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1 2018 : msjmeeting-2018mar-02i002 () 1. 1:. (= ),,., Z, 1 π1 ab () 0 Chow ( CH 0 () := Coker iv : κ(x) ) Z x 1 x 0 2., x, x Frobenius ϱ : CH 0 () π ab 1 (), ϱ, π ab 1 () 3 (, Artin, Lang [23], Bloch [2], [19]). 1 ϱ?, Poincaré,. 2:., n, j, n 1, i 0. F x = Spec(F ) Z/n(i) = Z/n(i) x. Z/n(i) := { µ i n W r Ω i x,log [ i] µ i m (n F ) (ch(f ) = p > 0, n = p r m, (p, m) = 1 ), µ n 1 n., W r Ω i x,log Hoge-Witt [14], x = Spec(F ) x [4]. (W r Ω i x,log) x = K M i (F )/p r (K M i i Milnor K ) ( F Galois ) H q (x, Z/n(i)), F.,. H 1 (x, Z/n) = Hom cont (Gal(F /F ), Z/n), H 2 (x, Z/n(1)) = n Br(F ), H 1 (x, Z/n(1)) = F /n, H i (x, Z/n(i)) = K M i (F )/n ([30], [51]),,, Lichtenbaum 1 R, R π1 ab (). 2 j x, {x} j. κ(x) O,x. iv, x 1 {x}. 3 Z, π1 ab(), Im(ϱ ), ϱ. 119

2 [18], 4 Bloch-Ogus () C q,i n ()., q = i + 1, x 0 [κ(x) : κ(x) p ] p i (, p := ch(x)), n. Cn q,i () : x j H q+j (x, Z/n(i + j)) H q+1 (x, Z/n(i + 1)) x 1 x 0 H q (x, Z/n(i)), Galois. x j j, C q,i n () (cochain complex). 1.3 ([18] Conjecture 0.5) Z, H q (Ĉ1,0 n ()) = 0 (q < 0), H 0 (Ĉ1,0 n ()) = Z/n 5. im() = 1, Hasse ( 1.4 ), im() = 2 ([18] Theorem 0.7). im() 3, [6], [15], [16], [21].,,,,. 2, C q,i n () Cousin, ()?, F,, F Cousin, E j,q 1 = x j H j+q x (Spec(O,x ), F ) = H j+q (, F ) E 1 6. E,q 1 (, F ) : E 0,q 1 E 1,q 1 E j,q K, O K. = Spec(O K ), F = G m ( ), Cousin E,q 1 (, G m ) q < 0 q = 1, q = 0, 2, 3,... (Hilbert 90 ). E,0 1 (, G m ) : K x 1 Z 4 ([40] 0.4 ),. excellent., Z. 5 Ĉn q,i (), Cn q,i (). 2-torsion, Ĉn q,i () = Cn q,i () 6 j, im(o,x ) = j x. E,q 1 (, F ) Ej,q 1 j. Cn q,i () E,q 1 (, F ), ( ). 120

3 E,2 1 (, G m ) : Br(K) x 1 H 1 (x, Q/Z) E,q 1 (, G m ) : H q (K, G m ) 0 (q 3), K n := G m L Z/nCousinE,q 1 (, K n ), q < 1, q 1. E, 1 1 (, K n ) : µ n (K) 0 E,0 1 (, K n ) : K /n x 1 E,1 1 (, K n ) : nbr(k) x 1 Z/n H 1 (x, Z/n) E,q 1 (, K n ) : H q+1 (K, µ n ) 0 (q 2),, C 1,0 n (), C 0,0 n (), C 1,0 n (), C q,0 n () (q 2), 2 = Spec(O K ) i = 0., Artin-Verier ([29], [31]), 7 Hc q (, Z/n) Ext 3 q (Z/n, G m) Hc 3 (, G m ) = Q/Z. K n = RH om(z/n, Gm )[1], H 1 (, K n ) = Ext 2 (Z/n, G m ) = Hom(H 1 c (, Z/n), Q/Z), ϱ. CH 0 () H 1 (, G m ) c 1 H 1 (, K n ) ϱ π ab 1 () /n = H 1 c (, Z/n), c 1 1 Chern (), K n, Lichtenbaum (Z ) 1, 2, Lichtenbaum.. [26], [27], [28], Lichtenbaum L0L7 {Z(i)} i 0, Z(2) K. 7 Z Y, H c (Y, ) Y. = Spec(O K ) Artin-Verier K. 8 Artin-Verier [46], [9]. 1 ( 1.4 ),.,, G m Lichtenbaum, Bloch (ualizing complex). 121

