advertisement

ISSN 1464-8997 (on line) 1464-8989 (printed) 123 Geometry & Topology Monographs Volume 3: Invitation to higher local fields Part I, section 15, pages 123–135 15. On the structure of the Milnor K -groups of complete discrete valuation fields Jinya Nakamura 15.0. Introduction For a discrete valuation field K the unit group K ∗ of K has a natural decreasing filtration with respect to the valuation, and the graded quotients of this filtration are written in terms of the residue field. The Milnor K -group Kq (K) is a generalization of the unit group and it also has a natural decreasing filtration defined in section 4. However, if K is of mixed characteristic and has absolute ramification index greater than one, the graded quotients of this filtration are known in some special cases only. Let K be a complete discrete valuation field with residue field k = kK ; we keep the notations of section 4. Put vp = vQp . A description of grn Kq (K) is known in the following cases: (i) (Bass and Tate [ BT ]) gr0 Kq (K) ' Kq (k) ⊕ Kq−1 (k). 1 (ii) (Graham [ G ]) If the characteristic of K and k is zero, then grn Kq (K) ' Ωq− k for all n > 1 . (iii) (Bloch [ B ], Kato [ Kt1 ]) If the characteristic of K and of k is p > 0 then 2 q−1 q−2 q−1 q−2 grn Kq (K) ' coker Ωq− −→ Ω /B ⊕ Ω /B s s k k k where ω 7−→ (C−s (dω), (−1)q m C−s (ω) and where n > 1 , s = vp (n) and m = n/ps . (iv) (Bloch–Kato [ BK ]) If K is of mixed characteristic (0, p), then 2 q−1 q−2 q−1 q−2 grn Kq (K) ' coker Ωq− −→ Ω /B ⊕ Ω /B s s k k k where ω − 7 → (C−s (dω), (−1)q m C−s (ω)) and where 1 6 n < ep/(p − 1) for e = vK (p), s = vp (n) and m = n/ps ; and ep K (K) gr p− q 1 2 q−1 q−2 q−1 q−2 ' coker Ωq− −→ Ω /(1 + a )B ⊕ Ω /(1 + a )B C C s s k k k c Geometry & Topology Publications Published 10 December 2000: 124 J. Nakamura where ω 7−→ ((1 + a C) C−s (dω), (−1)q m(1 + a C) C−s (ω)) and where a is the residue class of p/π e for fixed prime element of K , s = vp (ep/(p − 1)) and m = ep/(p − 1)ps . (v) (Kurihara [ Ku1 ], see also section 13) If K is of mixed characteristic (0, p) and 1 q−1 absolutely unramified (i.e., vK (p) = 1 ), then grn Kq (K) ' Ωq− k /Bn−1 for n > 1. (vi) (Nakamura [ N2 ]) If K is of mixed characteristic (0, p) with p > 2 and p - e = vK (p), then as in (iv) (1 6 n 6 ep/(p − 1)) grn Kq (K) ' q−1 q−1 (n > ep/(p − 1)) Ωk /Bln +sn where ln is the maximal integer which satisfies n − ln e > e/(p − 1) and sn = vp (n − ln e). (vii) (Kurihara [ Ku3 ]) If K0√is the fraction field of the completion of the localization Zp [T ](p) and K = K0 ( p pT ) for a prime p 6= 2 , then grn K2 (K) ' as in (iv) k/k p (1 6 n 6 p) kp 0 (n = lp, l > 3) (otherwise). l−2 (n = 2p) (viii) (Nakamura [ N1 ]) Let K0 be an absolutely unramified complete√ discrete valuation field of mixed characteristic (0, p) with p > 2 . If K = K0 (ζp )( p π) where π is a prime element of K0 (ζp ) such that dπ p−1 = 0 in Ω1OK (ζ ) , then grn Kq (K) are 0 p determined for all n > 1 . This is complicated, so we omit the details. (ix) (Kahn [ Kh ]) Quotients of the Milnor K -groups of a complete discrete valuation