p : prime ideal of A
q : p-primary ideal
として
p ∩ S = φ
⇒
1.
AS p : prime ideal of AS
AS q : AS p-primary ideal
2.
AS p ∩ A = p , AS q ∩ A = q
proof :
1)
無条件で！
AS p ⊂ rad( AS q )
2)
AS p ⊃ rad( AS q )
3)
AS p = rad( AS q )
4)
AS p : prime ideal of AS
AS q : AS p-primary ideal
5)
AS p ∩ A = p
AS q ∩ A = q
q.e.d.
theorem 2.8
q : primary ideal of A
⇒
rad q : prime ideal of A
theorem 2.10
p : prime ideal of A
q : p-primary ideal ( ⇔ p = rad q )
として
ab ∈ q
and
a not ∈ p
⇒
b ∈ q
proof :
if
b not ∈ q
⇒ : 定義
a ∈ rad q = p
で矛盾！
q.e.d.
theorem 2.11
p , q : ideals of A
p = rad q
として
「 ab ∈ q and a not ∈ p ⇒ b ∈ q 」: ＊
⇒
p : prime ideal of A
q : p-primary ideal
proof :
ab ∈ q
and
a not ∈ q
として
if
b not ∈ rad q = p
⇒ : ＊
a ∈ q
で矛盾！
だから
b ∈ rad q = p
⇒
q : primary ideal of A
⇒ : p = rad q
p : prime ideal of A
q.e.d.
I : ideal of A ( ≠ I )
⇒
AS I ∩ A ⊃ I
AS ( AS I ∩ A ) = AS I
であるが
AS I ∩ A = I
は出て来ない！
if
I ∩ S ≠ φ
⇒
AS I = AS
⇒
AS I ∩ A = AS ∩ A = A ≠ I
となって
I ∩ S = φ
でなければならない！
そこで
I = P : prime ideal of A
としてみると
AS I = AS P : prime ideal of AS
で
AS I ∩ A = AS P ∩ A = P = I
になってくれる！
以上まとめて
I : prime ideal of A
I ∩ S = φ
⇒
AS I ∩ A = I
q.e.d.
Nikon D700
+
AIS 180mm F2.8
+
close up lens
で撮りました.
Technics
SL-1200MK4
+
Denon
PRA-2000 ( PHONO-3 )
+ : rec out
Denon
PMA-390Ⅳ ( TAPE-2 )
+ : pre out
Denon
POA-1500
Nikon D40
+
AIS 28mm F2.8
で撮りました.
Nikon D700
+
AIS 180mm F2.8
で撮りました.
( A : Noetherian ring , m : unique maximal ideal of A )
: local ring
A/m = F : field
a ( ≠ A ) : ideal of A
として
1.
{ unit element of A } = A-m
2.
( A/a , m/a ) : local ring
3.( Krull intersection theorem )
∩ m^n ( n : 自然数 ) = ( 0 ) = { 0 }
4.
∩ ( a+m^n ) ( n : 自然数 ) = a
5.
a/ma ( : A-module )
= : identify
a/ma ( : F-module = vector space over F )
6.
1)
2)
a = ( a1 , ・・・ , ar ) ( [ a1 , ・・・ , ar ] : minimal base )
⇒
r = dim F a/ma
3)
a
= ( a1 , ・・・ , ar ) ( [ a1 , ・・・ , ar ] : minimal base )
= ( b1 , ・・・ , bs ) ( [ b1 , ・・・ , bs ] : minimal base )
⇒
r = s
n = { x ∈ A | xs = 0 となる s ∈ S が存在する } : ideal of A
f : A ∋ x → x+n ∈ A/n
n = Ker f
AS : ring of quotients of A with respect to S
= : 定義
( A/n )f( S )
AS a
= : 定義
( A/n )f( S )f( a )
a’ ∩ A
= : 定義
f-inverse( a’ ∩ ( A/n ) )
として
16.
1)
f( S ) not ∋ zero diviser of f( A ) = A/n
2)
f( S ) : 乗法的に閉じている
3)
AS a = { f( a )/f( b ) | a ∈ a , b ∈ S }
a = Aa1 + ・・・ + Aar ⇒ AS a = ASf( a1 ) + ・・・ + ASf( ar )
f( a ) ∩ f( S ) ≠ φ ⇒ AS a = AS
( AS a ) ∩ A ⊃ a
17.
a’ ∩ A : ideal of A
AS ( a’ ∩ A ) = a’
p : prime ideal of A
q : p-primary ideal
として
18.
