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鐃緒申鐃緒申鐃� II 鐃緒申鐃緒申 No.11

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{環の局所化}

鐃緒申鐃� 11.1   鐃緒申 $R$ 鐃塾醐申 $a$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申鐃緒申鐃� $R[X]/(aX-1)$ 鐃塾わ申鐃夙わ申 $R$ 鐃緒申 $a$ 鐃祝わ申鐃宿所化鐃夙わ申鐃緒申鐃緒申 $R_a$ 鐃叔わ申鐃緒申錣刻申鐃�

鐃緒申鐃緒申 11.1   $R$ 鐃塾醐申 $a$ 鐃祝わ申鐃宿所化 $S=R_a$ 鐃緒申佑鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃�
1.
$R$ 鐃緒申鐃緒申 $S$ 鐃舜の種申鐃緒申鐃淑環緒申同鐃緒申 $\phi :R\to S$ 鐃緒申存鐃淳わ申鐃暑。
2.
$\phi(a)$ 鐃緒申 $S$ 鐃塾可逆醐申鐃叔わ申鐃暑。
3.
$S$ 鐃緒申 $\phi(R)$ 鐃夙￥申 $\phi(f)^{-1}$ 鐃夙わ申鐃緒申鐃緒申鐃緒申鐃緒申襦�
4.
$S$ 鐃熟常申鐃�3鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申里里鐃緒申鐃緒申悩任鐃緒申鐃緒申鐃重�鐃緒申物鐃叔わ申鐃暑。 鐃緒申鐃淑わ申鐃緒申鐃緒申發件申未隆鐃� $T$ 鐃緒申 鐃緒申同鐃緒申 $\psi: R\to T $ 鐃叔￥申鐃緒申鐃緒申鐃緒申4鐃緒申鐃緒申 (鐃緒申 $S,\phi$ 鐃薯それぞ鐃緒申 $T,\psi$ 鐃祝わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃�)鐃緒申鐃淑わ申弔鐃塾わ申 鐃緒申鐃緒申鐃緒申鐃緒申譴随申箸鐃緒申鐃夙￥申$S$ 鐃緒申鐃緒申 $T$ 鐃舜の環緒申同鐃緒申鐃緒申鐃緒申

\begin{displaymath}\rho: S\to T \end{displaymath}

鐃叔￥申 $\psi=\rho\circ\phi$ 鐃緒申鐃淑りた鐃緒申物鐃緒申鐃緒申鐃緒申鐃緒申鐃渋醐申澆鐃緒申襦�

鐃緒申 $R$ 鐃薯、わ申鐃塾醐申 $a$ 鐃叔局所化鐃緒申鐃緒申鐃緒申蓮鐃緒申鐃獣縁申砲鐃緒申鐃緒申鐃� $R$ 鐃緒申 $\frac{1}{a}$ 鐃緒申 鐃春わ申鐃獣わ申鐃銃わ申鐃緒申隼廚辰討茲わ申鐃�(鐃緒申鐃緒申鐃緒申鐃緒申鐃縮にわ申 $\phi$ 鐃緒申単鐃粛とは限わ申覆鐃緒申鐃�)

鐃緒申鐃緒申 11.2   鐃緒申鐃重�鐃緒申鐃緒申 $V \subset k^n$ 鐃緒申与鐃緒申鐃緒申譴随申箸鐃緒申鐃� 鐃緒申鐃塾ワ申鐃春ワ申鐃緒申弔鐃� $A(V)=k[X_1,\dots,X_n]/I(V)$, 鐃夙書く￥申 $f\in A(V)$ 鐃祝わ申鐃緒申鐃緒申鐃銃￥申 $k[X_1,\dots,X_n,Y]$ 鐃塾ワ申鐃叔ワ申鐃緒申 $J=(I(V), Yf-1)$ 鐃緒申佑鐃緒申鐃夙￥申鐃緒申鐃塾誌申鐃緒申鐃緒申鐃緒申立鐃縦￥申
1.
$W=V(J)$ 鐃緒申鐃緒申鐃緒申 $\{x \in V; f(x)\neq 0\}$ 鐃塾醐申鐃夙逸申鐃出逸申鐃緒申弍鐃緒申鐃緒申襦�
2.
$I(W)=J$
3.
$A(W)$ 鐃緒申 $A(V)$ 鐃緒申 $f$ 鐃祝わ申鐃宿所化 $A(V)_f$ 鐃緒申同鐃緒申鐃叔わ申鐃暑。

鐃緒申鐃緒申 11.1  

$A=\mbox{${\Bbb R}$ }[X,Y]$ 鐃塾ワ申鐃叔ワ申鐃緒申 $I=(X^2-Y^2-1)$ 鐃緒申 $B=\mbox{${\Bbb R}$ }[T]_{(T^2-1)} $ 鐃夙につわ申鐃銃￥申 鐃縦緒申同鐃緒申 $\phi:A\to B$ 鐃緒申

\begin{displaymath}\phi(X)=\frac{1+T^2}{1-T^2},\quad \phi(Y)=\frac{2T}{1-T^2} \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃�
1.
$V(I)$ 鐃緒申渕鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�
2.
$\phi(I)=0$ 鐃叔わ申鐃緒申鐃緒申鮗┐鐃緒申覆鐃緒申鐃緒申鐃�
3.
鐃緒申鐃緒申鐃縦の誌申鐃緒申鐃初、 $\phi$ 鐃緒申 $\mbox{${\Bbb R}$ }\setminus\{\pm 1\}$ 鐃緒申鐃緒申 $V(I)$ 鐃舜の種申鐃緒申 $f$ 鐃緒申鐃緒申鐃暑こ鐃夙わ申分鐃緒申鐃暑が鐃緒申 鐃緒申鐃緒申 $f$ 鐃祝つわ申鐃銃￥申

\begin{displaymath}f(0), f(\pm 2),f( \pm 3) \end{displaymath}

鐃薯それぞ鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�



2001-07-03