Ke=C3=9Fler?= Date: Wed, 16 Feb 2022 01:13:38 +0100 Subject: [PATCH] replace fs --- 2021_Algebra_I.tex | 6 +++--- algebra.sty | 2 -- 2 files changed, 3 insertions(+), 5 deletions(-) diff --git a/2021_Algebra_I.tex b/2021_Algebra_I.tex index 42bbe9d..2c6b005 100644 --- a/2021_Algebra_I.tex +++ b/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 \[ diff --git a/algebra.sty b/algebra.sty index 161de40..dd01e2d 100644 --- a/algebra.sty +++ b/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}}