Hartshorne《代数几何》的习题

设有概形间的态射$X\overset{\varphi}{\rightarrow}\mathrm{Spec}A, \forall f\in A$.
$$\begin{array}[c]{ccc}
A&\stackrel {\lambda}{\rightarrow}&\mathcal{O}(\varphi^{-1}(\mathrm{Spec}A))\\
\downarrow&&\downarrow\\
A_f&\stackrel {\mu}{\rightarrow}&\mathcal{O}(\varphi^{-1}(\mathrm{Spec}A_f))\\
\end{array}$$
$$\begin{array}[c]{ccc}
A&{\rightarrow}&A/\mathrm{ker}\lambda\\
\downarrow&&\downarrow\\
A_f&\stackrel {\nu}{\rightarrow}&A_f/\mathrm{ker}\mu
\end{array}$$

$(A/\mathrm{ker}\lambda)_{f+\mathrm{ker}\lambda}\overset{\kappa}{\rightarrow} A_f/\mathrm{ker}\mu$
$\frac{a+\mathrm{ker}\lambda}{f^n+\mathrm{ker}\lambda}\mapsto \frac{a}{f^n}+\mathrm{ker}\mu$

$\frac{a+\mathrm{ker}\lambda}{f^n+\mathrm{ker}\lambda}=\frac{b+\mathrm{ker}\lambda}{f^m+\mathrm{ker}\lambda}\Rightarrow f^k(af^m-bf^n)\in \mathrm{ker}\lambda \Rightarrow \frac{a}{f^n}+\mathrm{ker}\mu=\frac{b}{f^m}+\mathrm{ker}\mu$

请问$\kappa$是同构吗?

$$\begin{array}[c]{ccc}
B&\stackrel{\zeta}{\rightarrow}&\mathcal{O}(\varphi^{-1}(\mathrm{Spec}B))\\
\downarrow&&\downarrow\\
B_g&\stackrel{\eta}{\rightarrow}&\mathcal{O}(\varphi^{-1}(\mathrm{Spec}B_g))\\
\end{array}$$

$A_f\cong B_g,\;\;A_f/\mathrm{ker}\mu\cong B_g/\mathrm{ker}\eta,\;\; \nu\cong \xi$
$$\begin{array}[c]{ccc}
A&{\rightarrow}&A/\mathrm{ker}\lambda\\
\downarrow&&\downarrow\\
A_f&\stackrel {\nu}{\rightarrow}&A_f/\mathrm{ker}\mu
\end{array}$$

$$\begin{array}[c]{ccc}
B&{\rightarrow}&B/\mathrm{ker}\zeta\\
\downarrow&&\downarrow\\
B_g&\stackrel {\xi}{\rightarrow}&B_g/\mathrm{ker}\eta
\end{array}$$

$(A/\mathrm{ker}\lambda)_{f+\mathrm{ker}\lambda}\to A_f/\mathrm{ker}\mu$
$\frac{a+\mathrm{ker}\lambda}{f^n+\mathrm{ker}\lambda}\mapsto \frac{a}{f^n}+\mathrm{ker}\mu$

$\frac{a+\mathrm{ker}\lambda}{f^n+\mathrm{ker}\lambda}=\frac{b+\mathrm{ker}\lambda}{f^m+\mathrm{ker}\lambda}\Rightarrow f^k(af^m-bf^n)\in \mathrm{ker}\lambda \Rightarrow \frac{a}{f^n}+\mathrm{ker}\mu=\frac{b}{f^m}+\mathrm{ker}\mu$
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