Jacobson Prime Avoidance

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2 Jacobson Prime Avoidance a a

3 1 Z 1 2 Z 2 Z ( 2 Z ) (2) 2 Z a (a) (2) = ( 2) 1.1 A a, b A (a) = (b) (0) A u b = au 1.2 A 1.1 b = au, a = bv A u, v a = auv A uv = 1 u A 3 A A A P(A) A/ P(A) a (a) 1.3 Z ±1 Z/ Z Z 1.1 Z 1 Z 2 Julius Wilhelm Richard Dedekind ( ) 3 a, b A b = au A u a b A 3

4 1.4 K K[X] K K[X] K = K \ {0} K[X]/ K[X] K[X] 1.1 K[X] A A U(A) Jacobson 2.1 A A A Jacobson J J(A) 2.2 A J A Jacobson (1) x J (2) A y 1 xy A x J A y 1 xy A m x J m xy m, 1 m m = A x / J x A m A/m x A/m x / m x 0 x y = 1 y A/m 1 xy m 1 xy A 4

5 2.3 A M M A a A x M ax M 4 (1) a A x, y M a(x + y) = ax + ay (2) a, b A x M (a + b)x = ax + bx (3) a, b A x M (ab)x = a(bx) (4) x M 1 A x = x 2.4 A A A A 2.5 ( ) M A a A A Jacobson J am = M M = 0 M 0 M u 1,..., u n M u n M = am u n = a 1 u 1 + a 2 u a n u n a 1, a 2,..., a n a (1 a n )u n = a 1 u 1 + a 2 u a n 1 u n 1 a n a J a n u n u 1, u 2,..., u n Jacobson 2.7 A A A 5

6 MMaxA M = A\U(A) A J A Jacobson J = A\U(A) 2.8 A a, b A (a) = (b) (0) b = au u A A b (b) = (a) u A b = au u (a) = (b) = (au) = (a)(u) (a) A 2.7 u J (u) J 2.5 (a) = (0) u 3 2 A A a, b A (a) = (b) (0) b = au A u Z/4Z, Z/3Z 2Z/4Z, 0 A = (Z/4Z) (Z/3Z) A m 1 = (2Z/4Z) (Z/3Z), m 2 = (Z/4Z) 0 a = (1, 0), b = (3, 0) A (a) = (b) (0) u = (3, 0) A b = au u m 2 u A 6

7 v = (0, 1) a(u + v) = au + av = b + 0 = b u m 1 v m 1 u + v m 1 u m 2 v m 2 u + v m 2 u + v A u av = 0 v u + v a(u + v) = b A (a) = (b) 0 a = bu u A 3.2 a = bu A u Prime Avoidance 3.1 Prime Avoidance A p xy p x, y A x p y p p A 3.4 a, b A p A a p b p ab p a a\p, b b\p p ab ab\p ab p 3.5 a 1,..., a r A p A a 1,..., a r p r i=1 a i p 3.6 (Prime Avoidance) x A a A p 1,..., p r A p i (x) + a p i r x + y y a p i p j p i p 1,..., p r i=1 p i 7

8 r = 1 y a x+y p 1 0 a x = x + 0 p 1 y a y = (x + y) x p 1 a p 1 (x) + a p 1 r > 1 x + y r 1 i=1 p i y a x + y p r x + y p r a p r y p r x p r (x) + a p r a, p 1,..., p r 1 p r p r a r 1 i=1 p i p r z p r z a r 1 i=1 p i y + z 1 i r 1 x + y p i z p i x + y + z p i x + y p r z p r x + y + z p r y, z a y + z a y + z 3.6 x = a A p 1,..., p r A p i a p i y r i=1 p i y a 3.8 A a, b (a : b) (a : b) = {x A xb a} A (0 : b) b Ann(b) Ann(b) = {x A xb = 0} A S 8

