1. No category

Title
A remark on Galois cohomology groups of algebraic tori.
Author(s)
AMANO, Kazuo
Citation
[岐阜大学教養部研究報告] vol.[11] p.[159]-[160]
Issue Date
1975
Rights
Version
岐阜大学教養部 (Dep. of Math., Faculty of General Education,
Gifu University)
URL
http://repository.lib.gifu-u.ac.jp/handle/123456789/46005
※この資料の著作権は、各資料の著者・学協会・出版社等に帰属します。
159
A remark on Galois cohomology groups of algebraic tori.
By K azuO A MANO
Dep. oJ M ath・, F αc. oy` Ge71. £ dtxcj, G φ x び戒 む.
( Recelved Sept・, 30, 1975)
L et K be an algebraic number field and K
1
Galois group G.
尺.
the Galois eχtension of
L£t T be an algebraic torus defined over 尺
which
W e set χ= H om ( Gm, 7 ) , where Gm is the multiplicative group of
paper ,
w e shal l g ive a pr oper ty of j ¯ 3 ( G, X )
w ith
universal
domain. ln my paper [ 1 1, we studied that the Galois cohom610gy 町 oup
χ) had important r01e to the validity of Hassej
K
splits over
示 3( G・
norm theoi・em of T.
ln this
as an anal ogy to the r esul t
as is
well known to the case of χ = Z .
2.
W e consider the fo110w ing situation :
尺
the union of the finite H ilbert class field tower,
finite H ilbert p- class field tower,
み
say Case( H ) ,
say Case ( H , p) ;
or the
し
the idele group of 尺;
e夕
Cχ the idele class group of K
びx the group of idele units;・
£ χ the group of units in 尺 ;
C& the ideal class group of K ;
71 = χ(召
) 尺)( the group of 尺-rational points of 刄 where K` is the multiplicative group of 尺 ;
n 。 = χ(呂)み the adele group of T.
By the follow ing eχact sequence;
O→
尺い
み→
Cχ→
0,
we have the eχact sequence;
O一 X⑧ 尺゛ →X⑧ み一 X⑧ Cこ → 0.
T herefor6 we can identify χ ⑧ Cχ w ith TAoZ Tχ。
set
Cg( T ) =
714. / 71 ,
the ad el e cl a s s g r oup of
71
二
F or the sake of simplicity,
S ince χ
is
Z - f r ee m odul e,
have χ(亘)£χ= χ⑧K゛nχ⑧ Uχ
. We set n = χ⑧ Eχ
, the unit group of T.
By the follow ing eχact sequences;
O→
O→
UgZEχ→
£ x→
Cχ→
UX→
Cな→
Uχ7ElC→
O
0,
we have the eχact sequences;
O一 X⑧( W /& ) 一 X⑧ G 一 X② Cなー→ O
O一 X⑧ & 一 X⑧ 脆一 X②( W/& ) 一 〇.
T herefore we have
X⑧ Ctχさ X ② Cg/χ⑧ ( UXyEX)
゛ X③ み/X② r /X(8)UXIχ(8)&
1
卜
we
we
160
Kazuo
A M ANO
= X⑧ み/( χ⑧ r ) ・ ( X⑧ UX)・
F or the sake of simplicity,
THEOREM.
C(1se( 珀
we set C & ( 7 ) = X ⑧ C な 尚a㎡
71 . = χ ⑧ 脆 .
£ d 尺 6e α71 aなe6r心c 71txm6eΓβeld a71d 尺/Å
; 疏e Galojs exle71sion oy
or Case( H, p) 仙i仏 Galois group G.
Then lj e hat
Xe 硫e JoUo切ing iso-
・ oTph18・ JoT eueT!J k tegeT 71,
Hil( G, TE。) 苫H71-3( G, X)
・ヽ
-
籾/1ere H is T (l te- co/10m㎡ og!y grotxps.
P R0 0 F.
S ince 尺/ i is unramified,
びg
is
cohomologically
hence X ③ びg is also cohomologically trivia1.
tr ivial
mo(! ule and
T herefore 万 ( G、T 、 ) = O and hence
j -1(G, χ②( UXyEχ)) 包H11( G, TE。) for every integer 71.
0n the other hand, we
have the follow ing eχact sequence;
O- → TUoTχZTχ→
TAoyTχ→
?11
TAoyTUoTχ- → O
11
TUoZTEo
?¦¦
C爪 T)
C& ( 7)
Passing to cohomology w e have
-
‥‥‥- → H71¯2 ( G, C& ( 7) ) →
→
r ¯1( G, Cな( 7) ) →
By N akayama- T atej
_
召 - 1( G, TUoyTE。) →
。 。
H゛ - 1( G, Cg( 77) )
‥●
‥●
theorem in class field theory, we have
- ヽ
一
H71¯1( G, G ( T) ) き H -3( G, X)
lf 尺μ is Case( H) , Ca = 1 and our theorem is trivial.
lI K Zk i s Case( H , p) , C a is of order prime to p.
conclude that C &
is cohomologically trivia1.
Since
G
is
a p- group,
wQ
T herefore we obtain the follow ing
isomorphism for every integer 7x;
- ヽ
一
召 ( G, TE。) さ H゛ -3( G, X)
R EFERENCE S
〔 1〕
¥
K. Aman0, 0n the Galois cohomology groups of algebraic tori andHassei normtheorem,
to appea r ・
〔 2〕
K. lwasawa, A note on the group of units of an algebraic number
appl.
35 ( 1956) ,
field,
189 - 192.
〔 3〕
J. - P. Serre, Corps locauχ, Herrnann Paris, 1962.
〔 4〕
J. - P. Serre, Cohomologie galoisienne,
Springer Verlag, Berlin, 1964.
2
J. Math. pures