1)
p ∩ S ≠ φ ⇒ q ∩ S ≠ φ
p ∩ S = φ ⇒ q ⊃ n
2)
p ∩ S ≠ φ ⇒ f( q )∩ f( S ) ≠ φ
p ∩ S = φ ⇒ f( p ) ∩ f( S ) = φ
3)
AS p : prime ideal of AS
AS q : ASp-primary ideal
p ∩ S = φ ⇒ AS p ∩ A = p , AS q ∩ A = q
19.
p’ : prime ideal of AS
q’ : p’-primary ideal
⇒
p’ ∩ A : prime ideal of A
q’ ∩ A : ( p’ ∩ A )-primary ideal
20.
p ∩ S = φ
⇒
AS p ∩ A ( S-component of p ) = p
AS q ∩ A ( S-component of q ) = q
p ∩ S ≠ φ
⇒
S-component of p = S-component of q = A
21.
A : Noetherian ring
p ( ≠ A ) : prime ideal of A
として
22.
AS : Noetherian ring
23.
( Ap , Ap p ) : local ring
24.
( A-p )-component of p^k ( k : 自然数 ) = AA-p p^k ∩ A = Ap p^k ∩ A
= : 定義
p^( k ) : p-primary ideal
proof :
Ap p^k = ( Ap p )^k : Ap p-primery ideal of AA-p p = Ap p
⇒ : 19
Ap p^k ∩ A : ( Ap p ∩ A )-primery ideal of Ap p
Ap p ∩ A = AA-p p ∩ A = p ( ← p ∩ ( A-p ) = φ )
⇒
Ap p^k ∩ A : p-primery ideal of Ap p
q.e.d.
25.
∩ p^( k ) ( k : 自然数 ) = n = AA-p ( 0 ) ∩ A
proof :
∩ p^( k )
= : 定義
∩ ( AA-p p^k ∩ A = ( AA-p p )^k ∩ A )
= : 定義
f-inverse( ∩ ( ( A/n )f( A-p )f( p ) )^k ∩ ( A/n ) )
= : 23
f-inverse( { 0A/n } ∩ ( A/n ) )
=
f-inverse( { 0A/n } )
= : 定義
Ker f
= : ？
n
q.e.d.
26.
1)
2)
3)
27.
1)
2)
A : ring
として
28.
p ∩ S = φ
⇒
height p = height AS p
A : Noetherian ring
として
29.( Krull-principal ideal theorem )
a : non unit
q : minimal prime ideal of ( a ) = aA
⇒
height q ≦ 1
30.
1)
( 0 ) : a = { x ∈ A | ax = 0 }
= ( 0 )
⇒
height q = 1
2)
height ( a ) ≦ 1
31.
1)
2)( Krull-height theorem )
a = Aa1 +・・・+ Aar = ( a1 , ・・・ , ar )
q : minimal prime ideal of a
として
a ≠ A
⇒
height q ≦ r
3)
height p ＜ +∞
4)
32.
a( ≠ A ) : ideal of A
height a = r
⇒
height ( Aa1 +・・・+ Aai )
= i ( = 1 , ・・・ , r )
となる
a1 , ・・・ , ar ∈ a
が存在する
A : ring
a , b( ≠ A ) : ideals of A
として
33.
p ⊃ a
⇒
depth p/a( : prime ideal of A/a ) = depth p
34.
a ⊃ b
⇒
depth a/b = depth a
35.
Dim A/a = depth a
36.
height p + depth p ≦ Dim A
37.
( A , m : maximal ideal of A ) : local ring
として
1)
Dim A = height m ＜ +∞
2)
depth a ＜ +∞
3)
a : m-primary ideal
⇔
depth a = 0
⇔
height a = Dim A
A : ring
E = { 非零因子( zero diviser ) of A 全体 }
として
x , y ∈ E ⇒ xy ∈ E ( : Eは乗法に関して閉じている )
( a , c ) , ( b , d ) ∈ A × E
に対して
( a , c ) 〜 ( b , d )
⇔ : 定義
ad-bc = 0
とすると
〜 : 同値関係 ( equivalence relation ) of A × E
で
a/c
= : 定義
{ ( b , d ) ∈ A × E | ( a , c ) 〜 ( b , d ) } : ( a , c )を含む同値類
になっている.
