R h R = R 1. φ : R 1 R 2 R 1. ker α = φ(ker α) R2. u R 1 R 2 R 1. φ(u) R 2. R 2 R 1 φ

Transcripción

1

2

3 R h (R, m, κ) κ = R C R h R ˆR R R h ˆR R h

4 R h R R h R = κ[x 1,..., X n ] X1,...,X n R 1 R 2 R 1 R 1 R 2 R 1 φ : R 1 R 2 R 1 R 2 R 1 R 2 R 1 R 2 R 1 a 1,..., a n R 1 α : R1 n R 1 α : R2 n R 2 x α(x) := a 1 x a n x n y α (y) := φ(a 1 )y φ(a n )y n, ker α = φ(ker α) R2 R 2 R 1 φ R 2 R 1 α : R n 1 R 1 r R 1 α (y) = φ(r) R n 2 α(x) = r Rn 1 R 2 R 1 φ R 1 R 2 R 1 R 2 R 2 R 1 n = 1 α = 0 α (y) = φ(r) = 0 R 2 0 = α(x) = r R 1 φ R 1 R 2 φ(a 1 ) φ(a 2 ) R 2 a 1 a 2 R 1 φ(u) R 2 u R 1

5 a ec = a a R 1 a ec a a ec a r a ec φ(r) a e x i R 1 a i a i = 1, 2,..., n φ(r) = n i=1 x iφ(a i ) α (x) = φ(r) R2 n α(x) = r R1 n r a φ : (A, m A ) (B, m B ) φ(a) B a A A = m A A B = m B B φ(a) / m B φ(a) B a A φ(a) φ(a ) B φ(a) / B \ φ(a ) φ(a \ A ) = φ(m A ) φ(a) / φ(m A ) φ φ(m A ) m B (R 1, m 1 ) (R 2, m 2 ) φ : (R 1, m 1 ) (R 2, m 2 ) R 2 R 1 R 2 R 1 φ : R 1 R 2 φ(m 1 ) m 2 x, a i R 1 n a i x i = x R 1 i=1 n φ(a i )x i = φ(x) R 2 i=1 y = (y 1,..., y n ) ỹ = (y 1,..., y n, y n+1 ) n φ(a i )x i + φ( x)x n+1 = 0 i=1 R 2 R 1 R 2 R2 n n i=1 a ix i +( x)x n+1 = 0 S j = (s j,1,..., s j,n, s j,n+1 ) R1 n+1 λ i R 1 j = 1,..., r r N (y 1,..., y n, 1) = r j=1 φ(λ j)s j n a i s j,i + ( x)s j,n+1 = 0 j = 1,..., r. i=1 S j n i=1 a ix i +( x)x n+1 = 0 1 = r j=1 φ(λ j)s j,n+1 λ j R 1 a = s 1,n+1,..., s r,n+1 R 1 R 1 s 1,n+1,..., s r,n+1 1 φ(a)r 2 a e = φ(a)r 2 = R 2 a m 1 a m 1 φ(a) φ(m 1 ) m 2 φ(a)r 2 m 2 R 2 = φ(a)r 2 m 2 R 2 µ j R 1 j = 1,..., r 1 = r j=1 µ js j,n+1 µ j n a i µ j s j,i + µ j s j,n+1 ( x) = 0 i=1 j = 1,..., r

6 j n r a i ( µ j s j,i ) + i=1 j=1 r µ j s j,n+1 ( x) = 0 j=1 x i := r j=1 µ js j,i n i=1 a i x i + 1( x) = 0 x = (x 1,..., x n ) n i=1 a ix i = x R 1 R 2 Spec(R 2 ) Spec(R 1 ) Spec(R 2 ) p Spec(R 1 ) Spec(R 2 R1 κ(p)) κ(p) (R 1 ) p R 1 p p V K n p V V p O p O h p O p (R, m, κ) f f(0) m f (0) R R m f(x) = X n a 1 X + a 0 a 1 R a 0 m m α m f f(x) = (X α)g(x) g(x) = X n b 1 X + b 0 R[X] b 0 αb 1 = a 1 R b 0 R β m f f(β) = 0 g(β) R α = β f(x) = a n X n a 1 X + a 0 a 1 R a 0 m m f f(x) = a n X n a 1 X + a 0 a 1 R a 0 m g(x) R[X] g(x) = X n b 1 X + b 0 b 1 R b 0 m R(X) a 0 g(x) = (X + 1) n f( a 0a 1 1 X + 1 ),

7 R(X) R[X] X n f( a 0a 1 1 X ) = a 0(X n X n 1 + a 0 n j=2 ( 1) j a j a j 2 0 a j 1 Xn j ) = a 0 h(x) h(x) = X n X n 1 + a 0 n j=2 ( 1) j a j a j 2 0 a j 1 Xn j = X n X n 1 + a 0 l(x) g(x) = h(x + 1) = X n b 1 X + b 0 b 0 = g(0) = h(1) = a 0 l(1) m b 1 = g (0) = h (1) = 1 + a 0 l (1) 1 + m (R, m, κ) f(x) = a n X n +...+a 1 X +a 0 R[X] f(x) κ[x] a κ α R f α = a h(x) = f(x + γ) γ = a f(a) = 0 γ m g(x) h(x) g(0) = h(1) = f(γ) m g (0) = h (1) = f (γ) R h R (X, f, 0) (X, f) R 0 κ f X (0) κ 0 u (X, f) u i = 0 i u R (X, f, 0) (Z, g, 0) g (R, m, κ) R (R, m, κ) κ R f(x) = X n a 1 X + a 0 R[X] R[X] := R[X] f(x) X X S R[X] S := {g(x) R[X] : g(x) R[X], g(0) R }. R f R[X] R \ S R f := S 1 R[X]. R f mr f κ f κ R[X] f(x) R R R f R R f

8 f B := R[X] f(x) R R S 1 (R[X] f(x) ) R[X] f(x) S 1 B B S 1 (R[X] f(x) ) R R f r(x)/s(x) r, s R[X] s(0) R r(x), s(x) R[X] a = r(x)/s(x) R f R f a R f a J R f r(0) R a R f r(0) m b = r 1(X)/s 1 (X) R f 1 + ab = (s(x)s 1 (X) + r(x)r 1 (X))/(s(X)s 1 (X)) = p(x)/q(x) p(0) R 1 + ab R f m R f r(x)/s(x) r(0) m m m Rf R f r(x)/s(x) r(0) R m Rf X/1 mr f = mr f Y = X n 1 + a n 1 X n a 2 X + a 1 Y R f Y X = a 0 X = a 0 Y 1 mr f m Rf = mr f R R f R f R R R f r(x)/s(x) = p(x)/q(x) R f (X)(p(X)s(X) r(x)q(x)) f(x) R[X] u(0) R p(0)s(0) r(0)q(0) m R f r(x)/s(x) r(0)/s(0) κ m Rf R f κ (R, m, κ) (X, f, 0) R R f R[X] := R[X] f(x) R f R[X] X X/1 R f f m Rf R[X/1] R f R[X] (R, m, κ) C J C ϕ : R C f(x) = X n a 1 X + a 0 R[X] ϕ(f) = X n ϕ(a 1 )X + ϕ(a 0 ) C[X] ξ J C ψ : R f C ψ(x) = ξ (R, m) C, J C ψ (R f, mr f ) C ϕ ψ ϕ

