[Oc, Proposition 2.1, Theorem 2.4] K X (a) l (b) l (a) (b) X [M3] Huber adic 1 Huber ([Hu1], [Hu2], [Hu3]) adic 1.1 adic A I I A {I n } 0 adic 2

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1 On the action of the Weil group on the l-adic cohomology of rigid spaces over local fields (Yoichi Mieda) Graduate School of Mathematical Sciences, The University of Tokyo 0 l Galois K F F q l q K, F K, F Fr q Gal(F /F ) F Frobenius q Fr q Fr q Gal(F /F ) φ: Gal(K/K) Gal(F /F ) W K W K K Weil σ W K n(σ) Z φ(σ) = Fr n(σ) q W + K = { σ W K n(σ) 0 } X K X H i c(x K K, Q l ) Gal(K/K) W K X K K 0 σ W + K i) ([M1, Theorem 1.1]) σ H i c(x K K, Q l ) σ α m ι: Q l C ι(α) = q m/2 ii) ([M2, Theorem 1.1]) σ 2 dim X i=0 ( 1) i Tr ( σ ; H i c(x K K, Q l ) ) l i) Weil ii) l 1

2 [Oc, Proposition 2.1, Theorem 2.4] K X (a) l (b) l (a) (b) X [M3] Huber adic 1 Huber ([Hu1], [Hu2], [Hu3]) adic 1.1 adic A I I A {I n } 0 adic 2

3 1.2 A B A B A f-adic 1.3 f-adic A S 0 U 0 V S V U S V {sv s S, v V } A a A {a n n Z, n 0} A A a A 0 U N n N a n U A A 1.4 f-adic Tate 1.5 K Tate K + π K 1.6 Tate A B B π π A = B π B π B π {π n B π } 0 πb B B π s A B 0 s n B n π = s n φ: B π A π B 0 A ( ) a A aπ n aπ n n B a = φ φ π n 3

4 πb B π n B 0 B π B I m n π n I m π n B I m πb B φ 1.7 Tate A { A T 1,..., T n = i 1,...,i n 0 a i1,...,i n T i 1 1 T i n n A[[T 1,..., T n ]] } a i 1,...,i n 0 A U { } U T 1,..., T n = a i1,...,i n T i 1 1 T i n n A T 1,..., T n a i 1,...,i n U i 1,...,i n 0 { U T 1,..., T n U 0 A } 0 A T1,..., T n A T 1,..., T n Tate A B B π A T 1,..., T n B[T 1,..., T n ] π B T 1,..., T n 1.8 Tate A Noether n A T 1,..., T n Noether 1.2 adic 1.9 A Γ v : A Γ {0} (Γ, v) v(ab) = v(a)v(b) v(a + b) max { v(a), v(b) } v(0) = 0, v(1) = 1 4

5 Γ {0} Γ 0 γ = 0, 0 γ (γ Γ) A (Γ, v) A v 1 (0) (Γ, v) v(a) \ {0} Γ Γ Γ A 2 (Γ 1, v), (Γ 2, v 2 ) f : Γ 1 Γ 2 v 2 = f v 1 v 1.10 A A Spv A Spv A { v Spv A v(a) v(b) 0 } (a, b A) 1.11 Spv A A Spv A v x a A v(a) a(x) 1.12 A (Γ, v) A v γ Γ A v 1({ δ Γ {0} δ < γ }) 1.3 adic adic V 1.13 V X O X 5

6 x X O X,x v x {v x x X} 3 ( X, O X, {v x x X} ) ( X, O X, {v x } ) ( Y, O Y, {v y} ) f : X Y φ: O Y f O X x X v f(x) = v x φ x 1.14 A f-adic A B A ring of integral elements 3 B A B A B A f-adic A ring of integral elements A + A = (A, A + ) (A, A + ) (B, B + ) A B A + B + A = (A, A + ) V Spa A 1.15 A = (A, A + ) Spv A Spa A Spa A = { x Spv A x a(x) 1 (a A + ) } Spv A 1.16 Spa A p K 1 D 1 K Spa( K T, K + T ) a D 1 K v a : K T R; v a (f) = f(a) va Spa A D 1 K Spa( K T, K + T ) O Spa A 6

7 1.17 A = (A, A + ) T 1,..., T n A T i A s 1,..., s n A ( T1 R,..., T ) n = s 1 s n n { x Spa A t Ti t(x) s i (x) } i=1 Spa A Spa A ( ) T T A T A s A R s [Hu1, Theorem 3.5] 1.18 A Tate T i A T i A = A 1.19 A f-adic A 0 I A 0 T 1,..., T n A T i A s 1,..., s n A [ 1 A s1,...,s n = A A ( s1,...,s ) n T1 A A s 1,..., T n ( T1 s n s 1,..., T n s n s 1,..., 1 s n ] [ ] t, B = A 0 t T i, i = 1,..., n s i {I n B} 0 A 0, I ) T1 A,..., T n s 1 s n 1.20 A = (A, A + ) T 1,..., T n A ( T i A s 1,..., s n A B = A T1,..., T ) n [ ] s 1 s n t A + t T i, i = 1,..., n B C s i ( T1 (B, C) A,..., T ) n s 1 s n 7

