Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ

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1 Galois ) Galois Galois Galois TaylorWiles Fermat [W][TW] Galois Galois Galois 1 Noether 2 1

2 Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ C Λ Λ Λ /m lim /a a a /a rtin /a C Λ Λ 2

3 Ob C Λ m /a rtin Ob C Λ n M M n n M M n G V n k ρ : G ut k V ) 1.1. ρ Ob C Λ n M G ρ : G ut M) k[g] ψ : M k V ρ, ψ ) D ρ : C Λ Sets Ob C Λ D ρ ) = { ρ } ρ, ψ ) ρ, ψ ) C Λ D ρ 1 ) D ρ 2 ) ρ 1, ψ 1 ) D ρ 1 ) ρ 1 1 2, ψ ) D ρ R ρ Ob C Λ R ρ ρ ρ R ρ ρ V V k ut k V ) = GL n k) G ρ ut k V ) = GL n k) ρ Ob C Λ p : k Hom ρ G, GLn ) ) = { ρ : G GL n ) ρ GLn p ) ρ = ρ } ρ Hom ρ G, GLn ) ) G ρ GL n ) = ut n ) n k = k n Hom ρ G, GLn ) ) D ρ ) 1.1) ρ, ρ Hom ρ G, GLn ) ) ρ, ρ 1.1) ρ = Hρ H 1 H Ker GL n p ) ρ ρ, ψ ) 1.1) ρ Hom ρ G, GLn ) ) ρ, ψ ) ρ ρ, ψ ) V k β 3

4 1.2. ρ Ob C Λ n M G ρ : G ut M) k[g] ψ : M k V ψ β M β ρ, ψ, β ) D ρ : C Λ Sets Ob C Λ D ρ ) = { ρ } D ρ R ρ Ob C Λ R ρ ρ ρ R ρ ρ V k β ut k V ) = GL n k) G ρ ut k V ) = GL n k) ρ Ob C Λ ρ Hom ρ G, GLn ) ) G ρ GL n ) = ut n ) n k = k n n Hom ρ G, GLn ) ) D ρ ) 1.2) ρ : G ut k V ) V k[g] End k[g] V = k ρ R ρ ρ R ρ. 1.2) ρ R ρ Ob C Λ ρ univ Hom ρ G, GLn R ρ ) ) Ob C Λ Hom CΛ R ρ, ) Hom ρ G, GLn ) ) ; f f ρ univ) 2.1) f ρ univ) G ρ univ GL n R ρ ) GL nf) GL n ) 4

5 G Λ[G, n] Λ X g ij g G, 1 i, j n. { 1 i = j Xij e = 0 i j n X gh ij = X g il Xh lj g, h G, 1 i, j n. e G Λ l=1 Hom Λ lg Λ[G, n], ) Hom G, GL n ) ) ; f ρ f 2.2) ρ f g G ρ f g) = fx g ij )) i,j 2.2) ρ : G GL n k) Λ Λ[G, n] k m ρ m ρ Λ[G, n] Λ[G, n] m ρ R ρ R ρ Noether C Λ 2.2) Λ[G, n] R ρ ρ univ : G GL n R ρ ) ρ univ Hom ρ G, GLn R ρ ) ) 2.1) 2.1) Ob C Λ ρ Hom ρ G, GLn ) ) 2.2) ρ f ρ Hom Λ lg Λ[G, n], ) ρ m ρ f ρ m ρ ) m /a rtin a Λ[G, n] f ρ R ρ f ρ,a /a Λ f ρ,a : R ρ /a a f ρ,a Λ ˆf ρ : R ρ Hom ρ G, GLn ) ) Hom CΛ R ρ, ); ρ ˆf ρ 2.1) G = lim H H G ρ GL n k) ρ H H 5

6 ρ H : H GL n k) G H H ρ H R ρ H Ob C Λ ρ H,univ Hom ρ H, GLn R ρ H ) ) G 2 G H H H H ρ H,univ GL n R ρ H ) C Λ R ρ H R ρ H C Λ R H ) H R ρ = lim H R ρ H G H ρ H,univ GL n R ρ H ) H ρ univ : G GL n R ρ ) R ρ H rtin R ρ H Λ[H, n] R ρ H R ρ H G 2 G H H Λ[H, n] R ρ H Λ[H, n] R ρ H Λ[H, n] R ρ H R ρ R ρ H R ρ H R ρ rtin R ρ rtin R ρ H rtin R ρ Ob C Λ C Λ rtin = i lim i Hom CΛ R ρ, ) = lim i Hom CΛ R ρ, i ) = lim i lim H Hom CΛ R ρ H, i ) = lim i lim Hom ρh H H, GLn i ) ) = lim i Hom ρ G, GLn i ) ) = Hom ρ G, GLn ) ) 2.1) 2.1 End k[g] V = k 2.1 V k ut k V ) = GL n k) G ρ ut k V ) = GL n k) ρ G g 1,..., g r ρg i ) M n k) ρg i ) E i M n Λ) 6

