Σ− Σ − (1 − C 1 ) R R− A N N A− X A ϕ : X −→ N ψ : A −→ N N N N− R R R R R A− X A ϕ : N −→ A/X ψ : N −→ A M (C 1 ) M M M M (C 2 ) A B M A M B M (C 3 ) A B M A ∩ B = 0 A ⊕ B M (1 − C 1 ) M M M CS− (1 − C 1 ) M (C 1 ) (1 −C 1 ) (C 1 ) (C 2 ) (C 1 ) (C 3 ) (C 2 ) =⇒ (C 3 ) =⇒ =⇒ =⇒ =⇒ =⇒ (1 − C 1 ) M Σ− Σ− Σ − (1 − C 1 )) M (I) M (N) ) (1 − C 1 ) I N M 1− M M M 1− R 1− R R R R 1− R R N A− A− − X A ϕ : X −→ N ψ : A −→ N N − N N− N A− A− − X A ϕ : N −→ A/X ψ : N −→ A N N N− A ⊆ M A ⊆ e M A ⊆ ⊕ M End(M) A M A M A M M Σ− M 1− S = End(M) M S 1− M S 1− M 1− M A M f : A −→ M, (f = 0) Kerf = 0 f M f g : M −→ M Kerf = 0 α = i − f i A M Kerf ∩ Kerα = 0 M 1− Kerf ∩ Kerα = Kerα Kerf ∩ Kerα = Kerf Kerf = 0 Kerf ∩ Kerα = Kerα = 0 α α g : M −→ M gi = α = i − f f = i − gi = i(id M − g) M ϕ, ψ ∈ S Kerϕ, Kerψ M Kerϕ ⊆ Kerψ M/Kerψ ∼ = Imψ ⊆ M M/Kerϕ ∼ = Imϕ ⊆ M ϕ ∗ : M/Kerϕ −→ M ϕ ∗ (m + Kerϕ) = ϕ(m) ψ ∗ : M/Kerϕ −→ M ψ ∗ (m + Kerϕ) = ψ(m) ϕ ∗ M h ∈ S hϕ ∗ = ψ ∗ m ∈ M hϕ ∗ (m + Kerϕ) = ψ ∗ (m + Kerϕ) hϕ(m) = ψ(m) hϕ = ψ ψ ∈< ϕ > < ψ >⊆< ϕ > S S S 1− M 1− M f : M −→ M/X f M f f ∗ : M −→ M α = π − f π M M/X Imf + Imα = M/X. M 1− Imf +Imα = Imf Imf +Imα = Imα Imf = M/X Imf +Imα = Imα = M/X α h hπ = α f = (1 − h)π M ϕ, ψ ∈ S Imϕ, Imψ M Imϕ ⊆ Imψ ψ : M −→ Imψ ϕ M Imψ M h ψh = ϕ ϕ ∈< ψ > < ϕ >⊆< ψ > S S S 1− M S = End(M) M S M S M = ⊕ n i=1 M i M i 1− i = 1, 2, , n End(M) = ⊕ n i=1 End(M i ) Σ M = ⊕ i∈I U i U i M F I A ⊕ (⊕ i∈F U i ) ⊆ e M. A = M F A = M A M i ∈ I A ∩ U i = 0 F I A ∩ ⊕ i∈F U i = 0 V 1 = ⊕ i∈F U i V 2 = ⊕ i∈K U i K = I\F F A ∩ (V 1 ⊕ U k ) = 0 k ∈ K a ∈ A, a = 0 a = x − u x ∈ V 1 , u ∈ U k u = 0 u = a − x ∈ A ⊕ V 1 U k ∩ (A ⊕ V 1 ) = 0 k ∈ K A ⊕ V 1 ⊆ e M M M Σ − (1 − C 1 ) M Σ− (i) ⇒ (ii) (i) M Σ− M M (C 2 ) M = ⊕ i∈I U i U i CS− A M F ⊆ I A ⊕ (⊕ i∈F U i ) ⊆ e M. V 1 = ⊕ i∈F U i V 2 = ⊕ i∈K U i K = I\F p 1 p 2 M V 1 V 2 p 2 | A h = p 1 (p 2 | A ) −1 p 2 (A) −→ V 1 A = {x +h(x) | x ∈ p 2 (A)} h V 2 g : B −→ V 1 p 2 (A) ⊆ B ⊆ V 2 h V 2 C = {x + g(x) | x ∈ B} A ⊕ V 1 ⊆ e M p 2 (A) = p 2 (A ⊕ V 1 ) ⊆ e p 2 (M) = V 2 p 2 (M) = V 2 p 2 (A) ⊆ e B ⊆ V 2 A ⊆ e C A M A = C p 2 (A) = B g = h k ∈ K X k = U k ∩ p 2 (A) X k = 0, ∀k ∈ K X k A k = {x +h(x) | x ∈ X k } X k ∼ = A k A k A A k ⊆ e P ⊆ U k ⊕V 1 A k ∩V 1 = 0 P ∩ V 1 = 0 p 2 | P h k = h | p 2 (A k ) h h k λ k = p 1 (p 2 | P ) −1 : p 2 (M) −→ V 1 λ k h k p 2 (P ) = p 2 (A k ) p 2 | P A k ⊆ e P A k = P A k M V 3 M V 3 = ⊕ i∈L U i L ⊂ I A k ⊕ V 3 ⊆ e M. M (1 − C 1 ) A k ⊆ ⊕ M V 3 ⊆ ⊕ M M (C 3 ) A k ⊕ V 3 ⊆ ⊕ M A k ⊕ V 3 = M V 4 = ⊕ i∈J U i J = I\L M = A k ⊕ V 3 = V 4 ⊕ V 3 A k ∼ = M/V 3 = V 4 ⊕ V 3 /V 3 ∼ = V 4 A k | J |= 1 A k ∼ = U j (j ∈ I) X k ∼ = A k ∼ = U j X k ⊆ ⊕ M M (C 2 ) X k ⊆ e U k ⊆ ⊕ M X k = U k , ∀k ∈ F p 2 (A) = V 2 A ∼ = V 2 = ⊕ i∈K U i A ⊆ ⊕ M M CS− M M Σ− (ii) ⇒ (i) (ii) M Σ −(1− C 1 ) M Σ− M M (1 − C 1 ) M Σ− M Σ − (1 − C 1 ) R Σ − (1− C 1 ) R R R R Σ− R (ii) ⇔ (iii) ⇒ (i) (i) ⇒ (ii) R R R = ⊕ n i=1 R i R i R R R R Σ− Σ−