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 A, B
Direct sum A ⊕ B
Definition pi ( (x1, x2 ) ) : = xi
Homomorphism p1: A ⊕ B → A p2: A ⊕ B → B
P1 ∈Spec A
ap1 ( P1 ) = P1 ⊕ B
P2 ∈Spec B
ap2 ( P2 ) = P2 ⊕ A
Image ap1 ∩ Image ap2 = 0
Prime ideal of A ⊕ B P
( 1, 0 ) ⊂ P Ideal J ⊂ B
P = A ⊕ J
J is prime ideal.
( 0, 1 ) ⊂ P is also seen logically.
Spec ( A ⊕ B ) = Image ap1 ∪ Image ap2
Homomorphism of commutative ring φ: A → B ψ: A → 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 f ⊗ 1N : L ⊗ N → M ⊗ N
Definition A module N is flat.
3
Homomorphism of commutative ring φ: A →A’
A module M ≠ 0
M ⊗ A’ ≠ 0
Definition φ is faithfully flat.
Tokyo August 27, 2007
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