9 R f ϕ(f)(ξ) = 0 ϕ : R[X] C ϕ ϕ R C 3 ϕ R[X] ξ J C ϕ(r ) C g(x) = g(0) + X h(x) S g(0) R ϕ(g(x)) = ϕ(g(0)) + ξ h(ξ) C ψ : S 1 R[X] C ϕ ϕ R[X] C ψ S 1 R[X] C ϕ ψ(g(x)/s(x)) C g(x)/s(x) R f ψ(g(x)) = ϕ(g(0)) + ξ h(ξ) C ξ m C ϕ(g(0)) C ϕ g(0) R g(x) S g(x)/1 R f S (Z, m Z ) (R, m) S (Z, m Z ) S f(x) = X n a 1 X + a 0 Z[X] a 1 Z a 0 m Z (Z f, m Zf ) S S R (Z, m Z ) S m Z = mz S (B, m B ), (C, m C ) S f 1,..., f k g 1,..., g l f 1,..., f k C g 1,..., g l B C C B B S R h R R h := lím Z. Z S

10 R h R mr h (B, m B ) ϕ : R B!ψ (R, m) (B, m B ) ψ (R h, mr h ) n ψ n (R, m) (B, m B ) ψ n (R f,n, mr f,n ) ψ n : (R f,n, mr f,n ) (B, m B ) (B, m B ) (R f,n+1, mr f,n+1 ) p 1,..., p r (R f,n, mr f,n ) ψ n+1 ϕ (R, m) (B, m B ) ψ n (R f,n, mr f,n ) ψ n+1 5 (R f,n+1, mr f,n+1 ) ϕ ϕ (Y, f, 0) f = (f 1,..., f n ) i f i R[Y 1,..., Y p ] f j (0) = 0 (f 1,...,f p) (Y 1,...,Y (0) p) R R f := (R[Y ] f ) (m, Y ) = (R[Y 1,..., Y p ] f 1,..., f p ) (m, Y1,...,Y p ). (Z, g, 0) R g := (R[Z] g(z) ) Z (R, m) κ (Y, f, 0) R

11 (Z, g, 0) ω R f z Z mód m (R[Z]/ g(z) ) (m,z) R f z ω R f (R, m) R := κ[x] X R K[X, Y ] H (Y, H, 0) X = {X 1,..., X n } Y = {Y 1,..., Y p } H = {H 1,..., H p } K X := {X 1,..., X n } Y := {Y 1,..., Y p } O := K[X] X H 1,..., H p O[Y ] H j (0, 0) = 0 (H 1,...,H p) (Y 1,...,Y p) (0, 0) = 0 H j K[X, Y ] (O[Y ] H X,Y = K[X, Y ] H e O C := K[X, Y ] H 1,..., H p m := m A = x C m I := H 1,..., H p (0, 0) m (0, 0) I I = p 0... p r p p m I e = I K[X, Y ] X,Y = p K[X, Y ] X,Y, A = C X,Y = (K[X, Y ]I) x,y = (K[X, Y ] X,Y )I e = (K[X, Y ]p) x,y. A (K[X, Y ]p) m K[X] K[X, Y ]p i (K[X, Y ]p) x,y = (K[X, Y ] H 1,..., H p ) x,y = A, i K[X]

12 K[X] X A n = dim A = dim (K[X, Y ]p) x,y = dim K[X, Y ]p = K[X, Y ]p K[X] K[X, Y ]p K[X] K[X, Y ]p n y Q(A) L K K(X) = Q(K[X] X ) K[X] X L O K[X] X L O K[X] X A L K[X] X O O A K[X] X O A, m O m A m = m A O O m = A. y j O g(x,y) A 1+h(x,y) h(0, 0) = 0 h m A O K x y h O h O m A = m h m 1 + h / m y j O K[X] X y y y K[X] X O A qf(a) = L y j L y j A y j O O V = O b y j V b 1 O K[X] X O V := K[X] X,a a := b K[X] X K(X) V V ϕ : O k(b)[[t]] K[X] X O L k(a)[[t]] k(b)[[t]] k(b)((t)) ϕ(x) k(b)[[t]] y(t) k(b)[[t]] H i (ϕ(x 1 ),..., ϕ(x n ), y) = 0, i = 1,..., p. ϕ

13 y j V O K[X] X K[X] X O q X K[X] X K[X] X Oq O K[X] X q = K O q O q k[x] X q O O q K[X] X O q m 1,..., m l m A m 1 = m K[X] X q = K Oq O = (O m q O m ) (O ml q O ml ) O mj q O mj = O mj m e j j O mj e j j = 1 m 1 = m e 1 = 1 O m = A Aq A K[X, Y 1,..., Y p ](X, H 1,..., H p ) X,Y = K[Y1,..., Y p ](F 1 (0, Y ),..., F p (0, Y )) Y = K. ξ O O m qo m ξ m e 2 2,..., ξ me l l ξ / m ξ := ξ mód q O (1, 0,..., 0) O m q O m = K[ξ] K[X] X K[X] X [ξ] K[X] X K[X] X [ξ] O. n m K[X] X [ξ] n := m K[X] X [ξ] m q X K[X] X R := K[X] X [ξ] n K[X] X R = K[X] X [ξ] n O m S := R \ n S 1 O R = K[X] X [ξ] n S 1 O R m S 1 O = O m R S 1 O = O m Rn e = S 1 On S 1 O ξ R = O m = A

14 R m R O m O m (R \ m) 1 O = O m P (T ) ξ K(X) ξ O ξ K[X] X K X K[X] X K[X] X [ξ] K[X] X [T ] P (T ) K[X] X [T ] K[X] X [ξ] T ξ P (T ) K[X] X K[X] X [T ] P (T ) = K[X] X [ξ] c K[X] X [T ] K[X] X (K[X] X [T ] P (T ) ) c = K[X] X [ξ] n = Om = A. c (X, a) a P (T ) mód X P (T ) X X X = (0,..., 0) q = X K[X] X c P (T ) mód X K p(t ) K[X] X q = K (K[X] X [T ] P (T ) ) c c e Am A = K (K[X] X [T ] P (T ) ) c c e = K c (X, a) a K p(t ) = P (T ) mód X A = (K[X, T ] P (T ) ) (X,a) a P (T ) K[X] X K[X] X { } H1 : X 1 + Y 1 + X 2 Y 1 Y 2 + 2X 2 Y1 2 = 0, H := H 2 : X 2 + Y 2 + X 1 Y1 2 + X 1 Y 1 Y 2 + X 2 Y2 2 = 0 ((Y 1, Y 2 ), (H 1, H 2 ), (0, 0)) h 1 (X 1, X 2 ) h 2 (X 1, X 2 ) B := (K[X, Y ] H ) X,Y K[X] X T K[X] X B = (K[X, Z] g(z) ) X,Z,

15 g H 1 T := X 1 /Y 1 = 1 + X 2 Y 2 + 2X 2 Y 1 / X, Y Y 2 T = 1 + X 2 Y 2 + 2X 2 Y 1 = 1 + X 2 Y 2 + 2X 2 X 1 T H 2 T 4 T 3 X 2 2T 2 + X 2 1T 2 4X 1 X 2 T 2 X 2 1T + 2X 1 X 2 T X 2 X X 2 2X 2 1 = 0 Y 2 T K[X] X z := T 1 X, Y g(z) := Z 4 + 3Z 3 + 3Z 2 + X 2 1Z 2 X 2 2Z 2 4X 1 X 2 Z 2 6X 1 X 2 Z + Z + X 2 1Z 2X 2 2Z X 2 2 2X 1 X 2 X 2 X X 2 2X 2 1 = 0 g(0) = X 2 2 2X 1X 2 X 2 X X2 2 X2 1 X g (0) = 1 K U := X 1 Y 1 V := X 2 Y 2 H 2 Y 2 (1 + U + V ) = X 2 UX 1 Y 2 (1 + U + V )T 2 = X 2 T 2 X 3 1 W := (1 + U + V )T 2 = (1 + U + V )(z + 1) 2 W := T X T 3 2X 1 X 2 T K[X, Z] W / X, z K[X, Z] X,z Y 1 = X 1 T = X 1 z + 1, Y 2 = Y 2W W = X 2 (z + 1) 2 X 3 1 (z + 1)X (z + 1)3 2X 1 X 2 (z + 1). B = (K[X, Z] g(z) ) X,z X := (X 1,..., X n ) T X := X X i T X X α := X α 1 1 Xn αn α := (α 1,..., α n ) N n 0 < T X w (R + ) n GL(n, R) (α) := n i=1 α i w i X α < X β (α) < (β) w i X i X α = α T X N n 0 l N 0 X α (X α ) = l 1 K K[[X]] h K[[X]] K[X] Q K[X, T ] Q(X, h(x)) = 0