4 L0. Z(0) = Z, Z(1) = G m [ 1] L1. i 1, 1 q i, H q (Z(i)) 0. L2. ϵ : ét Zar Zariski, R i+1 ϵ Z(i) = 0. (i = 1 Hilbert 90, i 2 ) L3. n,, Z(i) n Z(i) µ i n Z(i)[1]. (i = 1 Kummer, i 2 ) L3. F p,,. (L3 p ) Z(i) pr Z(i) W r Ω i,log[ i] Z(i)[1] L4.,. Z(i) L Z(j) Z(i + j) L5. q [ H] q (Z(i)), K () gr i γ K 1 2i q, Z 9. ( K ) (i 1)! L6. i H i (Z(i)), Milnor K K M i. (Milnor K ) L7. α : Y, (Y ) Z(i c) Y [ 2c] = τ i+c Rα! Z(i) (c := coim (Y )). (), Bloch [3] 10 Z(i) cyc : ( ) U zi (U, )[ 2i], L0, L2L4, L7, L1 q > i H q (Z(i) cyc ) ([3], [7], [10], [37], [45]). L5, Q 11. L6 [20]., Deekin, Z(i) Z(i) cyc, L i 2 q 0 L5 L1, Beilinson-Soulé. 10, Suslin-Voevosky [ ] [47], [51]. 11 1, Z (, N i,q, = max{+i q+1, i 1}) N i,q,! [25]. 12 Deekin, Z(i) cyc L2, L3, L7, L1 q > i [8]. 122

5 ,, p, Y := ( Z F p ) re, U := Y = [p 1 ] (re ). Lichtenbaum, Z/p r (i) := Z(i) L Z/p r., F p ( = Y ), L3 Z(i) L Z/p r = Wr Ω i,log[ i],, Y ( U ). T1. t : Z/p r (i) U µ i p r. (Z(i) U L3 ) T2. 0 q i, H q (Z/p r (i) ) = 0. (L1 ) T3. α : T, T F p, T α! : W r Ω i c T,log [ i c] τ i+c Rα! Z/p r (i) (c := coim (T )). (L7, Z(i) T L3 ) T4. Z, x {y}, ch(x) = p 2 x Z c, y Z c 1, ( ( 1) ). H i c+1 (y, Z/p r (i c + 1)) y,x H i c (x, Z/p r (i c)) 2.1 H i+c 1 β y! = = β x! y (Spec(O Z,y ), Z/p r (i)) δy,x Hx i+c (Spec(O Z,x ), Z/p r (i)), w Z, β w : w Spec(O Z,w ). ch(w) = p, β w! T3, ch(w) = 0, β w! T1 t ([7], [37]). y,x Galois, δ y,x. (1) CousinE,i 1 (, Z/p r (i) )δ y,x. (2) ch(y) p, ch(x) p T4 [17]. (3) T4 Lichtenbaum, F = Z/p r (i) 2 q = i, n = p r. Milne ([32] Remark 2.7),. 2.2 j : U, α : Y, T1 T4 Z/p r (i)., (1). R i j µ i p β r y W r Ω i 1 y,log y Y 0 β x W r Ω i 2 x,log, x Y 1, z Y β z z Y, () Galois. () 123

6 (2),. α ν i 1 g Y,r [ i 1] Z/p r (i) t τ i Rj µ i p r σ r,i α ν i 1 Y,r [ i], t T1 tt2, ν i 1 Y,r () ( Y ). σ r,i (). 2.3 T1 T4 Z/p r (i) T1 t (Z/p r (i), t), T1 T q = i, n = p r. 3.1 ([41], [42]) A Deekin, A, ( ). ( ) Spec(A) p Σ, Spec(A) Σ ( Cartier Y ), 13., T1 T4 T r (i),. T5. Z/p r,, U µ i p r. (L4 ) T r (i) L T r (j) T r (i + j) µ j p r T6. f : = µ i+j p r T7. f : (T r (i) 3.1 ) Y normal crossing ivisor ( ( ) ), 2.2 (1) ()., Y (p) Hoge-Witt, Gros-Suwa [11]. () 2.2 (2) σ r,i. 2.2, T r (i). T r (i) := Cone ( σ r,i : τ i Rj µ i p r α ν i 1 Y,r [ i]) [ 1] (), T r (i) T2. T1. T3 α!, T r (i) T3, T4. T3, p α R q j µ q p (q i) (Bloch-Kato-Hyoo [4], [12]). 13 D, D x g : V v g 1 (x), g 1 (D) V O V,v ( m v, m v /m 2 v ). 124