8 (B, C T1 ) A,..., T n s 1 s n ( T1 Spa A U = R,..., T ) n T1 A,..., T n s 1 s n s 1 s n U T i, s i T1 ([Hu2, Proposition 1.3]) F A (U) = A,..., T n s 1 s n 1.21 Spa A V O Spa A (V ) = lim F A (U) O Spa A U V Spa A U O Spa A (U) = F A (U) x Spa A O Spa A,x = lim U x F A (U) U F A (U) x U A x: A Γ x {0} F A (U) F A (U) Γ x {0} x Spa A x: O Spa A,x Γ x {0} O Spa A ( Spa A, O Spa A, {v x } ) V Spa A 1.22 Spa A V adic adic V adic adic X adic adic X X O + X O + X (U) = { s O X (U) vx (s) 1 (x U) } Γ(Spa A, O Spa A ) = (A ), Γ(Spa A, O + Spa A ) = (A+ ) ([Hu2, Proposition 1.6 (iv)]) V Spa A A, A + A B = (B, B + ) 1 Spa A Spa B B A ([Hu2, Proposition 2.1 (i)]) 8

9 O Spa A 1.23 [Hu2, Theorem 2.2] A = (A, A + ) A Noether Tate A Noether O Spa A Spa A 1.4 K K R {0} K K K K + adic K 1.24 A Tate A B n A T 1,..., T n B A Tate (A, A + ) A T 1,..., T n A + T 1,..., T n C n A T 1,..., T n = ( A T 1,..., T n, C n ) A B A f : A T 1,..., T n B f(c n ) B B A Tate ([Hu2, 3]) adic X x X O X (U) Tate x U X adic adic f : Y X y Y f(y) U y V O X (U) Tate 9

10 f(v ) U ( O Y (V ), O + Y (V )) ( O X (U), O + X (U)) K 1.27 Spa(K, K + ) adic K 1.28 [Hu2, Lemma 4.4] A K A + ring of integral elements (A, A + ) (K, K + ) A + = A 1.29 A K Sp A = Spa(A, A ) K Noether ([BGR, 5.2.6/Theorem 1]) A Noether 1.5 adic 1.30 A = (A, A + ) I A A + /A + I A /I (A + /A + I) c A/I A/I = ( A /I, (A + /A + I) c) A /I A 1.31 adic f : X Y f adic Spa A I 2 = 0 A 10

11 I Hom Y (Spa A, X) Hom Y (Spa A/I, X) 1.32 A Noether Tate adic f : X Spa A i) f ii) x X x Spa B f 1,..., f m A T 1,..., T n (m n) A B = A T 1,..., T n /(f 1,..., f m ) ( ) fi det A T 1,..., T n /(f 1,..., f m ) T j 1 i,j m 1.6 Raynaud K K + Raynaud K + X K X rig K + / K X rig Raynaud adic K + adic 1.33 [Hu2, Proposition 4.1] K + Fsch /K + Spa(K +, K + ) adic adic /K + t: Fsch /K + adic /K + 11

12 Z adic /K + Hom adic/k + ( Z, t(x) ) = Hom ( (Z, O + Z ), (X, O X) ) Hom Spf K + t 1.34 X = Spf A t(x ) = Spa(A, A) 1.35 X adic x X x U O X (U) Tate X X adic X a adic adic 1.36 X adic Spa A x Spa A Supp x A Supp x A x U O X (U) x O X (U) O X (U) Γ x {0} p Supp x p O X (U) p p O X (U) Supp x A A B B I I Supp x a I \ Supp x a A a(x) ( ) 0 a a U = R U x a A a a 12

13 1.37 f : X Y X a = f 1 (Y a ) Y X 1.38 K + X X rig = t(x ) a X Raynaud 1.39 π K + B K + A = B π = B K + K B = Im(B A) X = Spf B X rig = Spa(A, B ) = Spa(A, A ) X = Spf K + T 1,..., T n /(f 1,..., f m ) X rig = Sp K T 1,..., T n /(f 1,..., f m ) x Spa(B, B) π(x) 0 x x B n π n (x) = 0 π(x) = 0 π(x) = 0 πb x ( x ) π Spa(B, B) a Spa(B, B) R ( ) π π π Spa(B, B) a = Spa B = Spa B ( π) π π B = (B π, B ) B B π Spa(B, B) a = Spa(A, B ) Spa(A, B ) = Spa(A, A ) B K + K + T 1,..., T n B ( K T 1,..., T n, K + T 1,..., T n ) (A, B ) 1.24 (A, B ) (K, K + ) 1.28 B = A X rig Spa(K, K + ) = Spa(K +, K + ) a adic K 13

