7. GALOI S COHOMOLOG Y O F LOCA L FIELD S

17

7. Galoi s cohomolog y o f local fields

7.1.

INERTI A GROUPS .

Le t K be a field with a discrete valuation v with residu e

field k(v) = k. Le t Ks b e a separable closur e of if, an d put IV = Gal(if s/K) an d

similarly fo r IV Assum e unti l 7.13 that K i s complete. Th e valuation v extend s

uniquely t o Ks; it s residue field is an algebraic closur e of k. Thi s give s a surjectio n

FK — » r^ wit h kerne l the inertia group I.

We have a natural exac t sequenc e

1 — • ^wild —/—/*— 1,

where

It = tam e inerti a grou p (se e below)

and

_ f l ifchar(/c ) = 0

wl

[th e unique p-Sylow subgrou p o f I i f char(/c) = p.

Let K

t

b e the subfield o f Ks fixed b y /wild? i t ha s residue field k s. Le t if Unr b e

the subfiel d o f Ks consistin g of the elements fixed by /. The n K

t

i s generated a s a

ifunr-algebra b y all n-th roots of a uniformizing elemen t TT of K fo r n not divisible

by the residue characteristic . Fo r such n, let /in be the group of n-th root s of unity

(in h* or if* nr, i t amount s t o the same). W e have a n isomorphism I

t

— * lim^n

given b y sending s G It t o £

s

G fin define d b y (

s

= S(TT 1 'ln)/'TT1 'ln. I t doe s not

depend o n the choice of TT no r on that o f 7r1//n.

REMARK

7.2 . Th e above description o f / an d It give s canonical isomorphisms :

(7.3) Hom(J , /x J = Hom(Jt, /in) = Z/nZ .

If on e identifie s Hom(7 , /in) with i^unr/^un r b y Kummer theory , th e isomorphism

Hom(7, /xn) = Z/n Z becomes the map

^unr/^unr~~ * Z/nZ give n by x \— v(x) (mo d n).

1 A.

COHOMOLOGY .

Fo r the rest o f this section , C denotes a finite T^-modul e

of order not divisible by the residue characteristic. W e also view C as a T^-modul e

with trivial action by /. (Mos t of what follow s also works for infinite torsion module s

when thei r element s hav e orde r prim e to the residue characteristic. )

LEMMA 7.5 . H l{I, C ) = 0 fori 2 .

PROOF.

Th e ^-Sylow subgroup s o f / (fo r £ ^ char(fc) ) ar e isomorphic t o Z^.

Hence the ^-cohomological dimensio n o f / i s 1; the lemma follows . •

LEMMA 7.6 . The sequence 1 — / — • YK — • T^— * 1 is split.

PROOF,

(cf . [Ax65] and [Se65, II.4.3, exerc.]) Fo r each m 1 and not divisible

by the residue characteristic , choos e a n ra-th roo t 7r

m

of TT S O that th e choices are

compatible wit h takin g power s (i.e. , (7r

mm/)m

= ^m)- Th e extension K' o f K

generated b y the 7rm's i s linearly disjoin t fro m K

nnr

an d we have K

unr

K' — K t.

This show s tha t th e exact sequenc e 1 —• It— Fx/^wiid ~ ^k -^ 1 splits. Thi s

proves the lemma if char(fc) = 0. If char(&) = p 0, the p-cohomological dimensio n

of Tk is 1 [Se65, II.2.2 , Prop . 3] , so that ever y homomorphis m T^ — • Tx/Iwiid

can be lifted t o TK [Se65 , 1.3.4, Prop. 16]. •