7 C Λ M n Λ) M n ) E i E i M n ) M 0 n) = M n )/ i : M 0 n) M n ) r ; M mod ME i E i M) r i=1 π Λ i Λ = id M 0 n ) Λ π Λ : M n Λ) r M 0 nλ) π : M n ) r M 0 n) M n ) r = Mn Λ) r Λ π Λ id M 0 n λ) Λ = M 0 n) Ob C Λ Hom ρ G,GLn ) ) M n ) r π M 0 n ) 2.3) ρ ρg i ) ) r i=1 ρ Hom ρ G, GLn ) ) π E 1,..., E r ) ρ Hom ρ G, GLn ) ) Hom ρ,well G, GLn ) ) 2.3. Ob C Λ ρ Hom ρ G, GLn ) ) m GL n k) H GL n ) Hρ H 1 H GL n ) 1 + m. m m m = 0 m m = 1 m 2 m m 1 M0 n) L m m 1 M n) L mod m m 1 L mod m m 1 m m 1 M0 n) Hom ρ G, GLn ) ) Hom ρ G, GLn ) ) Hom ρ G, GLn ) ) ; ρ 1 + L)ρ 1 + L) 1 L ρ m m L)ρ 1 + L) 1 L mod m m 1 m m 1 M0 n) L mod m m 1 M 0 n) M 0 n) M 0 n); M mod M + L mod m m 1 M0 n) 2.3) m m 1 M0 n) m m = /m m ρ m m ρ m : G GL n m ) 7

8 m m GL n k) H m GL n m ) H m ρ Hm 1 H m 1 + m m m 1 > m 2 H m1 m m 2 H m2 H m m H m GL n ) H H GL n ) 1 + m Hom ρ,well G, GLn ) ) Hom ρ G, GLn ) ) 1.1) 2.3 Hom ρ,well G, GLn ) ) D ρ ) 2.4) 2.1. ρ ρ univ 2.3 H GL n Runiv) ρ well = Hρ univh 1 g G ρ well g) Runiv Λ R ρ R ρ C Λ ρ well Hom ρ,well G, GLn R ρ ) ) ρ univ 2.4) Ob C Λ Hom CΛ R ρ, ) Hom ρ,well G, GLn ) ) ; f f ρ univ ) 2.5) f ρ univ ) G ρ univ GL n R ρ ) GL nf) GL n ) 2.5) R ρ 2.5) ρ Hom ρ,well G, GLn ) ) 2.1) ρ f Hom CΛ R ρ, ) f R ρ f f ρ univ ) = f ρ well) = f Hρ univh 1 ) = GL n f)h)ρ GLn f)h) ) 1 f ρ univ ) ρ 2.3 f ρ univ ) = ρ n = 1 End k[g] V = k n = 1 k p Λ m Λ Λ G ab,p G bel p ρ R ρ Λ G ab,p Λ[[G ab,p ]] 8

9 . ρ ut k V ) = k G ρ ut k V ) = k ρ ρg) k 0 k 0 k p k 0 Witt W k 0 ) Λ W k 0 ) Teichmüller k 0 W k 0 ) W k 0 ) Λ s Λ : k 0 Λ Ob C Λ GL 1 ) π : G G ab,p ρ univ : G GL 1 Λ[[G ab,p ]]); g s Λ ρg) ) πg) rtin Ob C Λ Hom CΛ Λ[[G ab,p ]], ) Hom ρ G, GL1 ) ) ; f f ρ univ ) 2.6) f ρ univ ) G ρ univ GL 1 Λ[[G ab,p ]]) GL 1f) GL 1 ) 2.6) rtin C Λ Λ s Λ : k Λ Λ s : k ρ Hom ρ G, GL1 ) ) h ρ : G ; g ρg)s ρg) ) 1 G h ρ π h ρ G ab,p h ρ : G ab,p Λ[G ab,p ] Λ G ab,p Λ f ρ : Λ[G ab,p ] g G ab,p Λ[G ab,p ] g h ρ g) f ρ Λ ˆf ρ : Λ[[G ab,p ]] Hom ρ G, GL1 ) ) Hom CΛ Λ[[G ab,p ]], ); ρ ˆf ρ 2.6) 9