16 T K[[X]] alg K[[X]] K[[X]] alg := {h K[[X]] : h K[X]}. h K[[X]] h = α N n h αx α h α K h 0 (i) := (α)=i h αx α h = h (i), i=0 (h) := {X α : h α 0}, h, (h) := mín < {Xα : h α 0}, h, (h) := h α X α, h X α = (h), (h) := h α, h X α = (h), (h) := h (i), h h (j) = 0 j < i, (h) = (h) := mín{(α) : h α 0}, h. R K K[X] R K[[X]] S := {f R \ {0} : (f) = 1}, S 1 R = { f g : f, g R, (g) = 1}, R m = { f : f, g R, g m g(0) = 0}, 1 + g m := X K[[X]] R h = (1 + g) 1 f R m I R m (h) : = (f) (h) : = (f) (I) : = (h) : h I K[X] K[[X]] alg p H 1,..., H p (K[X])[Y ] = K[X 1,..., X n, Y 1,..., Y p ]

17 H 1,..., H p Y 1,..., Y p det((c i,j ) i,j=1,...,p ) 0 c i,j := H i Y j (0, 0) i, j = 1,..., p H i H i (X, Y ) = p c i,j Y j + A i (X, Y ), A i X + Y 2, i = 1,..., p; j=1 h 1,..., h p K[[X]] alg h j (0) = 0 j = 1,..., p H i (X, h 1,..., h p ) = 0 i = 1,..., p H 1,..., H p (c i,j ) c i,j = 0 i < j C := (c i,j ) D := (d i,j ) K D C =: (t i,j ) Hi (X, Y ) := r j=1 d i,jh j (X, Y ) Hi (X, Y ) = r j=1 t i,jy j + A i (X, Y ) A i X + Y 2 Hi (X, h 1,..., h p ) = 0 i = 1,..., p H := {H 1,..., H p } det(c i,j ) 0 (i < j c i,j = 0). h 1,..., h p K[[X 1,..., X n ]] alg H 1 = 0,..., H p = 0 H h j h j H (Y, H, 0) h 1,..., h p K[[X]] alg K[X 1,..., X n, h 1,..., h p ] = (K[X])[h 1,..., h p ] =: P [H], P := K[X] K[X 1,..., X n, h 1,..., h p ] X,h =: P [h 1,..., h p ] X,h K[[X]] alg. σ H : (K[X, Y ] H ) X,Y K[[X 1,..., X n ]] Y j σ H (Y j ) := h j, j = 1,..., p ker(σ H ) H (σ H ) = P [H] ker(σ H ) P [Y 1,..., Y p ] X,Y = H P [Y 1,..., Y p ] X,Y

18 P [H] X,Y = P [h 1,..., h p ] X,h = P [Y1,..., Y r ] X,Y H H P [Y ] X,Y (n + p) (0, 0) V(H 1,..., H p ) ker(σ H ) = H P [Y 1,..., Y p ] X,Y P [H] X,Y = P [Y1,..., Y p ] X,Y H σ H P [Y 1,..., Y p ] X,Y P [h 1,..., h p ] X,h σ H (Y, H, 0) h := (h 1,..., h p ) H K[X, Y ] X,Y (K[X, Y ] H ) X,Y = K[X, h] X,h. (Y, H, 0) A H A H := (K[X, Y ] H ) X,Y. H := {H 1,..., H p } K[X, Y ] h 1,..., h p K[[X]] alg d i := (H i ) d := p i=1 d i j = 1,..., p Q j P [T ] (Q j ) d Q(X, h j (X)) = 0 G P [Y 1,..., Y p ] m g = σ H (G) Q P [T ] (Q) dm Q(X, g(x)) = 0. g(x) = G(X, h 1 (X),..., h p (X)) P [H] G := G 0 1+G 1 P [Y 1,..., Y p ] X,Y (G 0 ) m (G 1 ) m G 1 (0, 0) = 0 g := σ H (G) P [H] X,Y Q P [T ] (Q) (m + 1)d Q(X, g(x)) = 0. H := {H 1,..., H p } h 1,..., h p K[[X]] alg d i := (H i ) i = 1,..., p d := p i=1 d i G P [Y ] m := (G) g = i=0 g (i) := σ H (G) g (i) i

19 Q(X, T ) := s i=0 q it i P [X, T ] s dm i = 0, 1,..., s q i P (q i ) dm i Q(X, T ) Q(X, g(x)) = 0 Q (X, T ) := T s + s i=1 qs i 1 s q i T i P [T ] u(x) = i=0 u (i) := q s g K[[X]] alg g = 0 g (i) = 0 i dm g P g (i) = 0 i : dm < i d 2 m 2 g P X u P u (i) = 0 i : (dm + 1) 2 /4 i (dm + 1) 4 /16. g P T g Q (T g) (Q) dm g = g : = g : = g (i) = g + g, i=0 dm g (i) i=0 i=dm+1 g (i) = i=d 2 m 2 +1 Q(X, T ) = s i=0 q it i s dm i = 0, 1,..., s q i P (q i ) dm i g (X) K[[X]] alg g (i). 0 = Q(X, g(x)) = Q(X, g (X) + g (X)) = Q(X, g (X)) + g (X)g (X) (Q(X, g )) d 2 m 2 (g ) > d 2 m 2 Q(X, g (X)) = 0 T g Q(X, T ) Q T g = Q = T g g = g

20 u = q s g Q (X, T ) Q (X, q s (X)g(X)) = q s sg s + = q s sg s + s i=1 s i=1 = q s 1 s (q s g s + = q s 1 s ( qs s i 1 q i qsg i i qs s 1 q i g i s q i g i ) i=1 s q i g i ) i=0 = q s 1 s Q(X, g(x)) = 0. (a + 1) 2 4b(a b + 1) = (a 2b + 1) 2 0 b(a b + 1) (a + 1) 2 /4 (Q ) s(dm s + 1) (dm + 1) 2 /4 g = (1 + G 1 ) 1 G 0 P X 1 + G 1 q s u = (1 + G 1 ) 1 G 0 q s P u Q (X, T ) g = u q s K(X) K[[X]] alg = P X. u P u (i) = 0 i : (dm + 1) 2 /4 i (dm + 1) 4 /16. G g := σ H (G) g 0 (g) Q P [T ] (Q) dm Q(X, g(x)) = 0. g g = dm i=0 g (i) g = 0 g = 0 g 0 (g) = (g ) g d 2 m 2 g (dm + 1) 4 /16 Q = dm i=0 q it i dm i=0 q i (g ) i