field K with perfect residue field are computed using symbols. Recall that the group of units U1,K can be described as a topological Zp -module. As a generalization of this classical result, there is an appraoch different from (i)-(ix) for higher local fields K which uses topological convergence and Kqtop (K) = Kq (K)/ ∩l>1 lKq (K) (see section 6). It provides not only the description of grn Kq (K) but of the whole top Kq (K) in characteristic p (Parshin [ P ]) and in characteristic 0 (Fesenko [ F ]). A top complete description of the structure of Kq (K) of some higher local fields with small ramification is given by Zhukov [ Z ]. Below we discuss (vi). Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields Part I. Section 15. On the structure of the Milnor K -groups of cdv fields 125 15.1. Syntomic complex and Kurihara’s exponential homomorphism 15.1.1. Syntomic complex. Let A = OK and let A0 be the subring of A such that A0 is a complete discrete valuation ring with respect to the restriction of the valuation of K , the residue field of A0 coincides with k = kK and A0 is absolutely unramified. Let π be a fixed prime of K . Let B = A0 [[X]]. Define X7→π J = ker[B −−−→ A] X7→π mod p I = ker[B −−−→ A −−−→ A/p] = J + pB. Let D and J ⊂ D be the PD-envelope and the PD-ideal with respect to B → A, respectively. Let I ⊂ D be the PD-ideal with respect to B → A/p . Namely, j x ; j > 0, x ∈ J , J = ker(D → A), I = ker(D → A/p). D=B j! Let J [r] (resp. I [r] ) be the r -th divided power, which is the ideal of D generated by j i j x x p ; j > r, x ∈ J , resp. ; i + j > r, x ∈ I . j! i! j! Notice that I [0] = J [0] = D . Let I [n] = J [n] = D for a negative n . We define the complexes J[q] and I[q] as d d [q−2] b1 − b 2 −→ · · · ] J[q] = [J [q] − → J [q−1] ⊗B Ω ⊗B Ω B →J B d d [q−2] b1 − b 2 −→ · · · ] I[q] = [I [q] − → I [q−1] ⊗B Ω ⊗B Ω B → I B b q is the p -adic completion of Ωq . We define D = I[0] = J[0] . where Ω B B Let T be a fixed set of elements of A∗0 such that the residue classes of all T ∈ T in k forms a p -base of k . Let f be the Frobenius endomorphism of A0 such that f (T ) = T p for any T ∈ T and f (x) ≡ xp mod p for any x ∈ A0 . We extend f to B by f (X) = X p , and to D naturally. For 0 6 r < p and 0 6 s , we get f (J [r] ) ⊂ pr D, b s ) ⊂ ps Ω bs , f (Ω B B since f (x[r] ) = (xp + py)[r] = (p!x[p] + py)[r] = p[r] ((p − 1)!x[p] + y)r , dT p dTs dT p dTs dT1 dT1 = z p1 ∧ · · · ∧ ps = zps ∧ ··· ∧ ∧ ··· ∧ , f z T1 Ts T1 Ts T1 Ts where x ∈ J , y is an element which satisfies f (x) = xp + py , and T1 , . . . , Ts ∈ T ∪ {X} . Thus we can define fq = f b q−r −→ D ⊗ Ω b q−r : J [r] ⊗ Ω B B pq Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields 126 J. Nakamura for 0 6 r < p . Let of S (q) and S 0 (q) be the mapping fiber complexes (cf. Appendix) 1−fq J[q] −−−→ D 1−fq and I[q] −−−→ D respectively, for q < p . For simplicity, from now to the end, we assume p is large 0 enough to treat (q) and (q). (q) is called the syntomic complex of A with 0 respect to B , and (q) is also called the syntomic complex of A/p with respect to B (cf. [ Kt2 ]). S S S S S Theorem 1 (Kurihara [ Ku2 ]). There exists a subgroup S q of H q ( (q)) such that cq (A) where K cq (A) = lim Kq (A)/pn is the p -adic completion of UX H q ( (q)) ' U1 K ←− Kq (A) (see subsection 9.1). S b q−1 ) be the subgroup of D ⊗ Ω b q−1 generated by Outline of the proof. Let UX (D ⊗ Ω B B q−1 q−2 q−1 b b b XD ⊗ ΩB , D ⊗ ΩB ∧ dX and I ⊗ ΩB , and let b q−1 )/((dD ⊗ Ω b q−2 + (1 − fq )J ⊗ Ω b q−1 ) ∩ UX (D ⊗ Ω b q−1 )). S q = UX (D ⊗ Ω B B B B P q q−1 n b b The infinite sum n>0 fq (dx) converges in D ⊗ ΩB for x ∈ UX (D ⊗ ΩB ). Thus we get a map S b q−1 ) −→ H q ( (q)) UX (D ⊗ Ω B ∞ X x 7−→ x, fqn (dx) n=0 q q S (q)). Let Eq be the map and we may assume S is a subgroup of H ( b q−1 ) −→ K b q (A) Eq : UX (D ⊗ Ω B dTq−1 dT1 x ∧ ··· ∧ 7−→ {E1 (x), T1 , . . . , Tq−1 }, T1 Tq−1 P where E1 (x) = exp ◦( n>0 f1n )(x) is Artin–Hasse’s exponential homomorphism. In [ Ku2 ] it was shown that Eq vanishes on b q−2 + (1 − fq )J ⊗ Ω b q−1 ) ∩ UX (D ⊗ Ω b q−1 ), (dD ⊗ Ω B B B hence we get the map b q (A). Eq : S q −→ K b q (A) by definition. The image of Eq coincides with U1 K b q (A) −→ S q by On the other hand, define sq : K sq ({a1 , . . . , aq }) q X dag daeq f (aei ) dae1 dag 1 i−1 i+1 ∧ · · · ∧ f1 (−1)i−1 log ∧ ··· ∧ ∧ f1 = p p aei ae1 ag ag aeq i−1 i+1 i=1 Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields Part I. Section 15. On the structure of the Milnor K -groups of cdv fields 127 (cf. [ Kt2 ], compare with the series Φ in subsection 8.3), where e a is a lifting of a b q (A). Note that if to D . One can check that sq ◦ Eq = − id . Hence S q ' U1 K b q (A) ' U1 K b q (K) (see [ Ku4 ] or [ N2 ]), thus we have ζp ∈ K , then one can show U1 K b q (K). S q ' U1 K Example. We shall prove the equality sq ◦ Eq = − id in the following simple case. b q−1 ) for T ∈ T ∪ {X} . Then Let q = 2 . Take an element adT /T ∈ UX (D ⊗ Ω B dT T a), T }) = sq ({E1 (e f (E1 (a)) 1 dT = log f1 p p E1 (a) T X X dT 1 f1n (a) − p log ◦ exp ◦ f1n (a) = log ◦f ◦ exp ◦ p T n>0 n>0 X X dT n n f1 (a) − f1 (a) = f1 T sq ◦ Eq a n>0 = −a n>0 dT . T 15.1.2. Exponential Homomorphism. The usual exponential homomorphism expη : A −→ A∗ x 7−→ exp(ηx) = X xn n! n>0 is defined for η ∈ A such that vA (η) > e/(p − 1). This map is injective. Section 9 contains a definition of the map b q−1 −→ K b q (A) expη : Ω A dyq−1 dy1 ∧ ··· ∧ 7−→ {exp(ηx), y1 , . . . , yq−1 } x y1 yq−1 for η ∈ A such that vA (η) > 2e/(p − 1). This map is not injective in general. Here is a description of the kernel of expη . Theorem 2. The following sequence is exact: (*) exp ψ 1 b q−2 −→ K b q (A). S 0(q)) −→ Ωq− A /pdΩA H q−1 ( p Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields 128 J. Nakamura Sketch of the proof. There is an exact sequence of complexes J[q] I[q] →MF →I[q] /J[q] → 0, 0 →MF −f −f 1 1 q q y y D k k S (q) D S 0 (q) where MF means the mapping fiber complex. Thus, taking cohomologies we have the following diagram with the exact top row S 0 (q)) −−−ψ−→ H q−1 (Ix[q] /J[q] ) −−−δ−→ H q (Sx (q)) H q−1 ( (1) Thm.1 exp p b q−1 /pdΩ b q−2 −−−− b q (A), Ω → U1 K A A where the map (1) is induced by b q−1 /J ⊗ Ω b q−1 = (I[q] /J[q] )q−1 . b q−1 3 ω 7−→ pe ω ∈I ⊗Ω Ω A B B We denoted the left horizontal arrow of the top row by ψ and the right horizontal arrow of the top row by δ . The right vertical arrow is injective, thus the claims are (1) is an isomorphism, (2) this diagram is commutative. First we shall show (1). Recall that ! [2] ⊗ Ω b q−2 b q−2 I I ⊗ Ω B B . H q−1 (I[q] /J[q] ) = coker −→ b q−2 b q−2 J [2] ⊗ Ω J ⊗Ω B B From the exact sequence 0 −→ J −→ D −→ A −→ 0, b q−1 /J ⊗ Ω b q−1 = A ⊗ Ω b q−1 and its subgroup I ⊗ Ω b q−2 /J ⊗ Ω b q−2 is we get D ⊗ Ω B B B B B b q−1 in A ⊗ Ω b q−1 . The image of I [2] ⊗ Ω b q−2 in pA ⊗ Ω b q−1 is equal to the pA ⊗ Ω B B B B image of b q−2 = J2 ⊗ Ω b q−2 + pJΩ b q−2 + p2 Ω b q−2 . I2 ⊗ Ω B B B B On the other hand, from the exact sequence 0 −→ J −→ B −→ A −→ 0, we get an exact sequence d b q−2 −→ b q−1 −→ Ω b q−1 −→ 0. (J/J2 ) ⊗ Ω A⊗Ω B B A Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields Part I. Section 15. On the structure of the Milnor K -groups of cdv fields 129 b q−2 vanishes on pA ⊗ Ω b q−1 , hence Thus dJ2 ⊗ Ω B B H q−1 (I[q] /J[q] ) = b q−1 b q−1 p−1 pA ⊗ Ω A⊗Ω B B b q−1 /pdΩ b q−2 , ' 'Ω A A q−2 q−2 q−2 q−2 2 b b b b pdJΩB + p dΩB dJΩB + pdΩB which completes the proof of (1). Next, we shall demonstrate the commutativity of the diagram on a simple example. b 1 for T ∈ T ∪ {π} . We want to Consider the case where q = 2 and take adT /T ∈ Ω A show that the composite of Eq (1) δ b 1 /pdA −→ b 2 (A) H 1 (I[2] /J[2] ) −→ S q −→ U1 K Ω A b 1 /J ⊗ Ω b 1 is coincides with expp . By (1), the lifting of adT /T in (I[2] /J[2] )1 = I ⊗ Ω B B pe a ⊗ dT /T , where e a is a lifting of a to D . Chasing the connecting homomorphism δ, 0 −−→ b1B )⊕D (J⊗Ω −−→ b1B )⊕D (I⊗Ω −−→ 0 −−→ b2B )⊕(D⊗Ω b1B ) (D⊗Ω −−→ b2B )⊕(D⊗Ω b1B ) (D⊗Ω −−→ dy dy S dy dy b1B )/(J⊗Ω b1B ) (I⊗Ω −−→ 0 0 −−→ 0 dy dy S (the left column is (2), the middle is 0 (2) and the right is I[2] /J[2] ); pe adT /T in the upper right goes to (pde a ∧ dT /T, (1 − f2 )(pe a ⊗ dT /T )) in the lower left. By E2 , this element goes d dT = E2 (1 − f1 )(pe a) ⊗ T T X n = {E1 ((1 − f1 )(pe a)), T } = exp ◦ f1 ◦ (1 − f1 )(pe a), T E2 (1 − f2 ) pe a⊗ n>0 = {exp(pa), T }. b 2 (A). This is none other than the map exp . in U1 K p By Theorem 2 we can calculate the kernel of expp . On the other hand, even though b q (A) and we already know expp is not surjective, the image of expp includes Ue+1 K b q (K) for 0 6 i 6 ep/(p − 1). Thus it is enough to calculate the kernel of exp gri K p b q (K). Note that to know gri K b q (K), we may assume that in order to know all gri K b q (A) = U0 K b q (K). ζp ∈ K , and hence K Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields 130 J. Nakamura 15.2. Computation of the kernel of the exponential homomorphism 15.2.1. Modified syntomic complex. We introduce a modification of calculate it instead of 0 (q). Let Sq be the mapping fiber complex of S S 0(q) and 1 − fq : (J[q] )>q−2 −→ D>q−2 . Here, for a complex C · , we put C >n = (0 −→ · · · −→ 0 −→ C n −→ C n+1 −→ · · · ). S By definition, we have a