10 3 k[ϵ] ϵ ϵ 2 = 0 k k[ϵ] k x, y x + yϵ k[ϵ] C Λ R C Λ R Zariski t R t R = Hom CΛ R, k[ϵ]) t R k k k[ϵ] k k[ϵ] k[ϵ] k k[ϵ] = { x + y 1 ϵ, x + y 2 ϵ) k[ϵ] k[ϵ] x, y1, y 2 k } h : Hom CΛ R, k[ϵ] k k[ϵ]) = Hom CΛ R, k[ϵ]) Hom CΛ R, k[ϵ]) = t R t R k f s : k[ϵ] k k[ϵ] k[ϵ]; x + y 1 ϵ, x + y 2 ϵ) x + y 1 + y 2 )ϵ t R t R t R h 1 Hom CΛ R, k[ϵ] k k[ϵ]) Hom C Λ R,f s ) Hom CΛ R, k[ϵ]) = t R a k k t R a m a : k[ϵ] k[ϵ]; x + yϵ x + ayϵ t R = Hom CΛ R, k[ϵ]) Hom C Λ R,m a) Hom CΛ R, k[ϵ]) = t R d ρ) End k V ) g G End k V ) End k V ); ϕ ρg)ϕ ρg) 1 G G d ρ) 3.1. End k[g] V = k R ρ ρ k t R ρ = H 1 G, d ρ) ). V k End k V ) M n k) ρ G GL n k) Hom ρ G, GLn k[ϵ]) ) D ρ k[ϵ]) = t R ρ 3.1) 10

11 Z 1 G, d ρ) ) G d ρ) 1 Z 1 G, d ρ) ) c : G M n k) Z 1 G, d ρ) ) Hom ρ G, GLn k[ϵ]) ) ; c 3.2) 3.1) g 1 + cg)ϵ ) ρg) ) 3.2) Z 1 G, d ρ) ) t R ρ 3.3) k 3.3) R C Λ k Hom k-cont mr /m 2 R + m Λ R), k ) = tr m R /m 2 R + m ΛR) k k m R /m 2 R + m ΛR) R. m 2 k[ϵ] = 0 Hom CΛ R/m 2 R + m Λ R), k[ϵ] ) Hom CΛ R, k[ϵ]) = t R k m R /m 2 R + m Λ R) R/m 2 R + m Λ R) m R /m 2 R + m ΛR) Hom CΛ R/m 2 R + m Λ R), k[ϵ] ) Hom k-cont mr /m 2 R + m Λ R), k ) R ρ Noether 3.3. End k[g] V = k R ρ ρ R ρ Noether H 1 G, d ρ) ) k. 3.1 H 1 G, d ρ) ) k t R ρ k R ρ rtin lim i R i 3.2 t R ρ = Hom CΛ R ρ, k[ϵ]) = lim i Hom CΛ R i, k[ϵ]) = lim i Hom k mri /m 2 R i + m Λ R i ), k ) 11

12 t R ρ k dim k mri /m 2 R i + m Λ R i ) ) i Λ Noether dim k m Λ /m 2 Λ ) dim k mri /m 2 R i + m Λ R i ) ) dim k m Ri /m 2 R i ) dim k m Λ /m 2 Λ ) dim k mri /m 2 R i +m Λ R i ) ) i dim k m Ri /m 2 R i ) i H 1 G, d ρ) ) k 1. K p G = G K H 1 G, d ρ) ) k 2. K S K K G S K Galois H 1 G, d ρ) ) k , 1 C Λ p 1,0 : 1 0 C Λ I = Ker p 1,0 Im 1 = 0 I k ρ 0, ψ 0 ) ρ 0 ρ 0, ψ 0 ) p 1,0 Oρ 0, ψ 0 ) H 2 G, d ρ) k I ) 1. Oρ 0, ψ 0 ) 2. ρ 1 ρ 1, ψ 1 )D ρ p 1,0 ) : D ρ 1 ) D ρ 0 )ρ 0, ψ 0 ). ρ 0, ψ 0 ) ρ 0 Hom ρ G, GLn 0 ) ) γ 1 : G GL n 1 ) GL n p 1,0 ) γ 1 = ρ 0 c : G G d ρ) k I g 1, g 2 G cg 1, g 2 ) = γ 1 g 1 g 2 )γ 1 g 2 )γ 1 g 1 ) 1 + M n k) k I = d ρ) k I c G d ρ) k I 2 c H 2 G, d ρ) k I ) Oρ 0, ψ 0 ) Oρ 0, ψ 0 ) ρ 0 γ End k[g] V = k R ρ ρ i = 1, 2 d i = dim k H i G, d ρ) ) d 1 Krull dimr ρ /m Λ R ρ ) d 1 d 2 d 2 = 0 R ρ Λ d 1 12