21 R K[X] R K[[X]] I R {g 1,..., g s } I g R R {g 1,..., g s } g R h i R g = i h ig i (h i )(g i ) (g), i h R R {g 1,..., g s } g h R {g 1,..., g s } h = 0 (h) / (g 1 ),..., (g s ) h (g, {g 1,..., g s }, R) (I) := (g) : g I K[X] {g 1,..., g s } R I {(g 1 ),..., (g s )} (I) R {g 1,..., g s } g R R g {g 1,..., g s } p I G := {g 1,..., g s } I 0 g G g I g G g / I g R g R I = g 1,..., g s {g 1,..., g s } 0 (g, {g 1,..., g s }, R) g I (g, {g 1,..., g s }, R) = {0} h (g, {g 1,..., g s }, R)\{0} h / I h (g, {g 1,..., g s }, R) h 0 (h) = (h ) (h) = máx{(f) : g f I} 0 (g, {g 1,..., g s }, R) g R {g 1,..., g s } I f (g, {g 1,..., g s }, R) f 0 g f R {g 1,..., g s } g f I g I f I (f) (I) (f) / (I) f h (g, {g 1,..., g s }, R) \ {0} h / I (h) / (I) g h I g / I 0 / (g, {g 1,..., g s }, R) h (g, {g 1,..., g s }, R) h 0 h h I (h) (h ) (h) (h ) (h h ) = c (h) c K \ {0} (h) (I) f h f I (f) > (h)

22 R g 1,..., g s R I = g 1,..., g s R {g 1,..., g s } R I g R g I g R {g 1,..., g s } g R g I 0 (g, {g 1,..., g s }, R) < X T X < Y T Y < := (< X, < Y ) T X,Y X α Y β < X α Y β X α < X X α (X α = X α Y β < Y Y β ) S < := {f K[X, Y ] : < (f) = 1} {f K[Y ] <Y : <Y (f) = 1} K[X, Y ] < : = S 1 < K[X, Y ] = { f g : f, g K[X, Y ], < (g) = 1} = { f g : f K[X, Y ], g (K[Y ] < Y ) } = (S 1 < Y K[Y ])[X] = (K[Y ] <Y )[X]. < X f K[X, Y ], < (f) K[Y ] f K[Y ]. f / K[Y ] i = 1,..., n; m (f) : X i m < (f) = t / K[Y ] I K[X, Y ] < G := {g 1,..., g s } I G := {g G : (g) K[Y ]} I := I K[Y ] {g 1 mód Y,..., g s mód Y } I f I I G I < i = 1,..., s < (g i ) < (f) < (f) K[Y ] < (g i ) K[Y ] g i G < (g i ) K[Y ] g i K[Y ] g i I G I G I I Y I K[Y ] = I

23 K[X] X K[X] X G, H 1,..., H p K[X] X U, W K[X] U K[X] X U(0) = 1 U 1 W K[X] X G {H 1,..., H p } F 1,..., F r {F 1,..., F r } K[X] X H 1,..., H p G H 1,..., H p R I {g 1,..., g s } h R R g {g 1,..., g s } g h R {g 1,..., g s } h = 0 (h) (g 1 ),..., (g s ) = R {g 1,..., g s } g R R g {g 1,..., g s } K[X] loc g, g 1,..., g s (g, {g 1,..., g s }, K[[X]]) = 0 K[[X]] g(x) Q(X, T ) Q(X, g(x)) = 0 Q Q(X, T ) = 0 g d d N H h := (h 1,..., h p ) h j H j X Y j Y j (0) = 0 α Y j X α α N n

24 h 1 H = 0 h h 1 h I K[X] E := θ P (θ + N n ), #P <. N n \ E E Λ(E) := {γ E : ( i {1,..., n})(x i X γ Xγ X i / E)} j > 0 Λ j (E) := Λ(E \ j 1 i=1 Λi (E)) Λ 0 (E) = T n X Nn := {X γ : γ E} Λ j () := {X γ : γ Λ j (E)} {Λ i (E)} i N E i N X α Λ i () j = 1,..., n X j X α Λ i+1 () 2 δ N n i N X α Λ i () j = 1,..., n X δ X α = X α+δ Λ i+ δ 1 () δ 1 = δ 1 + δ δ n E \ j 1 i=1 Λi (E) N n Λ j (E) = {γ N n : ( δ N n, δ 1 = j)(x δ X γ Xγ X δ / E)} F j := {γ N n : ( δ N n, δ 1 = j)(x δ X γ Xγ / E)} X δ j = 0 j k : Λ j (E) = F j j = k + 1 Λ k+1 (E) = F k+1 ξ Λ k+1 () = Λ(E \ k i=1 Λi (E)) ξ E ξ / k i=1 Λi (E) i = 1,..., n X i X ξ Xξ X / \ k i i=1 Λi () X ξ X i Λ k () = F k.

25 δ N n : δ 1 = k X δ X ξ X i X δ := X δ X i X ξ Xξ X δ X i = Xξ X δ Xξ /X δ X i / / δ 1 = k + 1 ξ F k+1 i = 1,..., k 1 X ξ X Λ i () = F i i δ N n : δ 1 = i, X δ Xξ X Xξ / X δ i X δ X i X ξ Xξ / X ξ Λ i+1 () ξ k X δ i=1 Λi (E) ξ F k+1 δ N n δ 1 = k + 1 X δ X ξ Xξ / X δ δ 1 = k + 1 > 0 i = 1,..., n : X δ = X i X δ δ N n δ 1 = k X X i ξ X δ X δ X ξ X δ Λ k+1 (E) = Xξ /X δ X i Xξ / Xξ Λ() 2 X δ X δ = X ξ Λ 1+ δ 1 () = Λ k+1 () ξ Λ k+1 (E) F k+1 Λ j (E) = F j j N d := #(N n \E) < X 2, Y 2, Z 2, XZ, Y Z B := {1, X, Y, Z, XY } Λ(B) := {X 2, X 2 Y, XY 2, Y 2, XZ, XY Z, Y Z, Z 2 } Λ 2 (B) : = {X 3, X 3 Y, X 2 Y 2, XY 3, Y 3, X 2 Z, X 2 Y Z, XY 2 Z, Y 2 Z, XZ 2, XY Z 2, Y Z 2, Z 3 } t Λ(B) B T X F t T X t B B (t) (t) i N t Λ i (B) p K[X] \ {0} p B B p B (p) := máx{ B (t) : t (p)} t B B (t) = 0 T X \ B

26 (V, m, κ) m κ := Vm X := (X 1,..., X n ) V[X] F V[X] F κ[x] F mód m m V I V[X] m,x I 0 = I κ[x] X F F I I 0 κ[x] X I 0 κ V[X] m,x I V < T X E E = s i=1(β i + N n ) β i E F := N n \ (E + N n ) F κ κ[x] (X) I 0 X β I 0 mód I 0 κ F X β I 0 X I 0 (X α, I 0 ) B := Λ(E) E I 0 g α {(g α )} I 0 {g α := X α (X α, I 0 ) : α B} g α = X α + γ F α c α,γ X γ c α,γ κ F α := {γ F : γ < α} α B q α,i κ[x] i = 1,..., r P α (0) = 0 (1 + P α )g α = r q α,i F i i=1 g α (1 + P α )g α I

27 G α G α : = = = r q α,i F i i=1 r q α (F i + m α,i Fα,i) i=1 r q α,i F i + i=1 r q α,i m α,i F α,i i=1 m α,i m, F α,i V[X] = g α (1 + P α ) + m α H α m α m, H α V[X] X α G α V[X] 1 + d α d α m G α X β β < α m A := {A α,γ } α B G α := X α + γ F α (c α,γ A α,γ )X γ. G α { G α } X β G α X α : α B X β = X µ X α X µ Gα A V X β β F β B G β γ F α A α,γ X γ β E \ B β 1 X β = X i X α X i Gα X γ X i X γ F F E 0 E m X Gα ; m X X m X v; v V m X m A ; m A A G α = Q α,α (A, X) F α + R α,γ (A, X)X γ α B γ F α Q α,α V[A, X] R α,γ V[A]