natural surjection H q−1 (Sq ) → H q−1 ( 0 (q)), hence ψ(H q−1 (Sq )) = ψ(H q−1 ( 0 (q))), which is the kernel of expp . To calculate H q−1 (Sq ), we introduce an X -filtration. Let 0 6 r 6 2 and s = q −r . b s ) be the subgroup of I [r] ⊗B Ω bs Recall that B = A0 [[X]]. For i > 0 , let fili (I [r] ⊗B Ω B B generated by the elements S ) pl bs aω : n + ej > i, n > 0, j + l > r, a ∈ D, ω ∈ Ω B j! l! n (X e j X e j l dX s−1 n (X ) p b ∪ X aυ ∧ : n + ej > i, n > 1, j + l > r, a ∈ D, υ ∈ ΩB . j! l! X bs → D ⊗ Ω b s preserves the filtrations. By using the latter The map 1 − fq : I [r] ⊗ Ω B B we get the following Proposition 3. H q−1 (fili Sq )i form a finite decreasing filtration of H q−1 (Sq ). Denote fili H q−1 (Sq ) = H q−1 (fili Sq ), gri H q−1 (Sq ) = fili H q−1 (Sq )/fili+1 H q−1 (Sq ). Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields Part I. Section 15. On the structure of the Milnor K -groups of cdv fields Then gri H q−1 (Sq ) 0 2e−1 dX ∧ Ω b q−3 /p X A 0 q− 2 i−1 b q−3 /p) b dX ∧ (Ω Xi Ω A0 /p ⊕ X A0 q− 2 e−1 b b q−3 p2 Ω b q−3 /p ⊕ X dX ∧ Z Xe Ω Ω 1 A0 A0 A0 2 q−3 e−1 q− 3 b b p Ω dX ∧ Z1 Ω A0 A0 = X 0 max(ηi −vp (i),0) bq−2 2 2 bq− bq− p ΩA ∩Zηi Ω +p2 Ω A A 0 0 0 X i q−2 2Ω b p A0 ! 3 2 bq−3 bq− Zηi Ω +p ΩA A 0 0 ⊕ X i−1 dX ∧ 3 bq− p2 Ω A 0 0 131 ( if i > 2e) ( if i = 2e) ( if e < i < 2e) ( if i = e, p | e) ( if i = e, p - e) ( if 1 6 i < e) ( if i = 0). 0 Here ηi and ηi0 are the integers which satisfy pηi −1 i < e 6 pηi i and pηi −1 i − 1 < 0 e 6 pηi i − 1 for each i, d b q+1 n b q −→ b q = ker Ω Zn Ω Ω /p A0 A0 A0 b q for n 6 0 . bq = Ω for positive n , and Zn Ω A0 A0 Outline of the proof. From the definition of the filtration we have the exact sequence of complexes: 0 −→ fili+1 Sq −→ fili Sq −→ gri Sq −→ 0 and this sequence induce a long exact sequence · · · → H q−2 (gri Sq ) → H q−1 (fili+1 Sq ) → H q−1 (fili Sq ) → H q−1 (gri Sq ) → · · · . The group H q−2 (gri Sq ) is b q−2 −→ (gri I ⊗ Ω b q−1 ) ⊕ (gri D ⊗ Ω b q−2 ) gri I [2] ⊗ Ω q−2 B B B H (gri Sq ) = ker . x 7−→ (dx, (1 − fq )x) b q−2 if i = 0 , thus they b q−2 → Ω The map 1 − fq is equal to 1 if i > 1 and 1 − fq : p2 Ω A0 A0 are all injective. Hence H q−2 (gri Sq ) = 0 for all i and we deduce that H q−1 (fili Sq )i form a decreasing filtration on H q−1 (Sq ). Next, we have to calculate H q−2 (gri Sq ). The calculation is easy but there are many cases which depend on i, so we omit them. For more detail, see [ N2 ]. Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields 132 J. Nakamura Finally, we have to compute the image of the last arrow of the exact sequence 0 −→ H q−1 (fili+1 Sq ) −→ H q−1 (fili Sq ) −→ H q−1 (gri Sq ) because it is not surjective in general. Write down the complex gri Sq : d b q−1 ) ⊕ (gri D ⊗ Ω b q−2 ) → b q ) ⊕ (gri D ⊗ Ω b q−1 ) → · · · , · · · → (gri I ⊗ Ω (gri D ⊗ Ω B B B B where the first term is the degree q −1 part and the second term is the degree q part. An element (x, y) in the first term which is mapped to zero by d comes from H q−1 (fili Sq ) b q−2 such that z ≡ y modulo fili+1 D ⊗ Ω b q−2 if and only if there exists z ∈ fili D ⊗ Ω B B and X b q−1 . f n (dz) ∈ fili I ⊗ Ω q B n>0 From here one deduces Proposition 3. 