ρ : A ∋ a → a/1A ∈ ( A × E )/〜
として
ρ : into isomorphism
⇒
a
= : identify
a/1A
( A × E )/〜 : ring , ⊃ A
( A × E )/〜 : Aの全商環 ( ring of total quotients of A )
E ⊃ S : 乗法に関して閉じている
Q : Aの全商環
AS
= : 定義
{ a/s | a ∈ A , s ∈ S }
として
AS : sub ring of Q
になっている.
AS : AのSによる商環 ( ring of quotients of A with respect to S )
a : ideal of A
a’ : ideal of AS
AS a : a-generated ideal of AS
として
theorem 3.7
AS a = { a/s | a ∈ a , s ∈ S }
a = Aa1 + ・・・ +Aar ⇒ AS a = AS a1 + ・・・ +AS ar
a ∩ S ≠ φ ⇔ AS a = AS
( AS a ) ∩ A ⊃ a
theorem 3.8
a’ ∩ A : ideal of A
AS( a’ ∩ A ) = a’
cor.
AS a ∩ A : ideal of A ( S-component of a )
AS( AS a ∩ A ) = AS a
Remark :
a ∩ S ≠ φ ⇒ AS a ∩ A = AS ∩ A = A
であるが
AS( AS a ∩ A ) = AS a ⇒ AS a ∩ A = a
は出て来ない！
p : prime ideal of A
q : p-primary ideal
p’ : prime ideal of AS
q’ : p’-primary ideal
として
theorem 3.9
AS p : prime ideal of AS
AS q : AS p-primary ideal
p ∩ S = φ
⇒
AS p ∩ A = p
AS q ∩ A = q
theorem 3.10
p’ ∩ A : prime ideal of A
q’ ∩ A : ( p’ ∩ A )-primary ideal
theorem 3.11
theorem 3.12
p ∩ S = φ
⇒
S-component of p = p
S-component of q = q
p ∩ S ≠ φ
⇒
S-component of p = S-component of q = A
theorem 3.13
A : Noetherian ring
として
theorem 3.14
AS : Noetherian ring
proof :
a’ : ideal of AS
⇒ : 3.8 ⅰ )
a’ ∩ A : ideal of A
A : Noetherian ring
⇒
a’ ∩ A = Aa1 + ・・・ + Aar ( a1 , ・・・ , ar ∈ A )
と書ける.
⇒
a’
= : 3.8 ⅱ )
AS ( a’ ∩ A )
= : 3.7 ⅱ )
ASa1 + ・・・ + ASar
⇔
AS : Noetherian ring
q.e.d.
theorem 3.15
A-p ⊂ E
⇒
( AA-p , : AA-p p ) : local ring
Remark :
AA-p
= : 定義
Ap
U( S )
= { p ∈ Spec( A ) | p ⊃ S }
として
a : S-generated ideal of A ( ⊃ S なる ideal of A の内最小のもの )
⇒
a ⊃ S
U( S ) = U( a ) = U( rad a )
次に
T
= { U( S ) | S ⊂ A }
として
Spec( A ) = U( { 0 } ) ∈ T
φ = U( { 1 } ) ∈ T
U( S1 ) ∩ ・・・ ∩ U( Sn )
= U( S1 ∪・・・∪ Sn ) ∈ T ( n : 自然数 )
U( S1 ) ∪ ・・・ ∪ U( Sn ) ∈ T ( n ＜ +∞ )
⇒
Spec( A ) : topological space
⇔ : 定義
T
= { U( S ) : closed set of Spec( A ) | S ⊂ A }
: Zariski topology in Spec( A )
Spec( A ) : affine scheme
R = C[ x1 , ・・・, xn ]
として
39.
Spec( R )
= : 位相同型( homeomorphism )
C^n/〜 : 剰余空間( 商空間 )
40.
p : prime ideal of R
⇒
V( p ) : affine algebraic manifold
⇒
Spec( R/p )
= : 位相同型( homeomorphism )
V( p )/〜 : 剰余空間( 商空間 )
Remark :
T : not Hausdorff
Spec( A )
= : 定義
{ p : prime ideal of A , ≠ A } : spectrum of A
S ⊂ A
U( S )
= : 定義
{ p ∈ Spec( A ) | p ⊃ S }
D( S )
= : 定義
Spec( A )-U( S )
とする.