28 (G α ) = α B (Q α,α )( G α ) G α m A u V Gβ β < α m A A u V V G α = g α + g α P α + m α H α g α m A m α H α m α H α m g α P α X σ g α P α G α u V σ < α σ α ux σ ux σ α 1 Gα1 um A X σ α 1+α 2 α 2 α 1 u m A Xσ G G α (1 + d α ) G α 1 + d α X α d α m (1+ α )m A Gβ α β (1 + d α ) G α X α m A (1 + d α )X σ σ < α σ < β β < σ S := {R α,γ = 0 : α B, γ F α } l := #{(α, γ) B F : γ F α } α B #F α #{(α, γ) B F : γ F α } l κ[a] S 0 κ l

29 G α = g α (1 + P α ) + m α H α = g α (1 + P α ) = Q α,α (0, X) g α + R α,γ (0)X γ α B γ F α = Q α,α (0, X)g α + R α,γ (0)X γ α B γ F α {g α } I 0 R α,γ (0) S 1 0 R α,γ γ α (α, γ) < (α, γ ) γ < γ (γ = γ α > α ). α B γ F α R α,γ A α,γ 1 A α,γ (α, γ ) < (α, γ) X α G α γ F α A α,γ X γ A α,γ X γ R α,γ X γ R α,γ X β G α E α 1 B µ 1 N n β = α 1 + µ 1 X µ 1 Gα1 A α1,γ Xµ 1+γ γ 1 : µ 1 +γ 1 = γ F α R α,γ R α,γ µ 1 0 γ 1 < γ µ 1 = 0 α 1 = β > α A X σ G α g α c α,γ X γ : γ F α G α R α,γ G α m A (A α,γ ) V l ξ := (ξ α,γ ) α B,γ Fα (α, γ) B F α : ξ α,γ V, ξ α,γ = A α,γ = 0 R α,γ (ξ) = 0. Q ξ α,α (X) := Q α,α (ξ, X) G ξ α V[X] G α ξ A α,γ α B, γ F α = 0 G ξ α = g α R α,γ (ξ) = 0

30 det(q ξ α,α ) V[X] X,m Q ξ α,α (0) = Q α,α (0, 0) = δ α,α α α Q ξ α,α (0) / m det(q ξ α,α ) V[X] X,m Q α,α (0, 0) α < α Q α,α (0, 0) = 1 G ξ α = α B Q ξ α,α (X) G ξ α α B G ξ α I α B G ξ α = X α + γ F α ξ α,γx γ I X V[X]I {X α : α F } V {X α : α F } V X,m [X]I H/(1 + Q) H, Q V[X] Q(0) = 0 Q m, X D := {D γ : γ F } X D γ V H (1 + Q)( γ F D γ X γ ) { G ξ α = X α + ξ α,γ X γ : α γ F α B} X E R γ X γ R γ D γ V {R γ = 0} D γ 1 (1 + Q) d := (d γ ) d γ V H = (1 + Q)( γ F d γx γ ) mód G ξ α mód I V K m V V

31 I F 1,..., F r I e K[X] (X) X m IK[X] (X) X α α α n = m u α X α = r Q i F i, i=1 u α Q i F i u α u α (0,..., 0) = 1 u α X α Q i /u α < (f) < (I e ) < F 1,..., F r < (I e ) = M(F 1 ),..., M(F r ) X m F i m X α X m α α n = m g x α I e x α u α = r F i Q i + h i=1 Q i h (Q i )(F i ) x α = x α M(h) / M(I e ) u α (0,..., 0) = 1 h 0 (h) (h) < x α x α M(Q i )M(F i ) M(h) (h) > x α (h) m h (h) m h X m = (I e ) h = 0 x α u α = r i=1 F iq i (Q i )(F i ) (x α ) = x α G(X) F i h 0 <

32 G = r H i F i + h 0 + h 1 i=1 (H i )(F i ) (G) (h 1 ) m (h 0 ) (I e ) h 1 I e G mód I e = h 0 h 1 {F i : i = 1,..., r} F i G h 0 F i F i h 0 G I e V (R := K[T ], m := T ) T := (T 1,..., T n ) R h R h = lím A[Z] q g(z) g(z) q A[Z] m A (Z, g, 0) G := {G 1,..., G s } V[X] (Y := (Y 1,..., Y p ), H := {H i (T, Y ) = 0 : i = 1,..., p}, 0) A H = K[T, Y ] T,Y H G j (T, Y ; X) K[T, Y ][X] X T Y Y y 1,..., y p K[[T ]] alg V V[X] V[X] A H I := G K[[T ]] K[[T, X]] K[T ] T (K[T, Y, X] H ) T,Y,X K[[T ]] K[[T, X]] K[T ] T (K[T, Y, X] H ) T,Y,X T T, X

33 K[[T ]] alg K[T ] T V[X] X I (V[Y, X] H, G ) T,Y,X A H (T, Y ) = (0, 0) j = 1,..., s : G j : = G j (T, Y ; X) mód T, Y = G j (0, 0; X) K[X] K D := V[X] X I C := K[X] X I 0 I 0 := {G j (0, 0; X) : j = 1,..., s}. D d := dim K C < C K B T X I 0 I 0 X α + γ F α ξ α,γ X γ, α B, F α I 0 ξ α,γ V = K[[T ]] alg A H A H ({A α,γ, Y i : (α, γ) B F α, i = 1,..., p}, H := {R α,γ (T, Y ; A), H j : (α, γ) B F α, j = 1,..., p}, 0). H Θ V[X] I X γ γ F α V Θ mód I = γ F v γ X γ. A H

34 (V, m, κ) := (K[z] z, z, K) V[X] := V[x, y] x, y, z F 1 := xy + xy 2 + y 3 + 2x 3 y + xyz 2 F 2 := x 2 + xy 2 + y 3 + x 3 y + xy 2 z 3 F 3 := x 2 + xy + 2xy 2 + x 3 y y 3 z 4 F 1 := xy + xy 2 + y 3 + 2x 3 y F 2 := x 2 + xy 2 + y 3 + x 3 y F 3 := x 2 + xy + 2xy 2 + x 3 y I := F 1, F 2, F 3 I 0 := F 1, F 2, F 3 I 0 F 1 F 2 + F 3 = 2xy(1 + x 2 + y) xy I }{{} 0 u F 3 F 2 = xy(1 + y) y 3 y 3 I }{{} 0 v F 3 = x 2 + xy(1 + 2y + x 2 ) x 2 I }{{} 0 w (I 0 ) := x 2, xy, y 3 Λ(I 0 ) = {y 3, xy 2, xy, x 2 } {1, x, y, y 2 } y 3 y 2 y 1 xy x 1 x 2 2u xy = F 1 F 2 + F 3 2u y 3 = vf 1 + (2u v)f 2 + (v 2u)F 3 2u x 2 = wf 1 + wf 2 + (2u w)f 3 2u xy 2 = y(f 1 F 2 + F 3 )