7 3.2 i = 1, T r (1) = Gm L Z/p r [ 1]., T3, Brauer p. Br( T ){p} = Br(){p} (coim (T ) 2) 3.3 Lichtenbaum Z(i),, Bloch Z(i) cyc. Z/p r (i) cyc := Z(i)cyc Z/pr, T r (i)., Z/p r (i) cyc = T r (i) ([42] Conjecture 1.4.1), A, Geisser ([8] Theorems 1.2 (2), (4), 1.3)., Z/p r (i) cyc T r (i) Z/p r (i) cyc. [43], Zhong [53]. Z/p r (i) cyc Gersten, [53]. 3.4 ( [42]) A. A, p, 3.1 ( ). = im(), Hc q (, T r (i)) H 2+1 q (, T r ( i)) Hc 2+1 (, T r ()) = Z/p r p. Y := ( Z F p ) re, U := Y, Artin-Verier, Hc q (U, µ i p ) r H2+1 q (U, µ i p ) r H2+1 c (U, µ p ) = Z/p r r., ( [42]) 3.4, Σ Spec(A) p, Σ := s Σ A A s (= Z Z p )., A s A s., H q ( Σ, T r (i)) H 2+1 q Y (, T r ( i)) H 2+1 Y (, T r ()) = Z/p r p. 3.5, p (explicit reciprocity law), Y Serre. 125

8 ,. CH 0 () cl H 2 (, T r ()) ϱ π ab 1 () /p r = H 1 c (, Z/p r ), cl T3, T4, Gabber. 3.4., p 3 Hom alg (A, R) =, , F = Z/p r () ( := im()) 2 (q, i) = (1, 0), n = p r (, x 0, κ(x) ) [17] Theorem (4). 4. p regulator p, T r (i) s r (i) [22]. p,, (Bloch- Kato Selmer ). T r (i),, p, Bloch-Kato Selmer., T r (i), Selmer Hf 1, H1 g (4.3, 4.5 ). K, O K. G K := Gal(K/K). O K, p, 3.1 ( ). Q p. V q (i) := Q p Zp lim r 1 H q ( K, µ i p ) r H q ( K, Q p (i)) := Q p Zp lim r 1 H q ( K, µ i p ) r H q (, T Qp (i)) := Q p Zp lim r 1 H q (, T r (i)) V q (i) H q (, T Qp (i)) Q p, H q ( K, Q p (i)) (, 2i q = 1 )., 2i q 1. Bloch-Kato Selmer [5], (). 4.1 ( [5] ) reg q,i K (1) p regulator : K 2i q ( K ) chq,i H q ( K, Q p (i)) H 1 (G K, V q 1 (i)) H 1 g (G K, V q 1 (i))., ch q,i Chern. (2) reg q,i H 1 f (G K, V q 1 (i)). : K 2i q() K 2i q ( K ) reg q,i K H 1 (G K, V q 1 (i)) 126

9 (1), Im(reg q,i K ) Hg 1 (G K, V q 1 (i)) [24], [34]. (2), v p v goo reuction, Niziol Im(reg q,i ) H1 (G K, V q 1 (i)) [36]. 4.1 T r (i), (A) Niziol (B) H 1 g (G K, V 2 (2))/Im(reg 3,2 K ) ( Q p /Z p ). (A).. Φ q,i : H q (, T Qp (i)) H q ( K, Q p (i)) H 1 (G K, V q 1 (i)) 4.2 ([43]) Y := ( Z F p ) re. p 2 i, H q 1 log-crys (Y/W ) [13], [33]., Im(Φq,i ) = H 1 f (G K, V q 1 (i))., Fontaine-Jannsen [49], p [50]. 4.2 Chern ch q,i : K 2i q () H q (, T Qp (i)) ([43], [1]), , Im(reg q,i ) H1 f (G K, V q 1 (i)). 4.4 Scholl [44] K 2i q ( K ) Q K 2i q ( K ) OK, 4.3 K K (reg q,i K K 2i q ( K ) OK ).,, K., (B). H 3 ur(k(), 2) δ. δ : H 3 (Spec(K()), Q p /Z p (2)) x 1 H 4 x(spec(o,x ), T Qp /Z p (2)),, Q p /Z p (2) := lim r 1 µ 2 p, r T Q p/z p (2) := lim r 1 T r (2)., H 3 ur(k(), K ; 2) := H 3 ur(k(), 2) Im ( H 3 ( K, Q p /Z p (2)) H 3 (Spec(K()), Q p /Z p (2)) ). H 3 ur(k(), 2) Brauer p,., T r (2), K p regulator Q p /Z p reg 3,2 Q p /Z p : K 1 ( K ) Q p /Z p H 1 g (G K, A 2 (2)) (A 2 (2) := H 2 ( K, Q p /Z p (2))). 127

