next up previous
Next: About this document ...

    

鐃緒申鐃緒申鐃� II 鐃緒申鐃緒申 No.2

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{イデアル}

鐃緒申鐃緒申廛鐃緒申箸砲禄颪�なわ申鐃獣わ申鐃緒申鐃緒申鐃緒申鐃塾講居申鐃叔はとわ申鐃緒申鐃叔わ申覆鐃緒申造鐃瞬環」とわ申鐃緒申鐃緒申 単鐃縮醐申鐃緒申鐃緒申腸調鐃緒申弔里鐃緒申箸鐃緒申味鐃緒申鐃暑。 (鐃淑わ申鐃緒申老鐃縮随申鐃淑わ申鐃初、鐃緒申鐃緒申痢鐃緒申鐃緒申1.1鐃竣はわ申鐃緒申鐃夙わ申鐃緒申鐃緒申鐃緒申任呂覆鐃緒申鐃� 鐃緒申鐃塾わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申C 鐃殉わ申鐃緒申鐃緒申鐃緒申鐃�I鐃塾講居申鐃叔学わ申任鐃緒申鐃熟わ申鐃叔わ申鐃暑。)

鐃緒申鐃� 2.1 (鐃緒申鐃緒申)  
1.
鐃緒申 $R$ 鐃緒申鐃緒申分鐃緒申鐃緒申 $I$ 鐃緒申 鐃緒申鐃緒申2鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $I$ 鐃緒申 $R$ 鐃塾ワ申鐃叔ワ申鐃緒申任鐃緒申鐃夙わ申鐃緒申鐃緒申
(a)
$I$ 鐃緒申 $(R,+)$ 鐃緒申鐃緒申分鐃緒申鐃叔わ申鐃暑。
(b)
$rx\in R \qquad (\forall x\in I, \forall r\in I)$
2.
鐃緒申 $R$ 鐃緒申鐃緒申分鐃緒申鐃緒申 $T$ 鐃緒申与鐃緒申鐃緒申譴随申鐃緒申鐃�$T$ 鐃緒申泙鐃叔常申鐃塾ワ申鐃叔ワ申鐃緒申里鐃緒申箸鐃� $T$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃� $R$ 鐃塾ワ申鐃叔ワ申鐃緒申噺討咫鐃� $\langle T\rangle_{\text ideal}$ 鐃叔わ申鐃緒申蕕刻申鐃�

(鐃緒申鐃緒申)

鐃緒申鐃縮に￥申鐃緒申 $R$ 鐃塾醐申 $a_1,a_2,\dots,a_n$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申鐃� $\langle\{a_1,a_2,\dots a_n\}\rangle_{\text{ideal}}$ 鐃塾わ申鐃夙わ申単鐃緒申

\begin{displaymath}(a_1,a_2,\dots,a_n) \end{displaymath}

鐃夙丸鰹申未鐃夙って書くわ申鐃夙わ申多鐃緒申鐃緒申鐃緒申鐃縮に随申鐃舜では丸鰹申未鐃緒申鐃緒申未憤鐃縮ｏ申鐃盾た鐃緒申鐃暑こ鐃夙わ申 鐃緒申鐃殉わ申覆鐃緒申鐃緒申鐃緒申鐃緒申痢鐃�...鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申襯わ申妊鐃緒申鐃竣わ申鐃純外鐃叔わ申鐃暑。 鐃緒申鐃舜ではよく鐃術わ申鐃緒申鐃暑し鐃緒申鐃准でもあ鐃緒申里如鐃緒申楾峙鐃緒申任發鰹申竜鐃緒申鐃緒申鐃術わ申鐃暑こ鐃夙わ申 鐃緒申鐃暑が鐃緒申鐃緒申鐃署しなわ申鐃処う鐃祝緒申分鐃緒申鐃春わ申鐃暑こ鐃夙￥申

鐃緒申鐃緒申 2.1   鐃緒申 $k$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $n$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� $k^n$ 鐃緒申鐃緒申分鐃緒申鐃緒申 $S$ 鐃祝わ申鐃緒申鐃緒申鐃銃￥申 $S$ 鐃緒申鐃� $0$ 鐃祝なわ申多鐃準式鐃緒申鐃緒申鐃緒申

\begin{displaymath}I(S)=\{f\in k[X_1,X_2,\dots X_n]; \quad f(x)=0 \forall x \in S\} \end{displaymath}

鐃緒申 $k[X_1,X_2,\dots,X_n]$ 鐃塾ワ申鐃叔ワ申鐃緒申任鐃緒申襦�

鐃緒申鐃緒申 2.2   鐃緒申 $k$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $n$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� $R=k[X_1,X_2,\dots,X_n]$ 鐃緒申鐃緒申分鐃緒申鐃緒申 $T$ 鐃緒申鐃出わ申鐃銃￥申 $T$ 鐃塾駈申鐃緒申鐃緒申鐃緒申鐃塾緒申鐃緒申 $V(T)$ 鐃緒申 $T$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃曙た $R$ 鐃塾ワ申鐃叔ワ申鐃緒申 $I=\langle T \rangle_{\text ideal}$ 鐃塾駈申鐃緒申鐃緒申鐃緒申鐃塾緒申鐃緒申 $V(I)$ 鐃夙逸申鐃竣わ申鐃暑。