13 . d R ρ Noether Zariski k f 0 : k[[x 1,..., X d1 ]] R ρ /m Λ R ρ m k[[x 1,..., X d1 ]] J = Ker f 0 f 0 : k[[x 1,..., X d1 ]]/mj R ρ /m Λ R ρ f 0 k 0 J/mJ k[[x 1,..., X d1 ]]/mj f 0 R ρ /m Λ R ρ 0 ρ R ρ R ρ /m Λ R ρ ρ 0, ψ 0 ) f 0 ρ 0, ψ 0 ) Oρ 0, ψ 0 ) H 2 G, d ρ) k J/mJ ) rtin- Rees J/mJ f Hom k J/mJ, k) T f : d ρ) k J/mJ G id f d ρ) k k = d ρ) Φ : Hom k J/mJ, k) H 2 G, d ρ) ) ; f H 2 G, T f ) Oρ 0, ψ 0 ) ) Φ f Hom k J/mJ, k) Φf) = 0 f 0 k[[x 1,..., X d1 ]]/mj Ker f R f 0 : R R ρ /m Λ R ρ f 0 k 0 k R f 0 R ρ /m Λ R ρ 0 Φf) = 0 f 0 ρ 0, ψ 0 ) ρ R ρ, ψ ) D ρ f 0) ρ 0, ψ 0 ) R k ρ, ψ ) R ρ R g 0 : R ρ /m Λ R ρ R f 0 g 0 = id R ρ /m Λ R ρ g 0 Zariski g 0 f 0 Φ k[[x 1,..., X d1 ]] J dim k J/mJ Krull dimr ρ /m Λ R ρ ) d 1 dim k J/mJ Φ dim k J/mJ d 2 d 2 = 0 Φ J = 0 f 0 Krull dimr ρ /m Λ R ρ ) = d 1 Λ[[X 1,..., X d1 ]] f R ρ k[[x 1,..., X d1 ]] 13 f 0 R ρ /m Λ R ρ

14 f : Λ[[X 1,..., X d1 ]] R ρ f Zariski C Λ g 1 : R ρ R ρ /m Λ R ρ f 1 0 k[[x 1,..., X d1 ]] k[x 1,..., X d1 ]/X 1,..., X d1 ) 2 g 1 Zariski d 2 = 0 g 1 k[x 1,..., X d1 ]/X 1,..., X d1 ) 2 Λ[[X 1,..., X d1 ]]/m Λ, X 1,..., X d1 ) 2 k[x 1,..., X d1 ]/X 1,..., X d1 ) 2 g 1 g 2 : R ρ Λ[[X 1,..., X d1 ]]/m Λ, X 1,..., X d1 ) 2 m 2 g m : R ρ Λ[[X 1,..., X d1 ]]/m Λ, X 1,..., X d1 ) m g m+1 : R ρ Λ[[X 1,..., X d1 ]]/m Λ, X 1,..., X d1 ) m+1 g m ) m 2 g : R ρ Λ[[X 1,..., X d1 ]] g g 1 Zariki g f : Λ[[X 1,..., X d1 ]] Λ[[X 1,..., X d1 ]] Ker f 0 Kerg f) m) m 1 Λ[[X 1,..., X d1 ]] Λ[[X 1,..., X d1 ]] Noether f k 4.2 d 1 d 2 Euler-Poincaré 5 S D ρ S R) 1. Sk) = D ρ k). 2. ρ, ψ ) Ob C Λ ρ, ψ ) S) a a ρ /a, ψ /a) S/a) 14

15 3. ρ, ψ ) Ob C Λ a b a, b ρ /a, ψ /a) S/a)ρ /b, ψ /b) S/b) ρ /a b), ψ /a b) ) S /a b) ) 4. ρ, ψ ) Ob C Λ rtin C Λ Λ ρ, ψ ) S) ρ, ψ ) S ) 5.1. End k[g] V = k R ρ ρ D ρ S R) R ρ a S R ρ /a S Ob C Λ Ob C Λ Hom CΛ R ρ, ) D ρ ) Hom CΛ R ρ /a S, ) S). ρ R ρ, ψ R ρ ) R ρ X S ρ R ρ R ρ R ρ /a, ψ R ρ R ρ R ρ /a) SR ρ /a) R ρ a a S = a X S a.3 R ρ lim R ρ a XS /a R ρ /a S lim R ρ a XS /a a S C Λ n M G ρ : G ut M) ρ n G ut M) = det ρ δ : G Λ C Λ δ : G δ Λ Λ Λ D ρ D ρ,δ C Λ { } D ρ,δ ) = ρ : G ut M), ψ ) D ρ ) det ρ = δ 5.2. End k[g] V = k D ρ,δ k) = D ρ k) D ρ,δ R ρ,δ Ob C Λ. D ρ,δ R) 5.1 I G n = 2 C Λ ρ ρ : G ut M), ψ ) I M I M I M 1 I D ρ D ord ρ 15

