M d (A, m) l M m l 1. A m k = A/m M A dim M = d H i m (M) i M N M N N N q = (x 1 , , x d )A M q M qM N A (q; M) M l M m l M H i m (M) M 0 i ∈ {0, r, d} 0  r  d l M m l M M m m.H i m (M) = 0, 0  i < d M M M m M M m 1 2. A m k = A/m M A x = (x 1 , . . . , x d ) M q = xA Q M (q) = Q M (x) :=  t>0  x t+1 1 , . . . , x t+1 d  M : M x t 1 x t d  , I(M) := d−1  i=0  d − 1 i  (H i m (M)), J(M) := d−1  i=1  d − 1 i − 1  (H i m (M)). x q I(q; M) = I(x; M) = (M/qM) − e(x; M), J(q; M) = J(x; M) = e(x; M) − (M/Q M (x)). 2.1 A m k = A/m M A M M Soc(M) Soc (M) = (0 : M m) Soc (M) ∼ = Hom A (k, M) Soc (M) k Socdim(M) N ⊆ M U(N) dim M/N N N 2.2 A m M A x = (x 1 , . . . , x d ) M I(x 2 1 , . . . , x 2 d ; M) = I(x 1 , . . . , x d ; M). a A M (x 1 , . . . , x d ) M a 1  r  d ((x 1 , . . . , x r−1 ) M : M x r ) = ((x 1 , . . . , x r−1 ) M : M a) . 2.3 M a M a M (H i m (M)) < ∞ (0  i  d − 1) x = (x 1 , . . . , x d ) M I(x; M) = I(M). n I(q; M) = I(M) q m n 2.4 M x = (x 1 , . . . , x d ) M J(x; M) = J(M). 2.5 M a M x 1 , . . . , x r (0  r  d) M (x 1 , . . . , x r )A ⊆ a U ((x 1 , . . . , x r−1 ) M) = ((x 1 , . . . , x r−1 ) M : M x r ) . 2.6 x 1 , . . . , x r ∈ m M i = 1, . . . , r (x 1 , . . . , x i−1 ) M : M x i = (x 1 , . . . , x i−1 ) M : M m. a A x 1 , . . . , x r ∈ m M a i = 1, . . . , r (x 1 , . . . , x i−1 ) M : M x i = (x 1 , . . . , x i−1 ) M : M a. 2.7 M A dim M = d > 0 M m 2 M M m 2 M m H i m (M) = 0 i 0  i  d − 1 2.8 M A A 2.9 M q M J(q; M) = J(M). 2.10 A A A A 3. 3.1 N ⊆ M N A (N; M) = dim k Hom A (k, M/N) = Socdim(M/N) = (0 : M m). 3.2 W = H 0 m (M) n q M m n q N A (q; M) = Soc dim(M) + N A (q; M/W ). 3.3 M a M x 1 , x 2 , . . . , x r (1  r  d) M (x 1 , x 2 , . . . , x r )A ⊆ a n 1 , n 2 , . . . , n r  1  x n 1 +1 1 , . . . , x n r +1 r  M : M x n 1 1 x n r r  = (x 1 , . . . , x r ) M + r  i=1 U ((x 1 , . . . , x i , . . . , x r ) M). 3.4 M d > 0 l q ⊆ m l q N A (q; M) = Soc dim  d  i=1 U i + qM qM  + Soc dim  H d m (M)  , U i = U ((x 1 , . . . , x i , . . . , x d )M) . M a M M a M a x 1 , . . . , x d M a n  1 (x n+1 1 , . . . , x n+1 d )M : M (x 1 x d ) n = (x 1 , . . . , x d )M + d  i=1 ((x 1 , . . . , x i , . . . , x d )M : M a). a (x 1 , , x d )M + d  i=1 ((x 1 , , x i , , x d )M : M a) = d  i=1 (((x 1 , , x i , , x d )M : M x i ) + x i M). U i = U((x 1 , . . . , x i , . . . , x d )M) U i = ((x 1 , . . . , x i , . . . , x d )M : M x i ) (x n+1 1 , . . . , x n+1 d )M : M (x 1 x d ) n = d  i=1 (U i +x i M). l q m l ϕ : M/qM −→ H d m (M) Hom A (k, . ) l m l ⊆ a q = (x 1 , . . . , x d ) M m l K ϕ M/qM H d m (M) K =  n1 (x n+1 1 , . . . , x n+1 d )M : M (x 1 x d ) n qM = d  i=1 U i + x i M qM . Hom A (k, . ) 0 −→ K −→ M/qM −→ H d m (M) ϕ 0 −→ Soc(K) −→ Soc (M/qM) −→ Soc(H d m (M)) −→ 0. N A (q; M) = (Soc(M/qM)) = (Soc(K)) + (Soc(H d m (M)) = Soc dim  d  i=1 U i + qM qM  + Soc dim  H d m (M)  . Λ 3.5 M l q = (x 1 , . . . , x d )A m l N A (q; M) = d  i=0  d i  Soc dim(H i m (M)). d = 0 M d > 0 depth M = 0 W = H 0 m (M) M = M / W n q m n N A (q; M) = Soc dim(M) + N A (q; M). H i m (M) ∼ = H i m (M) i > 0 l  n M M depth M > 0 N A (q; M) = Soc dim  d  i=1 U i + qM qM  + Soc dim  H d m (M)  . m 2 M l  2 d  i=1 U i + qM qM = d  i=1 U i + qM qM = Q M (q) qM . M U i = U ((x 1 , . . . , x i , . . . , x d ) M) = ((x 1 , . . . , x i , . . . , x d ) M : M x i ) = ((x 1 , . . . , x i , . . . , x d ) M : M m) ⊆ (qM : M m) . m U i ⊆ qM, ∀i = 1, . . . , d m d  i=1 U i ⊆ qM m Q M (q) qM = 0 Soc dim  d  i=1 U i + qM qM  = Soc dim  Q M (q) qM  =   0 : Q M (q) qM m  =   Q M (q) qM  . I(x; M ) = I(M) J(x; M) = J(M) I(q; M ) + J(q; M ) = I(M ) + J(M ) = d−1  i=0  d − 1 i  (H i m (M)) + d−1  i=1  d − 1 i − 1  (H i m (M)) = d−1  i=0  d i  (H i m (M)). I(q; M ) + J(q; M ) = (M/qM ) − e(q; M ) + e(q; M ) − (M/Q M (q)) = (M/qM) − (M/Q M (q)) = (Q M (q)/qM). (Q M (q)/qM) = d−1  i=0  d i  (H i m (M)). m H i m (M) = 0, i = 1, , d − 1 Soc( H i m (M)) = H i m (M) N A (q; M) =   Q M (q) qM  + Soc dim  H d m (M)  = d  i=0  d i  Soc dim(H i m (M)). 3.6 M l m l M = M  H 0 m (M) N A (q; M) = const 3.7 M (x 1 , . . . , x d ) (x 2 1 , . . . , x 2 d ) M l q ⊆ m l q N A (q; M) = d  i=0  d i  Soc dim(H i m (M)). 3.8 A A m A A A m m d  i=0  d i  Soc dim(H i m (A)); 1 H d m (A) Soc(H i m (A)) = 0 (i < d) H i m (A) 0 H i m (A) = 0 A A A 27(3) 30 85 1 I 2 = QI 110 278/2 ¨a 102 M d (A, m) l M m l