9 (1) x, y S xy S (2) 1 S S M A S A (m, s), (m, s ) M S t(s m sm ) = 0 t S (m, s) (m, s ) M S 1 S 1(sm sm) = 0 (m, s) (m, s) (m, s) (m, s ) t S t(s m sm ) = 0 t(sm s m) = 0 (m, s ) (m, s) (m, s) (m, s ) (m, s ) (m, s ) t, u S t(s m sm ) = 0, u(s m s m ) = 0 tus S tus (s m sm ) = 0 (m, s) (m, s ) M S S 1 M (m, s) m/s S 1 A a s + b t ta + sb =, st a s b t = ab st well-defined a/s = a /s, b/t = b /t u, v S u(s a sa ) = 0, v(t b tb ) = 0 uv(s t (ta+sb) st(t a +s b )) = 0 (ta+sb)/st = (t a +s b )/s t uv(s t ab sta b ) = 0 ab/st = a b /s t S 1 A S 1 A 0/1 1/1 S 1 A A S S 1 M m s + n t tm + sn =, st a s m t = am st well-defined S 1 A S 1 M S 1 A S 1 M M S 9

10 3.10 M A a A (1) S 1 a = a(s 1 A) (2) S 1 (am) = (S 1 a)(s 1 M) 3.11 p A S = A\p (1) p S (2) p S 1 A S 1 p (1) (2) 1/1 S 1 p s (a s) = 0 s, s S, a p s a = s s s a p s s S S 1 p S 1 A b/t S 1 A (b A, t S) b/t S 1 p b p b, t S b/t S 1 A S 1 p S 1 A S 1 A S 1 p S 1 A S 1 p 3.12 A M A p M S = A\p S 1 M p S 1 M M p b = au u A a = (a) = (b), u = (u) a = ua A Max(A) = {m 1,, m r } i S i = A\m i m i S i A S 1 i u m i m i S 1 i A S 1 u S 1 i m i i m i S 1 i A Jacobson 10

11 a = ua S 1 i a = (S 1 i u)(s 1 i a) S 1 i a S 1 i A 2.5 S 1 i a = 0 sa = 0 s S i s Ann(a) s m i Ann(a) m i u + Ann(a) m i u m i m i u + Ann(a) m i 3.6 r u + v v Ann(a) u + v A u + v A i=1 m i a(u + v) = au + av = b + 0 = b u + v A u a bu A 4.1 A K A a, b 1.1 a, b A, (a) = (b) (0) A u a bu B = K[X, Y, Z]/(X XY Z) a = x, b = xy 1.1 π : K[X, Y, Z] B x = π(x), y = π(y ) A a, b A (a) = (b) (0) A u a = bu (a) = (b) b = ac, a = bd c, d A a = acd K[X, Y, Z] A γ K[X, Y, Z] γ A X a Y c Z d 11

12 a = acd γ(x XY Z) = 0 B = K[X, Y, Z]/(X XY Z) π : K[X, Y, Z] B γ γ π = γ γ K[X, Y, Z] A B π B (x) = (xy) (0) (a 0 x 0 ) v U(B) xy = xv B γ γ A x a xy ac = b γ(xy) = ac = b γ(xv) = γ(x) γ(v) = a γ(v) b = a γ(v) γ γ(v) U(A) A u a bu B A B B A A A i (i Z 0 ) A 4 (1) i, j Z 0 A i A j A i+j (2) A = nz 0 A i A i A i 4 (A, {A i } iz 0 ) 4.5 (2), (3) A {A i } iz 0 12

13 4.3 A I A a I a = a 0 + a a t ( 0 i t a i A i a ) i a i I I 4.4 A, A A i, A i f : A A (1) f (2) i f(a i ) A i 4.5 K (1) A := K[x] A 0 = K, A 1 = Kx, A 2 = Kx 2,..., A i = Kx i,... A (2) A := K[x 1, x 2 ] A 0 = K, A 1 = Kx 1 + Kx 2, A 2 = Kx Kx 1 x 2 + Kx 2 2,..., A i = Kx u 1 x v 2,... A u+v=i u,v 0 (3) A := K[x 1, x 2 ] A 0 = K[x 1 ], A 1 = K[x 1 ]x 2, A 2 = K[x 1 ]x 2 2,..., A i = K[x 1 ]x i 2,... A 4.6 A I A (1) I (2) I A A = iz 0 A i (1) I a a = a 0 +a 1 + +a k (1) a i A i I [ (Ai {0}) I ] T T I iz 0 T A (T ) (T ) I 13

