Geometry & Topology Monographs Volume 3: Invitation to higher local fields

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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
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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 )· ) → · · · .
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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
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