replace fs
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2 changed files with 3 additions and 5 deletions
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@ -2664,7 +2664,7 @@ If $X$ is a set, then $\cB \se \cP(X)$ is a base for some topology on $X$ iff $X
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\end{tikzcd}
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\hspace{50pt}
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\begin{tikzcd}
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&Y \arrow[bend right, swap]{ld}{\pi_0} \arrow[bend right, swap]{d}{\pi}&\cO_Y(Y) \cong A_\lambda \arrow{d}{\fs}& \\
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&Y \arrow[bend right, swap]{ld}{\pi_0} \arrow[bend right, swap]{d}{\pi}&\cO_Y(Y) \cong A_\lambda \arrow{d}{\mathfrak{s}}& \\
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X \arrow[hookrightarrow]{r}{}& U \arrow[swap]{u}{\sigma} & \cO_X(U)
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\end{tikzcd}
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\end{figure}
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@ -2673,8 +2673,8 @@ If $X$ is a set, then $\cB \se \cP(X)$ is a base for some topology on $X$ iff $X
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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 $1 - Z\ell(X_1,\ldots,X_n)$.
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Then $\cO_Y(Y) \cong \mathfrak{k}[Z,X_1,\ldots,X_n] / J \cong A[Z] / (1 -\lambda Z) \cong A_\lambda$.
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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 $U \xrightarrow{\sigma} Y$.
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We have $\fs(Z \mod J) = \lambda\inv$ and $\fs(X_i \mod J) = X_i \mod I$.
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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 $U \xrightarrow{\sigma} Y$.
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We have $\mathfrak{s}(Z \mod J) = \lambda\inv$ and $\mathfrak{s}(X_i \mod J) = X_i \mod I$.
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Thus $\sigma(x) = (\lambda(x)\inv, x)$ for $x \in U$.
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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
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\[
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@ -36,8 +36,6 @@
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\DeclareMathOperator{\hght}{ht}
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\newcommand{\Wlog}{W.l.o.g. }
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\newcommand{\fl}{\ensuremath\mathfrak{l}}
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\newcommand{\fs}{\ensuremath\mathfrak{s}}
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\newcommand{\fri}{\ensuremath\mathfrak{i}}
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\newcommand{\fm}{\ensuremath\mathfrak{m}}
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\newcommand{\Vspec}{\ensuremath V_{\mathbb{S}}}%\Spec}}
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