14 I A i I a i I (T ) I = (T ) I I I a a = b λ1 a λ1 + + b λt a λt b λi A a λi I A di a A a = c 0 + c c k c i A i i c i I b λi A b λi = e i0 + e i1 + + e iki e ij A j a a = (e e 1k1 )a λ1 + + (e t0 + + e tkt )a λt = e 10 a λ1 + + e 1k1 a λ1 + + e t0 a λt + + e tkt a λt c l = j+d u =l e uj a λu a λk I c l I I 4.7 A I A (1) A/I = iz 0 A i /(I A i ) (2) A A/I φ A i ψ i A φ A/I µ i A i /kerµ i A i A ψ i A A/I φ φ ψ i µ i Kerφ A i = Kerµ i φ A i φ(a i ) A i /Kerµ i 14

15 (1) A/I φ 0 i φ(a i ) A/I A/I φ(a i ) 4.2 A A = A i A = A i φ(a) = φ(a i ) iz 0 iz 0 iz 0 φ(a i ) a i k a i = 0 i a i = 0 A i b i a i = φ(b i ) k φ(b i ) = 0 φ φ( k b i ) = 0 i=0 k b i = 0 Kerφ Kerφ = I I i=0 i b i I i a i = 0 A/I iz 0 φ(a i ) φ 0 i, j φ(a i A j ) = φ(a i )φ(a j ) A φ(a i )φ(a j ) = φ(a i A j ) φ(a i+j ) A/I (2) φ A A/I 0 i A i /(A i I) = φ(a i ) φ i=0 i=0 4.2 K B = K[X, Y, Z] (X XY Z) B K \ {0} B 4.8 B x, y, z 1, 0, 0 15

16 R = K[X, Y, Z] R X, Y, Z 1, 0, 0 R i = K[Y, Z]X i R = R i iz 0 I = (X XY Z) I A X XY Z R 1 I R 4.7 ζ : R R/I = B R/I = B B B = R i /(I R i ) iz 0 S = K[X, Y, Z] (1 Y Z) 4.9 S x, y, z 1, 0, 0 R = K[X, Y, Z] X, Y, Z 1, 0, 0 1 Y Z R 0 (1 Y Z) S = K[X, Y, Z]/(1 Y Z) 4.10 S S = ( ) K[X, Y, Z] K[Y, Z] (1 Y Z) = [X] (1 Y Z) K[Y, Z] (1 Y Z) τ : K[Y, Z] K[Y, 1/Y ] Y Y Z 1/Y ker τ = (1 Y Z) (1 Y Z) ker τ ker τ (1 Y Z) u ker τ. u (1 Y Z) u (1 Y Z)v = a 0 + a 1 Y + + a n Y n + a 1 Z + a 2 Z a m Z m 16

17 ( a i K) u (1 Y Z)v ker τ 0 = τ(u (1 Y Z)v) = n i= m a i Y i i a i = 0 u = (1 Y Z)v ker τ = (1 Y Z) K[Y, 1/Y ] (1 Y Z) K[Y, Z] 4.11 S x, y, z 1, 0, 0 U(S) = U(S 0 ) U(S) a = a 0 +a 1 + +a t (a i S i a t 0) d = d 0 + d d s S ( d i S i ) (a 0 + a a t )(d 0 + d d s ) = 1 d s 0 0 a t d s S t+s S t + s = 0 t, s 0 t = s = 0 a d S 0 U(S) = U(S 0 ) 4.12 B S x, y, z 1, 0, 0 1 n B n S n I = (X XY Z) J = (1 Y Z) I J I = (I R i ) J = (J R i ) B n = R n /I n iz 0 iz 0 S n = R n /J n 1 n I n = J n R x, y, z 1, 0, 0 R (X XY Z)R (X XY Z)R = {f(x, Y, Z)(X XY Z) f(x, Y, Z) R } f(x, Y, Z) R f(x, Y, Z) = f 0 (X, Y, Z) + + f l (X, Y, Z) 17

18 f(x, Y, Z)(X XY Z) = f 0 (X, Y, Z)(X XY Z)+ +f l (X, Y, Z)(X XY Z) i f i (X, Y, Z)(X XY Z) R i+1 1 n (X XY Z)R R n = R n 1 (X XY Z) = K[Y, Z]X n 1 (X XY Z) = K[Y, Z](1 Y Z)X n = (1 Y Z)R R n 1 n I n = J n B n S n 4.13 U(B) = K K = K \ {0} B B = B i U(B) iz 0 a a = a 0 + a a k ( a i B i ) a B b ab = 1 b b = b 0 + b b k ( b i B i ) B S η η(a) = η 0 (a 0 ) + η 1 (a 1 ) + + η k (a k ) η n η B n B n S n a B η(a) S 4.11 U(S) = U(S 0 ) η(a) S 0 η 0 (a 0 ) + η 1 (a 1 ) + + η k (a k ) S 0 1 i η i (a i ) = i B i S i a i = 0 B b a 0 b 0 = 1 18