14 2 K 1 K X K x X x U K + X i) K (X ) rig = U ii) X Elkik ([El, III, Theorem 7]) [Hu3, Proposition 1.7.1] Tate 2.2 A Noether Tate f 1,..., f n A T 1,..., T n ( ) fi det B = A T 1,..., T n /(f 1,..., f n ) T j 0 A T 1,..., T n U g i f i + U (i = 1,..., n) i) A B = B = A T 1,..., T n /(g 1,..., g n ) ( ) gi ii) det B T j 1 K

15 x U X K x X U = Sp K T 1,..., T n /(f 1,..., f m ) (m n), ( ) fi = det K T 1,..., T n /(f 1,..., f m ) T j 1 i,j m A = K T m+1,..., T n A Noether K + T m+1,..., T n Tate K T 1,..., T n /(f 1,..., f m ) = A T 1,..., T m /(f 1,..., f m ) A T 1,..., T m /(f 1,..., f m ) A T 1,..., T m U g i f i + U i) A T 1,..., T m /(f 1,..., f m ) = A T 1,..., T m /(g 1,..., g m ) A ( ) ii) gi = det A T 1,..., T m /(g 1,..., g m ) T j 1 i,j m K[T 1,..., T n ] K T 1,..., T n = A T 1,..., T m g i K[T 1,..., T n ] g i f i + U g 1,..., g m i) K T 1,..., T n /(f 1,..., f m ) = K T 1,..., T n /(g 1,..., g m ) ii) K T 1,..., T n /(g 1,..., g m ) g i K + [T 1,..., T n ] π K T 1,..., T n /(g 1,..., g m ) l πl K + T 1,..., T n /(g 1,..., g m ) K T 1,..., T n, T n+1 /(g 1,..., g m, T n+1 π l ) K T 1,..., T n /(g 1,..., g m ) T n+1 πl B = K + [T 1,..., T n, T n+1 ]/(g 1,..., g m, T n+1 π l ), X = Spec B B K + K = K T 1,..., T n, T n+1 /(g 1,..., g m, T n+1 π l ) = K T 1,..., T n /(g 1,..., g m ) = K T 1,..., T n /(f 1,..., f m ) (X ) rig = U X K 15

16 2.2 Elkik ([El, I, Theorem 1]) 2.3 A π A A π ( π-torsion ) free fi f 1,..., f n A T 1,..., T n = det h T j N > 2h a A n f i (a) π N A (i = 1,..., n) π h (a)a a a (mod π N h ) f i (a ) = 0 (i = 1,..., n) a A n f i (a) 0 π h (a)a a f 1 = = f n = 0 y = (y 1,..., y n ) A n y i π N h A, f i (a y) π 2N 2h A N 1 = 2N 2h N 1 > N + 2h 2h = N f i (a y) π N 1 A π h (a y)a (a y) (a)+π N h A N h > h y i πn 1 h A, f i (a y y ) π 2N 1 2h A y A n y, y,... a ( = a ) y y y + fi M = y 1,..., y n π N h A T j f 1 (a y). f n (a y) f 1 (a) y 1. M(a). f n (a) f 1 (a) y 1. = M(a). f n (a) y i π N h A y n y n (mod π 2N 2h ) 16

17 z 1 f 1 (a) M N. = N(a). z 1,..., z n z n f n (a) M(a) z 1 f 1 (a) M(a). = (a). f n (a) z n f i (a) π N A z i π N A z i = π N z i z i A π h (a)a π h = (a)δ δ A δπ N h z 1 f 1 (a) π h M(a). = π h. δπ N h z n f n (a) δπ N h z 1 A π-torsion free M(a). δπ N h z n y 1 δπ N h z 1. =. = f 1 (a). f n (a) y n δπ N h z n 2.4 A π-torsion free Noether A π-tors = { a A π m a = 0 m } Artin-Rees A π-tors π k A = 0 k N max{2h + 1, h + k} N 2.3 I A A I 2.5 A π A A π B A b 1,..., b n B 17

