Processing math: 0%
ochiai/Grobner
をテンプレートにして作成
開始行:
Becker, Weispfenning, GTM, 141, p247--p249. Proof of Theo...
--------------------
Notation. Let
be the set of matrices of size $...
where
R=K[\underline{X}]
.
F=(f_1,\ldots,f_m) \in M(1,m,R)
,
G=(g_1,\ldots,g_r) \in M(1,r,R)
.
There exists
C \in M(m,r,R)
and
D \in M(r,m,R)
such t...
G=FC
and
F=GD
.
Put
A=I_m - CD \in M(m,m,R)
.
Then
FA=F-FCD=F-GD=F-F=O_{1,m}
shows $\langle A \rangle...
Suppose
B \in M(r,s,R)
be a generating system of
S_G
.
Put
B^* = CB \in M(m,s, R)
.
Then
F B^*=FCB=GB=O_{1,s}
shows $\langle B^* \rangle \s...
Now take
\mathbf{h} \in S_F \subset M(m,1,R)
.
Put
\mathbf{h}_* = D \mathbf{h} \in M(r, 1, R)
and
\mathbf{k} = C \mathbf{h}_* \in M(m,1,R)
.
Then $\mathbf{h} - \mathbf{k} = (I_m -CD) \mathbf{h}= A \...
On the other hand,
G \mathbf{h}_* =GD \mathbf{h} = F \mathbf{h}=0
shows that
\mathbf{h}_* \in S_G
.
Then there exists
\mathbf{\alpha} \in M(s,1,R)
such that
\mathbf{h}_* = B \mathbf{\alpha}
.
This shows $\mathbf{k} = CB \mathbf{\alpha}
= B^* \mathbf{\alpha} \in \langle B^* \rangle$.
Summing up these two consequences,
we obtain
$\mathbf{h} = \mathbf{h}-\mathbf{k} + \mathbf{k} \in \lan...
= \langle A \cup B^* \rangle$.
--------------------
Consider the following five maps $F: R^m \to R, G: R^r \t...
G=FC, F=GD
and ker(
G
)= Im(
B
).
We claim that ker(
F
) = Im(
I-CD
) + Im(
CB
).
The inclusion
\supset
follows from
F(I-CD)=F-GD=F-F=O
...
The other inclusion
\subset
is discussed as follows:
GD(
ker
F
)
=F
(ker
F)=0
shows
D(
ker
F) \subset
ker
G=
Im
B
.
Then
CD(
ker
F) \subset
Im
(CB)
.
On the other hand,
(I-CD)(
ker
F) \subset
Im
(I-CD)
.
This completes the proof of claim.
Note that we can come back to the book if we put
A=I-CD
...
終了行:
Becker, Weispfenning, GTM, 141, p247--p249. Proof of Theo...
--------------------
Notation. Let
M(p,q,R)
be the set of matrices of size $...
where
R=K[\underline{X}]
.
F=(f_1,\ldots,f_m) \in M(1,m,R)
,
G=(g_1,\ldots,g_r) \in M(1,r,R)
.
There exists
C \in M(m,r,R)
and
D \in M(r,m,R)
such t...
G=FC
and
F=GD
.
Put
A=I_m - CD \in M(m,m,R)
.
Then
FA=F-FCD=F-GD=F-F=O_{1,m}
shows $\langle A \rangle...
Suppose
B \in M(r,s,R)
be a generating system of
S_G
.
Put
B^* = CB \in M(m,s, R)
.
Then
F B^*=FCB=GB=O_{1,s}
shows $\langle B^* \rangle \s...
Now take
\mathbf{h} \in S_F \subset M(m,1,R)
.
Put
\mathbf{h}_* = D \mathbf{h} \in M(r, 1, R)
and
\mathbf{k} = C \mathbf{h}_* \in M(m,1,R)
.
Then $\mathbf{h} - \mathbf{k} = (I_m -CD) \mathbf{h}= A \...
On the other hand,
G \mathbf{h}_* =GD \mathbf{h} = F \mathbf{h}=0
shows that
\mathbf{h}_* \in S_G
.
Then there exists
\mathbf{\alpha} \in M(s,1,R)
such that
\mathbf{h}_* = B \mathbf{\alpha}
.
This shows $\mathbf{k} = CB \mathbf{\alpha}
= B^* \mathbf{\alpha} \in \langle B^* \rangle$.
Summing up these two consequences,
we obtain
$\mathbf{h} = \mathbf{h}-\mathbf{k} + \mathbf{k} \in \lan...
= \langle A \cup B^* \rangle$.
--------------------
Consider the following five maps $F: R^m \to R, G: R^r \t...
G=FC, F=GD
and ker(
G
)= Im(
B
).
We claim that ker(
F
) = Im(
I-CD
) + Im(
CB
).
The inclusion
\supset
follows from
F(I-CD)=F-GD=F-F=O
...
The other inclusion
\subset
is discussed as follows:
GD(
ker
F
)
=F
(ker
F)=0
shows
D(
ker
F) \subset
ker
G=
Im
B
.
Then
CD(
ker
F) \subset
Im
(CB)
.
On the other hand,
(I-CD)(
ker
F) \subset
Im
(I-CD)
.
This completes the proof of claim.
Note that we can come back to the book if we put
A=I-CD
...
ページ名:
![[PukiWiki]](image/pukiwiki.png "[PukiWiki]")