10 4.5 ([39], Theorem 7.1.1) p 5,. (i) Spec(O K ), Tate. (ii) CH 2 ( K ){p} (CH 2 ( K ){p}) Div,, p., Z p,. 0 CH 2 ( K ){p} H1 g (G K, A 2 (2)) Im ( reg 3,2 Q p /Z p ) H 3 ur(k(), K ; 2)., H 3 ur(k(), K ; 2) Div. H 1 (G K, V 2 (2)) H 1 (G K, A 2 (2)), H 1 g (G K, V 2 (2)) H 1 (G K, A 2 (2)) Div ([39] Lemma 2.4.1). 4.5 (), Im(reg 3,2 K ) = H 1 g (G K, V 2 (2)), (CH 2 ( K ){p}) Div = 0 H 3 ur(k(), K ; 2) Div = 0., Ker(CH 2 () CH 2 ( K )), ([39] Remark 3.2.5). [1] Asakura, M., Sato, K.: Chern class an Riemann-Roch theorem for cohomology theory without homotopy invariance. [2] Bloch, S.: Algebraic K-theory an classfiel theory for arithmetic surfaces. Ann. of Math. (2) 114, (1981) [3] Bloch, S.: Algebraic cycles an higher K-theory. Av. Math. 61, (1986) [4] Bloch, S., Kato, K.: p-aic étale cohomology. Inst. Hautes Étues Sci. Publ. Math. 63, (1986) [5] Bloch, S., Kato, K.: L-functions an Tamagawa numbers of motives. In: Cartier, P., Illusie, L., Katz, N. M., Laumon, G., Manin, Yu. I., Ribet, K. A. (es.) The Grothenieck Festscherift I, (Progr. Math. 86), pp , Boston, Birkhäuser, 1990 [6] Colliot-Thélène, J.-L.: On the reciprocity sequence in the higher class fiel theory of function fiels. In: Goerss, P. G., Jarine, J. F. (es.) Algebraic K -theory an algebraic topology, Lake Louise, 1991, NATO Av. Sci. Inst. Ser. C Math. Phys. Sci., 407, pp , Kluwer, Dorrecht, 1993 [7] Fujiwara, K.: A proof of the absolute purity conjecture (after Gabber). In: Usui, S., Green, M., Illusie, L., Kato, K., Looijenga, E., Mukai, S., Saito, S. (es.) Algebraic Geometry 2000, Azumino, (Av. Stu. Pure Math. 36), pp , Tokyo, Math. Soc. Japan, 2002 [8] Geisser, T.: Motivic cohomology over Deekin rings. Math. Z. 248, (2004) [9] Geisser, T.: Duality via cycle complexes. Ann. of Math. (2) 172, (2010) [10] Geisser, T., Levine, M.: The p-part of K-theory of fiels in characteristic p. Invent. Math. 139, (2000) 128

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.5.1. G K O E, O E T, G K Aut OE (T ) (T, ρ). ρ, (T, ρ) T. Aut OE (T ), En OE (F ) p..5.. G K E, E V, G K GL E (V ) (V, ρ). ρ, (V, ρ) V. GL E (V ), En

.5.1. G K O E, O E T, G K Aut OE (T ) (T, ρ). ρ, (T, ρ) T. Aut OE (T ), En OE (F ) p..5.. G K E, E V, G K GL E (V ) (V, ρ). ρ, (V, ρ) V. GL E (V ), En p 1. 1.1., 01 8 3, 57,,.. 1.., Gal(Q p /Q p ), 1. Wach,,. 1.3. Part I,,. Part II, Part III. 1.4.., Paé. Part 1. p.. p p.1. p Q p p (Q p p )... E Q p, E p Z p E, O E. O E E. E Q p, O E. v p : E Q Q E, v