16 5.3. End k[g] V = k D ord ρ k) = D ρ k) D ord ρ Ob C Λ R ord ρ. ρ : G ut M), ψ ) ρ C Λ ρ k I ρ, ψ ) I ψ x 1) 0 x M I V g 0 I [G] G g I M g 1) g 0 det ρ g 0 ) ) [G] C I ) ρ, ψ ) I C I ) C I ) y 0, g 0 det ρ g 0 ) ) y 0 ) k y 0 V ψ y 1) = y 0 y M x = g 0 det ρ g 0 ) ) y) x M I ψ x 1) 0 ρ, ψ ) I ρ, ψ ) I C I ) D ord ρ R) 5.1 n K p G = G K k p rtin C Λ ρ, ψ ) ρ O K G K C Λ ρ, ψ ) rtin ρ, ψ ) ρ, ψ ) D ρ Dfl ρ 5.4. End k[g] V = k D fl ρk) = D ρ k) D fl ρ Rfl ρ Ob C Λ. Dfl ρ R) C Λ rtin lim i i 1 j 3 M j i ) i M j i M 1 i ) M 2 i ) M 3 i ).1) i ) i i 1 j 3 M j i M j i i.1) lim Mi 1 lim i Mi 2 lim i i M 3 i 16

17 . i 1 j 3 M j i rtin rtin Mittag-Leffler.2. C Λ rtin lim i i 1. Noether 2. dim k m i /m 2 i ) i m Noether m m m m lim i m m i m m = 0 m m m lim i m m i m m+1 lim i m m+1 i 0 m m+1 i m m i m m i /m m+1 i 0.2) N = i lim m m i /m m+1 i 2 dim k m m i /m m+1 i ) i i m m i+1 /m m+1 i+1 m m i /m m+1 i N k.2) i.1 0 lim i m m+1 i m m N 0.3) l = dim k N N k m m a 1,..., a l a j i a j,i i l i m m i ; x 1,... x l ) x 1 a 1,i + + x l a l,i i.1 m m a 1,..., a l l dim k m m /m m+1 ) dim kn) = l m m+1.3) m m N mm+1 lim i mm+1 i Noether m b 1,..., b h m Λ Λ[[X 1,..., X h ]] ; X j b j 17

18 Noether m m m m m m 0 m m i i i /m m i 0.1 m m = lim i mm i /m m = lim i i /m m i 2 ) dim k i /m m i i i i+1 /m m i+1 i /m m i i /m m i m m.3. Ob C Λ X F = a X a lim a X /a /F lim a X /a. a i a i a 0 a 0 i a 0,i a i aa 0,i a 0,i rtin i a 0,i lim a a a 0,i.1 a 0 = lim i a 0,i lim i lim a a 0,i = lim lim a a 0,i = lim a0 + a)/a ) i a a a 0 = lim a /a a 0 lim a /a lim a0 a + a)/a ).1 a 0 lim a0 + a)/a ) lim /a lim /a 0 + a) 0 a a lim a /a 0 + a) rtin lim a0 a + a)/a ) lim /a lim /a a a a [Ma1] [Ma2] B. Mazur, Deforming Galois representations, Galois groups over Q Berkeley, C, 1987), , Math. Sci. Res. Inst. Publ. 16, Springer, New York, B. Mazur, n introduction to the deformation theory of Galois representations, Modular forms and Fermat s last theorem Boston, M, 1995), , Springer, New York, [Sch] M. Schlessinger, Functors on rtin rings, Trans. mer. Math. Soc ),

19 [SL] [TW] [W] B. de Smit, H. W. Lenstra, Jr. Explicit construction of universal deformation rings, Modular forms and Fermat s last theorem Boston, M, 1995), , Springer, New York, R. Taylor,. Wiles, Ring-theoretic properties of certain Hecke algebras nn. of Math. 2) ), no. 3, Wiles, Modular elliptic curves and Fermat s last theorem nn. of Math. 2) ), no. 3, [],,,

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