19 a 0 B 0 a = a 0 a B 0 U(B 0 ) = U(K[Y, Z]) B 0 = R 0 /I 0 R 0 = K[Y, Z] I 0 = (X XY Z)R R 0 (X XY Z)R 0 0 I 0 = (0) B 0 = K[Y, Z]/(0) = K[Y, Z] U(B 0 ) = U(K[Y, Z]) U(K[Y, Z]) = K K[Y, Z] Y Z 1 K K[Y, Z] 4.11 U(K[Y, Z]) = U(K) U(K) = K U(K[Y, Z]) = K U(B 0 ) = K B B = K[X, Y, Z]/(X XY Z) (x) = (xy) (0) B u x xyu x, y, z B (x) = (xy) (0) xy (x) (x) (xy) x xyz = 0 x = xyz x = xyz (xy) (x) (xy) (x) = (xy) (0) (xy) = (0) XY (X XY Z) f(x, Y, Z) K[X, Y, Z] XY = f(x, Y, Z)(X XY Z) K[X, Y, Z] K[X] ξ K[X, Y, Z] ξ K[X] X X Y 1 Z 1 X = f(x)(x X) = 0 (xy) (0) u U(B) x xyu u U(B) x = xyu 4.13 U(B) = K u K X XY u = g(x, Y, Z)(1 Y Z)X 19

20 g(x, Y, Z) K[X, Y, Z] X 1 1 Y u = g(1, Y, Z)(1 Y Z) Z 0 1 x = xyu B u 5 B = K[X, Y, Z]/(X XY Z) 1.1 B (x), (1 yz) 5.1 A N A A 5.2 A i) A ii) N A i) ii) A q q A p q q {q i } ii q = ii q i {q i } ii p q A p p N A = p ii) i) N A = qspec(a) q A p N A p p N A = p 5.3 u A u A A/N A ū 20

21 i) u U(A) ii) ū U(A/N A ) i) ii) ii) i) u / U(A) A m u m m m A/N A m A/N A ū m ū A/N A 5.4 C a, b C (a) = (b) (0) a = bu C u a 5.1 a 5.5 C a, b C (a) = (b) a u U(C) a = bu a (b) u C a = bu ā = bū ā(c/n C ) = b(c/n C ) N C C/N C 1.2 ū U(C/N C ) 5.3 u U(C) a a C K C (a) = (b) 0 a = bc, b = ad c, d C a = acd 21

22 K[X, Y, Z] C ξ K[X, Y, Z] ξ C X a Y c Z d a acd = 0 X XY Z Ker ξ a 0 n > 1 ξ(x n ) = a n = 0 X n Ker ξ D n := K[X, Y, Z]/(X XY Z, X n ) C ξ X, Y, Z D n x, y, z x = xyz xd n = xyd n D n x = xyu u U(D n ) ξ(u) C a = ab ξ(u) C 5.4 C a, b 5.4 x = xyu u U(D n ) D n ( D n /xd n K[X, Y, Z]/(X) K[Y, Z] K[Y, Z] xd n xd n pspec(d n) p = N D n x xd n N Dn N Dn 5.2 D n D n D n 5.6 x = xyu D n u v K[X, Y, Z] xy v = x v U(D n ) K[X, Y, Z] K[Y, Z] η K[X, Y, Z] η K[Y, Z] X 0 Y Y Z Z (X XY Z, X n ) Ker η D n K[Y, Z] η v U(D n ) η(v) = η( v) U(K[Y, Z]) 22

23 η(v) X XY v (X XY Z, X n ) l, m K[X, Y, Z] K[Y, Z] X XY v = (X XY Z)l + X n m 1 Y v = (1 Y Z)l + X n 1 m η(1 Y v) = η(1 Y Z)η(l) 1 Y η(v) = (1 Y Z)η(l) Z D n 5.4 a 5.4 [1] M.F.Atiyah; I.G.MacDonald, Introduction to Commutative Algebra (Westview Press, 1969) [2], (, 2000) 23

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11