18 A T 1,..., T n B T 1,..., T n φ: B B φ(b i ) b i + πb φ B A π B π φ: π n B/π n+1 B π n B/π n+1 B φ(π n b i ) = π n φ(b i ) π n b i + π n+1 B φ = id φ A π A A π-tf = A/A π-tors torsion free 2.6 A Noether π A A π ( ) fi f 1,..., f n A T 1,..., T n = det A T 1,..., T n, B = ( T j A T1,..., T n /(f 1,..., f n ) ) π-tf m π m (, f 1,..., f n ) g i f i + I 2m+1 A T 1,..., T n g 1,..., g n ( ) gi = det, B = ( A T 1,..., T n /(g 1,..., g n ) ) T π-tf j A B = B π m (, g 1,..., g n ) A π Noether B, B A T 1,..., T n B, B π T i B, B a i, b i a = (a 1,..., a n ), b = (b 1,..., b n ) φ: B B π m (, f 1,..., f n ) g i f i + I 2m+1 A T 1,..., T n π m (, g 1,..., g n ) π m (b)b g i (b) = 0 f i (b) I 2m b b (mod π m+1 ) f i (b ) = 0 b B n T i b i A T 1,..., T n /(f 1,..., f n ) B B π-torsion free φ: B B ψ : B B (mod π 2m+1 ) π m (, f 1,..., f n ) π m (a)b f i (a) = 0 g i (a) I 2m a a (mod π m+1 ) g i (a ) = 0 a B n T i a i A T 1,..., T n /(g 1,..., g n ) B 18

19 ψ : B B ψ(φ(a i )) = ψ(b i ) ψ(b i) = a i a i (mod π m+1 ) ψ(φ(a i )) a i + πb 2.5 ψ φ φ ψ φ φ(π n B) = π n φ(b) = π n B φ φ π m (, g 1,..., g n ) f i g i, (mod π 2m+1 ) 2.7 ([Fu, Proposition 2.1.1]) m φ ψ 2.2 A 0 A π A 0 A 0 π π ( ) f 1,..., f n A 0 T 1,..., T n fi = det A T 1,..., T n /(f 1,..., f n ) T j A T 1,..., T n (, f 1,..., f n ) 1 A 0 T 1,..., T n (, f 1,..., f n ) π π m U = I 2m+1 A 0 T 1,..., T n A 0 T 1,..., T n A T 1,..., T n 0 g i f i + U g 1,..., g n ( A 0 T 1,..., T n /(f 1,..., f n ) ) π-tf = ( A 0 T 1,..., T n /(g 1,..., g n ) ) π-tf π m (, g 1,..., g n ) A = (A 0 ) π A T 1,..., T n /(f 1,..., f n ) = ( A 0 T 1,..., T n /(f 1,..., f n ) ) π ( (A0 = T 1,..., T n /(f 1,..., f n ) ) π-tf ( (A0 = T 1,..., T n /(g 1,..., g n ) ) π-tf) = ( A 0 T 1,..., T n /(g 1,..., g n ) ) π = A T 1,..., T n /(g 1,..., g n ) A T 1,..., T n /(f 1,..., f n ) = A T 1,..., T n /(g 1,..., g n ) π m (, g 1,..., g n ) A T 1,..., T n /(g 1,..., g n ) ) π π 19

20 [BGR] S. Bosch, U. Guntzer, R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften, 261. Springer-Verlag, Berlin, [El] R. Elkik, Solutions d équations a coefficients dans un anneau hensélien, Ann. Sci. École Norm. Sup. (4) 6 (1973), [Fu] K. Fujiwara, Theory of tubular neighborhood in étale topology, Duke Math. J. 80 (1995), no. 1, [Hu1] R. Huber, Continuous valuations, Math. Z. 212 (1993), no. 3, [Hu2] R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), no. 4, [Hu3] R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30. Friedr. Vieweg & Sohn, Braunschweig, [M1] Y. Mieda, On the action of the Weil group on the l-adic cohomology of rigid spaces over local fields, Int. Math. Res. Not. 2006, Art. ID [M2] Y. Mieda, On l-independence for the étale cohomology of rigid spaces over local fields, to appear in Compositio Math. [M3] l Weil (2005) [Oc] T. Ochiai, l-independence of the trace of monodromy, Math. Ann. 315 (1999), no. 2,

Noether [M2] l ([Sa]) ) ) ) ) ) ( 1, 2) ) ( 3) K F = F q O K K l q K Spa(K, O K ) adc adc [Hu1], [Hu2], [Hu3] K A Spa(A, A ) Sp A A B X A X B = X Spec

Noether [M2] l ([Sa]) ) ) ) ) ) ( 1, 2) ) ( 3) K F = F q O K K l q K Spa(K, O K ) adc adc [Hu1], [Hu2], [Hu3] K A Spa(A, A ) Sp A A B X A X B = X Spec l Wel (Yoch Meda) Graduate School of Mathematcal Scences, The Unversty of Tokyo 0 Galos ([M1], [M2]) Galos Langlands ([Ca]) K F F q l q K, F K, F Fr q Gal(F /F ) F Frobenus q Fr q Fr q Gal(F /F ) φ: Gal(K/K)