鐃緒申 2.1   鐃緒申 $k$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $n$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� $R=k[X_1,X_2,\dots,X_n]$ 鐃緒申鐃緒申分鐃緒申鐃緒申 $T_1,T_2$ 鐃緒申鐃緒申鐃獣て￥申鐃盾し $T_1$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃曙た $R$ 鐃塾ワ申鐃叔ワ申鐃緒申鐃� $T_2$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃曙た $R$ 鐃塾ワ申鐃叔ワ申鐃暑が 鐃緒申鐃竣わ申鐃暑、鐃緒申鐃淑わ申鐃�

\begin{displaymath}\langle T_1 \rangle_{\text{ideal}} =\langle T_2 \rangle_{\text{ideal}} \end{displaymath}

鐃緒申鐃緒申鐃緒申立鐃縦なわ申弌鐃�$T_1$ 鐃塾駈申鐃緒申鐃緒申鐃緒申鐃塾緒申鐃緒申鐃� $T_2$ 鐃塾駈申鐃緒申鐃緒申鐃緒申鐃塾緒申鐃緒申鐃� 鐃緒申鐃竣わ申鐃暑。鐃緒申鐃淑わ申鐃� $V(T_1)=V(T_2)$ 鐃叔わ申鐃暑。

鐃緒申鐃緒申鐃粛鰹申鐃緒申蕁�$V(T)$ 鐃緒申 $T$ 鐃緒申鐃緒申鐃叔ワ申鐃緒申里箸鐃緒申里澆鐃粛わ申鐃緒申个鐃緒申紊�鐃瞬わ申任鐃緒申襦� 鐃緒申鐃縮に￥申$V(I)$ 鐃塾件申鐃塾緒申鐃緒申鐃緒申鐃緒申的鐃緒申鐃緒申噺討屐鐃�

(鐃緒申鐃緒申) $V(I)=V(J)$ 鐃緒申鐃緒申鐃緒申箸鐃緒申辰鐃� $I=J$ 鐃夙は限わ申覆鐃緒申鐃�

鐃緒申鐃緒申 2.3   鐃緒申 $k$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $n$ 鐃薯きめ、 $R=k[X_1,\dots X_n]$ 鐃夙わ申鐃緒申鐃緒申 鐃緒申鐃塾とわ申鐃緒申$R$ 鐃塾ワ申鐃叔ワ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃出わ申鐃銃種申鐃塾緒申鐃緒申鐃緒申鐃緒申鐃緒申立鐃縦￥申
1.
$V(I+J)=V(I)\cap V(J)$
2.
鐃緒申辰醗鐃緒申未鐃�

\begin{displaymath}V(\langle \cup_\lambda I_\lambda\rangle_{\text{ideal}})= \cap_\lambda V(I_\lambda) \end{displaymath}

3.
$V(I\cap J)=V(I)\cup V(J)$
4.
$V(R)=\emptyset$
5.
$V(0)=k^n$

鐃緒申 2.2   $k^n$ 鐃熟￥申鐃緒申鐃重�鐃緒申鐃緒申鐃緒申鐃緒申里鐃緒申捗鐃緒申鐃夙わ申鐃銃￥申鐃緒申鐃緒申鐃緒申屬鐃淑わ申鐃緒申 鐃緒申鐃塾逸申鐃緒申里鐃緒申箸鐃� Zariski 鐃緒申鐃緒申噺討屐鐃�

Zariski 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃舜で器申鐃緒申的鐃緒申鐃緒申鐃緒申未鐃緒申鐃緒申鐃� 鐃緒申鐃塾逸申鐃緒申鐃�(鐃処く鐃夙わ申鐃緒申鐃緒申鐃塾なわ申鐃叔わ申鐃緒申鐃緒申鐃緒申)Hausdorff 鐃緒申鐃瞬ではなわ申鐃緒申

鐃緒申鐃緒申 2.1  
1.
$V(Y-X^2, X-Y^2, X^2+Y^2-2)\in \mbox{${\Bbb R}$ }^2$ 鐃緒申渕鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�
2.
鐃緒申鐃叔ワ申鐃緒申鐃緒申鐃緒申鐃� $(Y-X^2,X-Y^2,X^2+Y^2-2)=(X-1,Y-1)$ 鐃緒申鐃緒申鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�

(鐃緒申鐃緒申)

1. 鐃緒申 2. 鐃緒申佑鐃緒申鐃緒申妊劵鐃夙にはなるが鐃緒申2.鐃塾常申鐃緒申鐃緒申 1. 鐃夙わ申鐃縮わ申 鐃春わ申鐃緒申必鐃竣わ申鐃緒申鐃暑。


Yoshifumi Tsuchimoto
2001-05-01