35 G 1 := F 1 F 2 + F 3 = 2xy + 2x 3 y + 2xy 2 + xyz 2 xy 2 z 3 y 3 z 4 G 2 := vf 1 + (2u v)f 2 + (v 2u)F 3 = 2y 3 + 2x 2 y 3 + 2y 4 + xyz 2 + xy 2 z 2 + xy 2 z 3 + 2x 3 y 2 z 3 + xy 3 z 3 + y 3 z 4 + 2x 2 y 3 z 4 + y 4 z 4 G 3 := wf 1 + wf 2 + (2u w)f 3 = 2x 2 + 2x 4 + 2x 2 y xyz 2 x 3 yz 2 2xy 2 z 2 + xy 2 z 3 + x 3 y 2 z 3 + 2xy 3 z 3 y 3 z 4 x 2 y 3 z 4 G 4 := y(f 1 F 2 + F 3 ) = 2xy 2 + 2x 3 y 2 + 2xy 3 + xy 2 z 2 xy 3 z 3 y 4 z 4 G 1 := xy (a 0 + a 1 x + a 2 y + a 3 y 2 ) G 2 := y 3 (b 0 + b 1 x + b 2 y + b 3 y 2 ) G 3 := x 2 (c 0 + c 1 x + c 2 y + c 3 y 2 ) G 4 := xy 2 (d 0 + d 1 x + d 2 y + d 3 y 2 ) a 0, a 1, a 2, a 3, b 0,..., d 3 16 G 1 G 2 G 3 G 4 G i G 1 G i G 1 G 2 G 4 G 3 G 3 G 4 G 1 G 2 z 4 b 0 d 0 z 3 + 2a 0 c a 0c 3 d 1 + a 0 z 2 + 2b 0 c 1 c 3 + 2b 0 c 3 d 3 + 2a 0 c 0 + 2c 2 d 0 + 2a 0 + 2d 0 + ( b 1 z 4 d 1 z 3 + 2a 1 c a 1c 3 d 1 + a 1 z 2 + 2b 1 c 1 c 3 + 2b 1 c 3 d 3 + 2a 1 c 0 + 2c 2 d 1 + 2a 1 + 2d 1 )x + ( b 2 z 4 d 2 z 3 +2a 2 c a 2c 3 d 1 +a 2 z 2 +2b 2 c 1 c 3 +2b 2 c 3 d 3 +2a 2 c 0 +2c 0 c 1 +2c 2 d 2 +2c 3 d 0 + 2a 2 + 2d 2 )y + ( b 3 z 4 d 3 z 3 + 2a 3 c a 3c 3 d 1 + a 3 z 2 + 2b 3 c 1 c 3 + 2b 3 c 3 d 3 + 2a 3 c 0 + 2c 1 c 2 + 2c 2 d 3 + 2c 3 d 2 + 2a 3 + 2d 3 )y 2 =: E 1 + E 2 x + E 3 y + E 4 y 2 G 2 G i G 2 G 2 G 2 G 4 G 4 G 4 G 3 G 3 G 4 G 2 G 4 G 1 G 2 2b 0 c 0 z 4 + b 1 c 0 z 3 + 2c 0 d 0 z 3 + 2b 3 c 0 d 1 + 2b 1 c 0 c 1 z 4 + 2c 0 c 1 d 1 z 3 + 2b 1 c 0 c 1 + 2d 0 b 1 c 3 + a 0 z 2 + 2a 0 b 1 +(2a 3 b 1 c 2 z 4 +2a 3 b 2 c 1 z 4 +2a 3 b 3 d 2 z 4 +2b 1 c 1 c 3 z 4 +2b 1 c 3 d 3 z 4 +2b 2 b 3 c 3 z 4 +2b 3 c 3 d 1 z 4 + 2b 3 d 2 3 z4 +a 3 b 1 z 4 +2a 3 c 1 d 2 z 3 +2a 3 c 2 d 1 z 3 +2a 3 d 2 d 3 z 3 +2b 0 c 3 z 4 +2b 2 c 2 z 4 +b 2 3 z4 +2b 3 c 3 d 2 z 3 + 2c 1 c 3 d 1 z 3 +4c 3 d 1 d 3 z 3 +2d 3 3 z3 +a 3 b 2 z 3 +b 1 c 3 z 3 +b 2 z 4 +b 3 d 3 z 3 +b 3 z 4 +2c 2 d 2 z 3 +2c 3 d 0 z 3 + d 3 z 3 +2a 3 b 1 c 2 +2a 3 b 2 c 1 +2a 3 b 3 d 2 +a 3 z 2 +2b 1 c 1 c 3 +2b 1 c 3 d 3 +2b 2 b 3 c 3 +2b 3 c 3 d 1 +2b 3 d 2 3 +d 3z 2 + 2a 3 b 1 +2b 0 c 3 +2b 2 c 2 +2b b 2 +2b 3 )y 2 +(2a 2 b 1 c 2 z 4 +2a 2 b 2 c 1 z 4 +2a 2 b 3 d 2 z 4 +2b 1 c 1 c 2 z 4 + 2b 1 c 3 d 2 z 4 +2b 2 2 c 3z 4 +2b 3 c 2 d 1 z 4 +2b 3 d 2 d 3 z 4 +a 2 b 1 z 4 +2a 2 c 1 d 2 z 3 +2a 2 c 2 d 1 z 3 +2a 2 d 2 d 3 z 3 + 2b 0 c 2 z 4 +b 2 b 3 z 4 +2b 2 c 0 z 4 +2b 2 c 3 d 2 z 3 +2c 1 c 2 d 1 z 3 +2c 2 d 1 d 3 z 3 +2c 3 d 1 d 2 z 3 +2d 2 d 2 3 z3 +a 2 b 2 z 3 + b 0 z 4 +b 1 c 2 z 3 +b 2 z 4 +b 3 d 2 z 3 +2c 0 d 2 z 3 +2c 2 d 0 z 3 +d 2 z 3 +2a 2 b 1 c 2 +2a 2 b 2 c 1 +2a 2 b 3 d 2 +a 2 z 2 + 2b 1 c 1 c 2 +2b 1 c 3 d 2 +2b 2 2 c 3+2b 3 c 2 d 1 +2b 3 d 2 d 3 +d 2 z 2 +2a 2 b 1 +2b 0 c 2 +2b 2 b 3 +2b 2 c 0 +2b 0 +2b 2 )y+ 2b 0 b 3 +2b 0 b 2 c 3 z 4 +2b 0 c 3 d 2 z 3 +2a 0 b 1 c 2 z 4 +2a 0 b 3 d 2 z 4 +2a 0 c 1 d 2 z 3 +2a 0 c 2 d 1 z 3 +2a 0 d 2 d 3 z 3 + 2a 0 b 2 c 1 z 4 +2d 0 b 3 d 3 z 4 +2b 3 c 0 d 1 z 4 +2c 0 d 1 d 3 z 3 +d 0 b 3 z 3 +(2a 1 b 1 c 2 z 4 +2a 1 b 2 c 1 z 4 +2a 1 b 3 d 2 z 4 +

36 2b 1 b 2 c 3 z 4 +2b 1 c 2 1 z4 +2b 1 c 3 d 1 z 4 +2b 3 c 1 d 1 z 4 +2b 3 d 1 d 3 z 4 +a 1 b 1 z 4 +2a 1 c 1 d 2 z 3 +2a 1 c 2 d 1 z 3 + 2a 1 d 2 d 3 z 3 +2b 0 c 1 z 4 +b 1 b 3 z 4 +2b 1 c 0 z 4 +2b 1 c 3 d 2 z 3 +2b 3 d 0 z 4 +2c 2 1 d 1z 3 +2c 1 d 1 d 3 z 3 +2c 3 d 2 1 z3 + 2d 1 d 2 3 z3 +a 1 b 2 z 3 +b 1 c 1 z 3 +b 1 z 4 +b 3 d 1 z 3 +2c 0 d 1 z 3 +2c 1 d 0 z 3 +2d 0 d 3 z 3 +b 0 z 3 +d 1 z 3 +2a 1 b 1 c 2 + 2a 1 b 2 c 1 + 2a 1 b 3 d 2 + a 1 z 2 + 2b 1 b 2 c 3 + 2b 1 c b 1c 3 d 1 + 2b 3 c 1 d 1 + 2b 3 d 1 d 3 + d 1 z 2 + 2a 1 b 1 + 2b 0 c 1 +2b 1 b 3 +2b 1 c 0 +2b 3 d 0 +2b 1 )x+2b 0 b 2 c 3 +a 0 b 2 z 3 +d 0 z 2 +2a 0 b 2 c 1 +2a 0 b 1 c 2 +2a 0 b 3 d 2 + a 0 b 1 z 4 + d 0 z 3 + b 0 b 3 z 4 + 2d 0 b 1 c 3 z 4 + 2d 0 c 3 d 1 z 3 + z 4 b 0 + 2d 0 b 3 d 3 + 2d 2 3 z3 d 0 + 2b 0 c 0 + 2b 0 =: E 5 + E 6 x + E 7 y + E 8 y 2 G 3 G i G 3 G 2 G 4 G 4 G 4 G 3 G 3 G 4 G 1 G 2 G 1 G 2 b 0 c 0 z 4 +2b 1 c 0 z 3 +c 0 d 0 z 3 b 0 b 2 c 3 z 4 +b 0 c 3 d 2 z 3 b 0 c 1 c 3 z 2 b 0 c 3 d 3 z 2 b 1 c 0 c 1 z 4 +c 0 c 1 d 1 z 3 + 4b 0 c 2 c 3 a 0 z 2 +( a 1 b 1 c 2 z 4 a 1 b 2 c 1 z 4 a 1 b 3 d 2 z 4 b 1 b 2 c 3 z 4 b 1 c 2 1 z4 b 1 c 3 d 1 z 4 b 3 c 1 d 1 z 4 b 3 d 1 d 3 z 4 + a 1 c 1 d 2 z 3 + a 1 c 2 d 1 z 3 + a 1 d 2 d 3 z 3 b 0 c 1 z 4 b 1 c 0 z 4 + b 1 c 3 d 2 z 3 b 3 d 0 z 4 + c 2 1 d 1z 3 + c 1 d 1 d 3 z 3 + c 3 d 2 1 z3 + d 1 d 2 3 z3 + 2a 1 b 2 z 3 a 1 c 2 1 z2 a 1 c 3 d 1 z 2 b 1 c 1 c 3 z 2 + 2b 1 c 1 z 3 b 1 c 3 d 3 z 2 b 1 z 4 +2b 3 d 1 z 3 +c 0 d 1 z 3 +c 1 d 0 z 3 +d 0 d 3 z 3 +2a 1 b 1 c 2 3 a 1c 0 z 2 +2b 0 z 3 +2b 1 b 3 c 2 3 c 2d 1 z 2 +d 1 z 3 + 4a 1 c 1 c 2 a 1 z 2 + 4b 1 c 2 c 3 + 2c c 1c 3 d 1 2d 1 z 2 + 2a 1 c 1 + 2b 1 c 3 + 4c 0 c 1 + 2c 1 )x + 2b 0 b 3 c 2 3 a 0 b 1 c 2 z 4 a 0 b 3 d 2 z 4 + a 0 c 1 d 2 z 3 + a 0 c 2 d 1 z 3 + a 0 d 2 d 3 z 3 a 0 b 2 c 1 z 4 + 2b 1 c 2 3 a 0 d 0 b 3 d 3 z 4 b 3 c 0 d 1 z 4 +c 0 d 1 d 3 z 3 a 0 c 3 d 1 z 2 +2c 0 c d 0b 3 z 3 +2a 0 b 2 z 3 2d 0 z 2 +2c 2 0 +d 0z 3 d 0 b 1 c 3 z 4 + d 0 c 3 d 1 z 3 a 0 c 2 1 z2 a 0 c 0 z 2 + 4a 0 c 1 c 2 + 2a 0 c 1 z 4 b 0 + 2b 0 c 3 + d 2 3 z3 d 0 c 2 d 0 z 2 + 4c 1 d 0 c 3 + 2c 0 +( a 3 b 1 c 2 z 4 a 3 b 2 c 1 z 4 a 3 b 3 d 2 z 4 b 1 c 1 c 3 z 4 b 1 c 3 d 3 z 4 b 2 b 3 c 3 z 4 b 3 c 3 d 1 z 4 b 3 d 2 3 z4 + a 3 c 1 d 2 z 3 +a 3 c 2 d 1 z 3 +a 3 d 2 d 3 z 3 b 0 c 3 z 4 b 2 c 2 z 4 +b 3 c 3 d 2 z 3 +c 1 c 3 d 1 z 3 +2c 3 d 1 d 3 z 3 +d 3 3 z3 + 2a 3 b 2 z 3 a 3 c 2 1 z2 a 3 c 3 d 1 z 2 + 2b 1 c 3 z 3 b 3 c 1 c 3 z 2 b 3 c 3 d 3 z 2 + 2b 3 d 3 z 3 b 3 z 4 + c 2 d 2 z 3 + c 3 d 0 z 3 +2a 3 b 1 c 2 3 a 3c 0 z 2 +2b 2 3 c2 3 c 1c 2 z 2 c 2 d 3 z 2 c 3 d 2 z 2 +d 3 z 3 +4a 3 c 1 c 2 a 3 z 2 +2b 2 c b 3 c 2 c 3 +2c 2 1 c 3 +4c 1 c 3 d 3 2d 3 z 2 +2a 3 c 1 +2b 3 c 3 +4c 0 c 3 +2c c 2 +2c 3 )y 2 +( a 2 b 1 c 2 z 4 a 2 b 2 c 1 z 4 a 2 b 3 d 2 z 4 b 1 c 1 c 2 z 4 b 1 c 3 d 2 z 4 b 2 2 c 3z 4 b 3 c 2 d 1 z 4 b 3 d 2 d 3 z 4 + a 2 c 1 d 2 z 3 + a 2 c 2 d 1 z 3 +a 2 d 2 d 3 z 3 b 0 c 2 z 4 b 2 c 0 z 4 +b 2 c 3 d 2 z 3 +c 1 c 2 d 1 z 3 +c 2 d 1 d 3 z 3 +c 3 d 1 d 2 z 3 +d 2 d 2 3 z3 + 2a 2 b 2 z 3 a 2 c 2 1 z2 a 2 c 3 d 1 z 2 +2b 1 c 2 z 3 b 2 c 1 c 3 z 2 b 2 c 3 d 3 z 2 b 2 z 4 +2b 3 d 2 z 3 +c 0 d 2 z 3 +c 2 d 0 z 3 + 2a 2 b 1 c 2 3 a 2c 0 z 2 +2b 2 b 3 c 2 3 c 0c 1 z 2 c 2 d 2 z 2 c 3 d 0 z 2 +d 2 z 3 +4a 2 c 1 c 2 a 2 z 2 +2b 0 c b 2c 2 c 3 + 2c 2 1 c 2 + 4c 1 c 3 d 2 2d 2 z 2 + 2a 2 c 1 + 2b 2 c 3 + 4c 0 c 2 + 2c 0 + 2c 2 )y =: E 9 + E 10 x + E 11 y + E 12 y 2 G 4 G i G 4 G 2 G 2 G 4 G 4 G 4 G 3 G 3 G 4 G 1 G 2 a 0 b 1 z 4 b 0 b 3 z 4 a 0 b 2 z 3 b 1 c 0 z 3 d 0 b 3 z 3 + 2a 0 c 1 d 2 + 2a 0 c 2 d 1 + 2a 0 d 2 d 3 + 2b 0 c 3 d 2 + 2c 0 c 1 d 1 +2c 0 d 1 d 3 +2c 3 d 1 d 0 +2d 2 3 d 0 +d 0 z 2 +2a 0 b 2 +2b 1 c 0 +2b 3 d 0 +2c 0 d 0 +2d 0 +( a 1 b 1 z 4 b 1 b 3 z 4 a 1 b 2 z 3 b 1 c 1 z 3 b 3 d 1 z 3 b 0 z 3 + 2a 1 c 1 d 2 + 2a 1 c 2 d 1 + 2a 1 d 2 d 3 + 2b 1 c 3 d 2 + 2c 2 1 d 1 + 2c 1 d 1 d 3 + 2c 3 d d 1d d 1z 2 + 2a 1 b 2 + 2b 1 c 1 + 2b 3 d 1 + 2c 0 d 1 + 2c 1 d 0 + 2d 0 d 3 + 2b 0 + 2d 1 )x+( a 2 b 1 z 4 b 2 b 3 z 4 a 2 b 2 z 3 b 0 z 4 b 1 c 2 z 3 b 3 d 2 z 3 +2a 2 c 1 d 2 +2a 2 c 2 d 1 +2a 2 d 2 d 3 + 2b 2 c 3 d 2 +2c 1 c 2 d 1 +2c 2 d 1 d 3 +2c 3 d 1 d 2 +2d 2 d 2 3 +d 2z 2 +2a 2 b 2 +2b 1 c 2 +2b 3 d 2 +2c 0 d 2 +2c 2 d 0 + 2d 2 )y + ( a 3 b 1 z 4 b 2 3 z4 a 3 b 2 z 3 b 1 c 3 z 3 b 2 z 4 b 3 d 3 z 3 + 2a 3 c 1 d 2 + 2a 3 c 2 d 1 + 2a 3 d 2 d 3 + 2b 3 c 3 d 2 +2c 1 c 3 d 1 +4c 3 d 1 d 3 +2d 3 3 +d 3z 2 +2a 3 b 2 +2b 1 c 3 +2b 3 d 3 +2c 2 d 2 +2c 3 d 0 +2d 3 )y 2 =: E 13 + E 14 x + E 15 y + E 16 y 2 H := {E i = 0 : i = 1,..., 16} H a 0, a 1, a 2, a 3, b 0,..., d 3 H h := (h 1, h 2,..., h 16 ) V 16.

37 G i G 1 := xy (h 1 + h 2 x + h 3 y + h 4 y 2 ) G 2 := y 3 (h 5 + h 6 x + h 7 y + h 8 y 2 ) G 3 := x 2 (h 9 + h 10 x + h 11 y + h 12 y 2 ) G 4 := xy 2 (h 13 + h 14 x + h 15 y + h 16 y 2 ) V[X] m,x (Vm)[X]I 0 V D 1 D 2 D z 1 + x 2 + z 2 D z 1 + x 2 + z 2 = D 0 + D 1 x + D 2 y + D 3 y 2, P := 1 + z (1 + x 2 + z 2 )(D 0 + D 1 x + D 2 y + D 3 y 2 ). P G 1 G2 G2 G3 G3 G3 G4 G4 G3 G3 G 1 D 1 h 1 h 12 h 16 D 1 h 16 h 10 h 9 D 2 h 1 h 2 12 D 2h 12 h 10 h 9 D 0 z 2 D 1 h 13 h 14 D 1 h 15 h 9 D 2 h 10 h 13 D 2 h 11 h 9 D 3 h 13 h 6 D 3 h 8 h 5 D 3 h 7 h 9 D 0 h 13 D 0 + z ( D 1 h 2 10 h 16 D 1 h 12 h 16 h 2 D 2 h 2 10 h 12 D 2 h 2 12 h 2 D 1 h 10 h 15 D 1 h 2 14 D 1z 2 D 2 h 10 h 11 D 2 h 10 h 14 D 3 h 10 h 7 D 3 h 14 h 6 D 3 h 6 h 8 D 0 h 14 D 1 h 13 D 2 h 9 D 3 h 5 D 1 )x+( D 1 h 10 h 11 h 16 D 1 h 12 h 16 h 3 D 2 h 10 h 11 h 12 D 2 h 2 12 h 3 D 1 h 11 h 15 D 1 h 14 h 15 D 1 h 16 h 9 D 2 h 10 h 15 D 2 h 2 11 D 2h 12 h 9 D 2 z 2 D 3 h 11 h 7 D 3 h 15 h 6 D 3 h 7 h 8 D 0 h 15 D 2 )y + ( D 1 h 10 h 12 h 16 D 1 h 12 h 16 h 4 D 2 h 10 h 2 12 D 2h 2 12 h 4 D 1 h 11 h 16 D 1 h 12 h 15 D 1 h 14 h 16 D 2 h 10 h 16 2D 2 h 11 h 12 D 3 h 12 h 7 D 3 h 16 h 6 D 3 h 2 8 D 3z 2 D 0 h 16 D 3 )y 2 =: S 1 + S 2 x + S 3 y + S 4 y 2. {S 1 = 0, S 2 = 0, S 3 = 0, S 4 = 0} D 0, D 1, D 2, D 3 h 1,..., h 12

38 A 1 m k := Am A A m dim k mm 2 = 1 1 A a A aa R Q(R) R Q(R) p Spec R (p) = 1 R p f R f f R f R p f p f p 1 p p = Anul b + f R f R p p e := pr p (p e ) 1 := {x Q(R) : x p e R p } R p (p e ) 1 b f Q(R) f (pe ) 1 (p e ) 1 t 1 t2 t 1 p t 2 / p b t 1 b = λf λ R b f t1 t2 = λ t 2 R p 1

39 b f (pe ) 1 (p e ) 1 p e R p b b = b f f b / p R p (p e ) 1 p e R p p e t p e \ (p e ) 2 p e = 0 p = 0 t p e t (p e ) 1 R p R p t (p e ) 1 p e t (p e ) 1 p e = tr p (p e ) 2 t t (p e ) 1 = R p tr p = p e p e R p p e = a R p a R p e f R p a R p f f 1 R p f R a R p p e a R p 1 p R p e R p R R = (p)=1 R p Q(R) R x := x 1 x 2 Q(R) x i R x 1 / x 2 R p 1 x / R p x 2 R x 2 R 1 x 2 R = r i=1q i p i q i x 1 / x 2 R x 1 / q 1 x 1 / x 2 R p1 = q 1 R p1 (A, m, k) A m ( ) A ( ) B= ( dimk=am mm 2) m m A m m m m B M A ξ 1,..., ξ l M ξ 1,..., ξ r (r l) MmM Am ξ 1,..., ξ l M N = ξ 1,..., ξ r A M N M N + mm = M x M x+mm Mm ξ 1,..., ξ r λ 1,..., λ r A x + mm = r i=1 λ iξ i x r i=1 λ iξ i mm N + mm = M N = M

40 (A, m, k := Am) m A d := dim A = m = dim k mm 2 m d (A, m, k) A m m (A) = k[x 1,..., X r ] m  = k[[x 1,..., X r ]] r A m A p 1 A p p e Âp k(p)[[t ]] k(p) = A p p e A p Âp = k(p)[[t ]] A p p e T h 1,..., h p K[[X]] alg h j (0) = 0 j = 1,..., p H i (X, h 1,..., h p ) = 0 i = 1,..., p m A A x i := X i i = 1,..., n m A = x 1,..., x n. A mód H 1,..., H p X : (K[Y ])[X] K n F X (F ) := ( F F (0, 0),..., (0, 0)) X 1 X n X X (0, 0)

41 X (X i ) = (0,..., 1 K n X, Y F := n a i X i + i=1 }{{},..., 0) =: e i { X (X 1 ),..., X (X n )} i X k (X i X j ) = 0 k i, k j, X i k = i, X j k = j, 2X k k = i = j, X k (X i X j ) (0) I := φ : K[X, Y ] K n p b j Y j + P (X, Y ) φ(f ) := (a 1,..., a n ) j=1 P X 2, Y 2 H i Y j K ker φ = I 2 II 2 φ K n dim A = dim k II 2 = n. A x 1,..., x n