15.2.2. Differential modules. Take a prime element π of K such that π e−1 dπ = 0 . We assume that p - e in this subsection. Then we have M dTiq dTi1 q b A ∧ ··· ∧ ΩA ' Ti1 Tiq i1 <i2 <···<iq M dTiq−1 e−1 dTi1 ⊕ A/(π ) ∧ ··· ∧ ∧ dπ , Ti1 Tiq−1 i <i <···<i 1 2 q−1 b q as where {Ti } = T . We introduce a filtration on Ω A ( q b Ω A bq = fili Ω A i bq b q−1 π ΩA + π i−1 dπ ∧ Ω A ( if i = 0) ( if i > 1). The subquotients are b q = fili Ω b q /fili+1 Ω bq gri Ω A A (A q ( if i = 0 or i > e) ΩF = q q−1 ΩF ⊕ ΩF ( if 1 6 i < e), where the map is bq ΩqF 3 ω 7−→ π i ω e ∈ πi Ω A 1 b q−1 . Ωq− 3 ω 7−→ π i−1 dπ ∧ ω e ∈ π i−1 dπ ∧ Ω F A b q /pdΩ b q−1 ) be the image of fili Ω b q in Here ω e is the lifting of ω . Let fili (Ω A A A b q−1 . Then we have the following: b q /pdΩ Ω A A Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields Part I. Section 15. On the structure of the Milnor K -groups of cdv fields 133 Proposition 4. For j > 0 , q ΩF b q−1 = Ωq ⊕ Ωq−1 b q /pdΩ grj Ω A A F F q ΩF /Blq (j = 0) (1 6 j < e) (e 6 j), where l be the maximal integer which satisfies j − le > 0 . b q /pdΩ b q−1 ) because pdΩ b q−1 ⊂ file Ω b q = grj (Ω b q . AsProof. If 1 6 j < e , grj Ω A A A A A b q−1 is generated by sume that j > e and let l be as above. Since π e−1 dπ = 0 , Ω A b q−1 . By [ I ] (Cor. 2.3.14), pπ i dω ∈ elements pπ i dω for 0 6 i < e and ω ∈ Ω A0 b q if and only if the residue class of p−n dω belongs to Bn+1 . Thus file(1+n)+i Ω A b q /pdΩ b q−1 ) ' Ωq /B q . grj (Ω A F A l b q−2 and b q−1 /pdΩ By definition of the filtrations, expp preserves the filtrations on Ω A A b q−1 /pdΩ b q−2 ) → gri+e Kq (K) is surjective and its b q (K). Furthermore, exp : gri (Ω K p A A b q−1 /pdΩ b q−2 ) in gri (Ω b q−1 /pdΩ b q−2 ). Now kernel is the image of ψ(H q−1 (Sq ))∩fili (Ω A A A A b q−2 and H q−1 (Sq ) explicitly, thus we shall get the structure b q−1 /pdΩ we know both Ω A A of Kq (K) by calculating ψ . But ψ does not preserve the filtration of H q−1 (Sq ), so it is not easy to compute it. For more details, see [ N2 ], especially sections 4-8 of that paper. After completing these calculations, we get the result in (vi) in the introduction. b q−2 is much more complicated. b q−1 /pdΩ Remark. Note that if p | e , the structure of Ω A A For example, if e = p(p − 1), and if π e = p , then pπ e−1 dπ = 0 . This means the torsion b q−1 is larger than in the the case where p - e . Furthermore, if π p(p−1) = pT part of Ω A for some T ∈ T , then pπ e−1 dπ = pdT , this means that dπ is not a torsion element. This complexity makes it difficult to describe the structure of Kq (K) in the case where p | e. Appendix. The mapping fiber complex. This subsection is only a note on homological algebra to introduce the mapping fiber complex. The mapping fiber complex is the degree −1 shift of the mapping cone complex. f Let C · → D · be a morphism of non-negative cochain complexes. We denote the degree i term of C · by C i . Then the mapping fiber complex MF(f )· is defined as follows. MF(f )i = C i ⊕ D i−1 , differential d: C i ⊕ D i−1 −→ C i+1 ⊕ D i (x, y) 7−→ (dx, f (x) − dy). Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields 134 J. Nakamura By definition, we get an exact sequence of complexes: 0 −→ D[−1]· −→ MF(f )· −→ C · −→ 0, where D[−1]· = (0 → D 0 → D 1 → · · · ) (degree −1 shift of D · . ) Taking cohomology, we get a long exact sequence · · · → H i (MF(f )· ) → H i (C · ) → H i+1 (D · [−1]) → H i+1 (MF(f )· ) → · · · , which is the same as the following exact sequence f · · · → H i (MF(f )· ) → H i (C · ) → H i (D · ) → H i+1 (MF(f )· ) → · · · . References [B] [BK] [BT] [F] [G] [I] [Kh] [Kt1] [Kt2] [Ku1] [Ku2] [Ku3] [Ku4] [N1] S. Bloch, Algebraic K -theory and crystalline cohomology, Publ. Math. IHES, 47(1977), 187–268. S. Bloch and K. Kato, p -adic étale cohomology, Publ. Math. IHES, 63(1986), 107–152. H. Bass and J. Tate, The Milnor ring of a global field, In Algebraic K -theory II, Lect. Notes Math. 342, Springer-Verlag, Berlin, 1973, 349–446. I. Fesenko, Abelian local p -class field theory, Math. Ann., 301 (1995), pp. 561–586. J. Graham, Continuous symbols on fields of formal power series, In Algebraic K -theory II, Lect. Notes Math. 342, Springer-Verlag, Berlin, 1973, 474–486. L. Illusie, Complexe de de Rham–Witt et cohomologie cristalline, Ann. Sci. Ecole Norm. Sup. 12(1979), 501–661. B. Kahn, L’anneau de Milnor d’un corps local à corps residuel parfait. Ann. Inst. Fourier 34, 4(1984), 19–65. K. Kato, A generalization of local class field theory by using K -groups. II, J. Fac. Sci. Univ. Tokyo 27(1980), 603–683. K. Kato, On p -adic vanishing cycles (applications of ideas of Fontaine–Messing). Adv. Stud. in Pure Math. 10(1987), 207–251. M. Kurihara, Abelian extensions of an absolutely unramified local field with general residue field, Invent. Math. 93(1988), 451–480. M. Kurihara, The Milnor K -groups of a local ring over a ring of the p -adic integers, to appear in the volume on Ramification theory for arithmetic schemes. Lminy 1999, ed. B. Erez. M. Kurihara, On the structure of the Milnor K -group of a certain complete discrete valuation field, preprint. M. Kurihara, A note on p -adic etale cohomology, Proc. Japan Acad. (A) 63 (1987), 275–278. J. Nakamura, On the structure of the Milnor K -groups of some complete discrete valuation fields, K -Theory 19 (2000), 269–309. Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields Part I. Section 15. On the structure of the Milnor K -groups of cdv fields [N2] [P] [Z] 135 J. Nakamura, On the Milnor K -groups of complete discrete valuation fields, Doc. Math. 5(2000), 151–200. A. N. Parshin, Local class field theory, Trudy Mat. Inst. Steklov (1984); English translation in Proc. Steklov Inst. Math. 165 (1985), no. 3, pp. 157–185. I. Zhukov, Milnor and topological K -groups of multidimensional complete fields, Algebra i Analiz (1997); English translation in St. Petersburg Math. J. 9(1998), 69–105. Department of Mathematics University of Tokyo 3-8-1 Komaba Meguro-Ku Tokyo 153-8914 Japan E-mail: [email protected] Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields