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1 0 1. 1.0. (1.0.1) - (1.0.2), B ϕ : B resp. B- M a m = ϕ(a) m (resp. m a = m ϕ(a)) resp. - M - B- resp. - M [ϕ] L - u : L M [ϕ] a x L u(a x) = ϕ(a) u(x) ϕ- L M (ϕ, u) u (, L) (B, M) - L (, L) (1.0.3) (1.0.2) J resp. ϕ(j) B resp. Bϕ(J)resp. ϕ(j)b BJresp. JB resp. B- B J Bresp. J B B (1.0.4) B B - ϕ : B ϕ() B J JB = BJ B B- M JM (BJ)M B- (1.0.5) - M p M 0 - M M p q p q M 0 - - B B B B -

2 0 B B B B - B B - B - - B (1.0.6) 0 1 p /p p {0} (1.0.7) B m n ϕ : B ϕ(m) n ϕ 1 (n) = m /m B/n 1.1. (1.1.1) a a n > 0 x n a x r(a) a r(r(a)) = r(a) a b r(a) r(b) ϕ a r(ϕ 1 (a)) = ϕ 1 (r(a)) a a (0) R R = (0) /R (1.1.2) R() /R() (0) 1.2. (1.2.1) S 1 S S S S

1. 3 1 f f n (n 0) S f 2 p p (1.2.2) S M - M S (m 1, s 1 ) (m 2, s 2 ) s(s 1 m 2 s 2 m 1 ) = 0 s S M S S 1 M (m, s) S 1 M m/s i S M : m m/1 is M S 1 M sm = 0 s S m M S 1 M S 1 M (m 1 /s 1 ) + (m 2 /s 2 ) = (s 2 m 1 + s 1 m 2 )/(s 1 s 2 ) S 1 (a 1 /s 1 )(a 2 /s 2 ) = (a 1 a 2 )/(s 1 s 2 ) (a/s)(m/s ) = (am)/(ss ) S 1 S 1 M S 1 S S 1 M S 1 - S s S s/1 S 1 1/s i S is M S 1 M i S : S 1 - (1.2.3) f S f = {f n } n0 S 1M S 1 f f M f f f = [1/f] f [T ]/(ft 1)[T ] f = 1 f M f M f f M f 0 p S = p S 1 S 1 M p M p p q i S (p) (i S ) 1 (q) = p i S /p p/q f p /q /p (1.2.4) S 1 i S B u u(s) B u : is S 1 u B u M - N B- v : M N - N u : B v v : M is M S 1 M v N

4 0 v S 1 - N S 1 - u (1.2.5) (a/s) m (am)/s S 1 - S 1 M S 1 M m/s (1/s) m (1.2.6) S 1 a a = (i S ) 1 (a ) a i S (a) S 1 S 1 a (1.3.2) p (i S ) 1 (p ) S 1 p S = p p (S 1 ) S 1 p (1.5.1) (1.2.7) K 0 S i S : S 1 S 1 K p p p p p p = p (1.2.8) S 1 x s S (x/s) n = 0 s x n = 0 s S (s x) n = 0 s x = 0 x/s = 0 1.3. (1.3.1) M N - u - M N S S 1 u S 1 - S 1 M S 1 N (S 1 u)(m/s) = u(m)/s S 1 M S 1 N S 1 M S 1 N (1.2.5) S 1 u 1 u P v - N P S 1 (v u) = (S 1 v) (S 1 u) S 1 M - S 1 - S M (1.3.2) S 1 M M u N v P S 1 M S 1u S 1 N S 1v S 1 P u : M N resp. S 1 u resp. N P M S 1 N S 1 P S 1 M S 1 (N + P) = S 1 N + S 1 P S 1 (N P) = (S 1 N) (S 1 P )

1. 5 (1.3.3) (M α ϕ βα ) - (S 1 M α S 1 ϕ βα ) S 1 - S 1 M α S 1 ϕ βα (1.2.5 1.3.1) S 1 lim M α lim S 1 M α M S 1 M (1.3.4) M, N - M N (S 1 M) S 1 (S 1 N) S 1 (M N) (m/s) (n/t) (m n)/st (1.3.5) M N S 1 Hom (M, N) Hom S 1 (S 1 M, S 1 N) u/s m/t u(m)/st M M r p q 0 M S 1 M Hom (M, N) - M 1.4. (1.4.1) S T S T S 1 T 1 ρt,s S 1 a/s T 1 a/s i T = ρt,s is - M S 1 M T 1 MT 1 M S 1 - S 1 - T 1 M m/s S 1 M m/s ρt,s M i T M = ρt,s M is M (1.2.5) ρt,s M ρt,s 1 S 1 M T 1 M M u : M N S 1 M S 1u S 1 N M N T 1 M T 1u T 1 N T 1 u S 1 u m Mt S (T 1 u)(m/t) = (t/1) 1 ((S 1 u)(m/1))

6 0 (1.4.2) - MN (1.3.4) (1.3.5) (S 1 M) S 1 (S 1 N) S 1 (M N) (T 1 M) T 1 (T 1 N) T 1 (M N) S 1 Hom (M, N) Hom S 1 (S 1 M, S 1 N) T 1 Hom (M, N) Hom T 1 (T 1 M, T 1 N) (1.4.3) T S S 1 M T 1 M S S S S S T S S 1 M (1.4.4) S T U S T U ρ U,S = ρ U,T (1.4.5) (S α )S α S β α β S α S α α β ρ βα = ρ S β,s α (1.4.4) ρ βα (Sα 1, ρ βα ) ρ α Sα 1 ϕ α = ρ S,S α (1.4.4) α β ϕ α = ϕ β ρ βα ϕ : S 1 Sα 1 ρ βα ρ α ϕα S 1 β ρ β ϕβ ϕ S 1 (α β) ϕ ϕ ρ α (a/s α ) ϕ(ρ α (a/s α )) = 0 S 1 a/s α = 0 sa = 0 s S s S β β α ρ α (a/s α ) = ρ β (sa/ss α ) = 0 ϕ

1. 7 - M lim Sα 1 (lim S α ) 1, lim Sα 1 M (lim S α ) 1 M M (1.4.6) S 1 S 2 S 1 S 2 S1 1 S 2 S 2 S 2 S1 1 - M S 1 2 (S 1 1 M) (S 1 S 2 ) 1 M m/(s 1 s 2 ) (m/s 1 )/(s 2 /1) 1.5. (1.5.1), ϕ S, S ϕ(s ) S, ϕ S 1 (1.2.4) S 1 ϕs S 1 ϕ S (a /s ) = ϕ(a )/ϕ(s ) = ϕ( ) S = ϕ(s ) ϕ S = ϕ ϕ S (1.4.1) ρ S,S (1.5.2) (1.5.1) M - S 1 - σ : S 1 (M [ϕ] ) (S 1 M) [ϕ S ] (S 1 M) [ϕ S ] m/ϕ(s ) S 1 (M [ϕ] ) m/s S = ϕ(s ) σ = ϕ σ (1.4.1) ρ S,S M M = ϕ - S 1 ( [ϕ] ) (ϕ(s )) 1 σ : S 1 ( [ϕ] ) S 1 S 1 - (1.5.3) M N - (1.3.4) (1.5.2) (S 1 M S 1 S 1 N) [ϕ S ] S 1 ((M N) [ϕ] ) ϕ(s ) = S (1.3.5) (1.5.2) S 1 ((Hom (M, N)) [ϕ] ) (Hom S 1 (S 1 M, S 1 N)) [ϕ] S

8 0 ϕ(s ) = S M (1.5.4) - N N [ϕ] a (n b) = n (ab) - S 1 - τ : (S 1 N ) S 1 (S 1 ) [ϕ S ] S 1 (N [ϕ] ) (n a)/(ϕ(s )s) (n /s ) (a/s) n /s a/s (n a)/(ϕ(s )s) (n a)/s (n /1) (s/a) τ S 1 (N [ϕ] ) (N [ϕ] ) S 1 (1.2.5) N (S 1 ) [ψ] ψ S 1 a ϕ(a )/1 (1.5.5) M N - (1.3.4) (1.5.4) S 1 M S 1 S 1 N S 1 S 1 S 1 (M N ) M (1.3.5) (1.5.4) Hom S 1 (S 1 M, S 1 N ) S 1 S 1 S 1 (Hom (M, N ) ) (1.5.6) (1.5.1) T T S T S T ϕ(t ) T S 1 ϕ S S 1 ρ T,S T 1 T 1 ϕ T M - S 1 (M [ϕ] ) σ (S 1 M) [ϕ S ] ρ T,S T 1 (M [ϕ] ) σ (T 1 M) [ϕ S ] N - (S 1 N ) S 1 (S 1 ) [ϕ S ] τ S 1 (N [ϕ] ) (T 1 N ) T 1 (T 1 ) [ϕ T ] τ S 1 (N [ϕ] )

1. 9 ρ T,S N S 1 N S 1 (1.5.7) ϕ : S ϕ (S ) S ϕ = ϕ ϕ ϕ S = ϕ S ϕ S M - M [ϕ ] = (M [ϕ] ) [ϕ ] (1.5.2) σ ϕ σ σ ϕ ϕ σ = σ σ N - - N [ϕ ] (N [ϕ ]) [ϕ] S 1 - (S 1 N ) S 1 (S 1 ) [ϕ S ] (S 1 N ) S 1 (S 1 ) [ϕ S ] S 1 (S 1 ) [ϕ S ] τ (1.5.4) ϕ τ τ ϕ ϕ τ = τ (τ 1) (1.5.8) B p B q p = q p B p (1.3.2) p (1.2.6)B p 0 q q p = p B q 1 q q 1 = p q 1 B q q = p 1.6. M f (1.6.1) M - f - (M n ) M n M m n ϕ nm M m M n z f n m z ((M n ), (ϕ nm )) N = lim M n N M f - n θ n : z z/f n M = M n M f m n θ n θ nm = θ m - θ : N M f ϕ n M n N n θ n = θ ϕ n M f n z/f n θ θ(ϕ n (z)) = 0 z/f n = 0 f k z = 0 k > 0

10 0 ϕ n+k,n (z) = 0 ϕ n (z) = 0 θ M f lim n (1.6.2) M n ϕ nm ϕ n M f,n ϕ f nm ϕf n g f n f n g n ρ fg,f : M f M fg (1.4.1 1.4.3) M f M fg lim M f,n lim M fg,n ρ fg,f ρ n fg,f (z) = gn z ρ n fg,f (z) = gn z : M f,n M fg,n f,n ρ n fg,f fg,n ϕ f n ϕ fg n f ρ fg,f fg 1.7. (1.7.1) - M M p 0 M Supp(M) M = 0 Supp(M) = p M p x M (1.7.2) 0 N M P 0 - Supp(M) = Supp(N) Supp(P ) p 0 N p M p P p 0 (1.3.2) M p = 0 N p = P p = 0 (1.7.3) M (M λ ) M p (M λ ) p 1.3.3 1.3.2 Supp(M) = λ Supp(M λ) (1.7.4) M - Supp(M) M M x M p = 0 s x = 0 s p p x M (x i ) 1in x i a i (1.7.3) Supp(M) a i p M a = i a i p (1.7.5) MN - Supp(M N) = Supp(M) Supp(N) p M p p N p 0 (1.3.4) M p 0 N p 0 P Q 0

1. 11 B P B Q 0 m B P/mP Q/mQ 0 (P/mP ) B/m (Q/mQ) = (P B Q) B (B/m) 0 M - a Supp(M/aM) a M n a + n

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