Monday 5 January 2015

Linguistic Focus 2 Faithfully Flat / August 27, 2007


Linguistic Focus

2

Faithfully Flat  


    TANAKA Akio

1
Prime ideal of commutative ring A     Spec A
Homomorphism     f : A  B
Prime ideal    U  B
Prime ideal of A     f-1 ( P )
P  Spec B
f-1 ( P )  Spec A
Map     af : Spec B → Spec A
Commutative ring     AB
Direct sum     A  B
Definition     pi ( (x1x2 ) ) : = xi
Homomorphism      p1A  B → A       p2A  B → B
P1 Spec A
ap1 ( P1 ) = P B   
P2 Spec B
ap2 ( P2 ) = P A
Image ap∩ Image ap= 0
Prime ideal of A  B     P
( 1, 0 )  P    Ideal     J ⊂ B
A  J
J is prime ideal.
( 0, 1 )  P is also seen logically.
Spec ( A  B ) = Image ap∪ Image ap2
Homomorphism of commutative ring     φA → B    ψA → C
Homomorphism     (φ, ψ) : A B  C
a (φ, ψ) : Spec ( B  C )  Spec A
Conclusion    Spec ( A1, …, An ) = i = 1 I = n Spec Ai
2
Injective homomorphism of A module     f : L  M
Homomorphism of A module      1N : L  N → M  N
Definition     A module is flat.
3
Homomorphism of commutative ring     φA →A
A module M ≠ 0
 A’  ≠ 0
Definition     φ is faithfully flat.

Tokyo August 27, 2007
Sekinan Research Field of Language

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