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��II �� No.14

��Υơ��:

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�� 14.1 ${\mathbb{C}}[x]$ �� ${\mathbb{C}}$ ��

��ɲä��ƤǤ��ġ��ʤ��

�ˤĤ��Ƥ� ${\mathbb{C}}$ ��¿�༰��ΤΤʤ��ġ� ${\mathbb{C}}[[x]]$ ��

�η��Ū�٤��Τʤ��ĤǤ��ä�� ${\mathbb{C}}[[x]]$ ��ʬ�� ${\mathbb{C}}\convergent{x}$ ��

$\displaystyle {\mathbb{C}}\convergent{x}=\{f=\sum_{i=0}^\infty a_i x^i\in {\mathbb{C}}[[x]];$ $f$ �μ�«Ⱦ�¤�� $\displaystyle \}$

�� ${\mathbb{C}}$ ��μ�«�٤��ĤȸƤ֡��

��ǥå��ε��ϡ��ؿ�� $\mathcal F$ ��Ƽ��Τ褦�˽񤤤Ƥ�� Τ��褤��

�� 14.2 (��13.2�Ʒ�) $P\in \mathcal D$ �Ȥ��롣 $M=\mathcal D/\mathcal D P$ �Ȥ��ơ�

�� $\mathcal F$ ��Υ��ǥå��

$\displaystyle \operatorname{ind}(P;\mathcal F) =\dim(\operatorname{Hom}_\mathcal D(M,\mathcal F))- \dim(\operatorname{Ext}^1_{\mathcal D}(M,\mathcal F))$

��롣(â��Ĥμ��Ȥ��ͭ�¼��ΤȤ��)

�ֵܹ��λ��ͽ�ֲ÷��áפˤϥޥ륰��󥸥��(161�ڡ��10.1)�� 񤤤Ƥ��롣

�� 14.1 (�ޥ륰��󥸥�) $P\in \mathcal D={\mathbb{C}}[x,\partial_x]$ �� $P=\sum_{i=0}^m a_i(t)\partial_x^i$ �Ƚ񤤤��Ȥ��

$\operatorname{ind}(P,{\mathbb{C}}\convergent{x})=m-v(a_m)$
$% latex2html id marker 838 $ \operatorname{ind}(P,{\mathbb{C}}[[x]])=\underset {0\leq i \leq m}{\max}(i-v(a_i))$$

(â�� $v(\bullet)$ �� $\bullet$ ��ΰ̿��򤢤�魯��)

��ιֵ��ǤϤ��ξ��ϽҤ٤ʤ�� ˤĤ��ƥ��ǥå��ɤ��ʤäƤ��뤫�� Ҥ٤��

�� 14.1 $P=x^2 \partial -1$ �ˤ��ơ�

$\operatorname{ind}(P,{\mathbb{C}}\convergent{x})=-1$
$\operatorname{ind}(P,{\mathbb{C}}[[x]])=0$

�ºݡ��ˤĤ��Ƥ�

$% latex2html id marker 851 $\displaystyle \dim(\operatorname{Hom}_\mathcal D(M,{... ...vergent{x}))=0,\quad \dim(\operatorname{Hom}_\mathcal D(M,{\mathbb{C}}[[x]])=0 $$

��

$\displaystyle \dim(\operatorname{Ext}^1_{\mathcal D}(M,{\mathbb{C}}[[x]]))=0$

�Ϥ��ˤ狼�롣

$% latex2html id marker 855 $\displaystyle \dim(\operatorname{Ext}^1_{\mathcal D}(M,{\mathbb{C}}\convergent{x}))\neq 0 $$

��ݥ��Ȥǡ��ʬ��η��Ū�ʲ�ɬ��«��ʤ��Ȥ� �б��Ƥ��롣

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