On the Galois cohomology groups of algebraic tori and Hasse's norm theorem

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Title On the Galois cohomology groups of algebraic tori and Hasse's norm theorem

Author(s) AMANO, Kazuo

Citation [岐阜大学教養部研究報告] vol.[13] p.[185]-[190]

Issue Date 1977

Rights

Version 岐阜大学教養部 (Dep. of Math., Fac. of Gen. Educ., Gifu Univ.)

URL http://hdl.handle.net/20.500.12099/47402

※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。

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K azuo A m ano

D叩 , of M ath・, F ae. j Gen. E dl, e. , G ㈲ U nil, .

On the Galois cohomology groups of algeblʻaic tori and H asseʼs norm theor em

algebraic torus T defined over ん which splits over K .

§2. GALoIs CoHoM01.0GY oF ToR1

2. 1 L et 苔 be the group oI K - rational points of T and 7¯y栓 the adele group of 7 0ver 瓦 W e denote by X = HO・ ( G 。, T ) the set of morphisms of G ointo T defined over 尺 and which are also group homomorphisms, where G o is the multiplicative group of universal domain. W e let (M operate on χ by the rule ( 舵 /) ( 8d) = 式/( わ) for

s E @ , y E X, and l E 7¯ I Thenitiswenknown that 几弓 X ⑧ 尺̀ and 不帰ミ X ⑧ 収

as (S5-modules, where み is the idele group of 瓦 Denoting by Cg= 収ZK̀ the idele class group of 瓦 we have the eχact sequence of (S5- modules since χ is Z -free; o → χ ⑧ 尺̀ → χ ⑧ み → χ ⑧ Cχ→ o and hence we can identify χ ⑧ Cχ with 八 / 几 as (S5-modules. Putting G ( ア) = T≒Z TX・we ca11Cバ T) the adele class group of T

oごr瓦 The nthec upm ultiplic atio nbythecano nical c lassof 万 2(@, (λ) induc e s an

isomorphism か明, C) ⊇か・2((55, G ( 勁) for every integers 7z〔5〕.

A nalogous result in the local field is the following, L et p be a place in ん and 雫 a place over p in 瓦 W e denote bj h and K、1, the completions of ん and K by the places, respectively, and by (55叩the Galois group of A74,/ 馳, T hen the group 栴 of 尺1- rational points of 7¯ʼ is isomorphic to X ⑧ 八万ヽ; as (55r module and 少 e cup mu1リplication by the canonical class り of か ((MI , χj ) induces an isomorphism 拓明 ,l。即 ⊇ ? ゛明い

/

Tり) for every integersI・71 〔3〕゜ / ・ʼ

2.2 J. T ate show ed the f0110wing results in て9 〕 . L et y be the fr ee abelian group genel̀ated by the places 畢 of 瓦 A n element s E (55 0perates on y by the rule

j(ミ|%雫)=回り(邱). yed e n o teb yWtりeke rn e lo fs u rje c tive叫h o m o m o rp h is m

y → Z defined 卜y :ミi ≒ 叩 → 2; 恥 . T hell the cup multiplication by the canonical classes α1 ∈ か ( 勁, Hom ( Z , C g) ) , Q2 ∈ か ( (翫 Hom(Y , み ) ) and α3 ∈ 會価 ,

(Received Oct. 15, 1977)

§|. INTRoDucTIoN

L et 几 be an ʻalgebraic number field of finite degree and K a finite Galois eχten- sion or だ w ith G alois group 既 lt is w ellknow n as the generalization of H asseʼs norm

˜

theorem that, if w e denote by N (k) the subgroup of the multiplicative group ん̀ of ん consisting of elements w hich are local norm from 尺 at every places of ん, then the

group凧ん )/倣y ぶ帽sisomorphictoafactor groupof 万一 3凹, Z).

T he purpose of the present paper is to generalize this theorem to the case of an

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ΣM 妬 gi(召)

μ ⑧ /1

戸“((S 5, X (8) .Z) ≧HO¯2( (M・ X (8) Cg), か (@ , X ⑧ 杓⊇刄o-2(@ , X ⑧ み), か(@ , X ⑧ W) ⊇が 2(@ , X ⑧ 7ぐ勺;

moreover there exists a commutative diagram with eχact rows:

・・・→ か (@ , X ⑧ W) → か (肌 X ⑧ y) → か (吼 X ⑧ Z) → ‥

↓ ↓ ↓

・・・→ 方o¯2((55, X ⑧ 尺勺→ H“ 2((S 5, X ⑧ み)→ HO2((S 5, X ⑧ Cχ )→ ‥

よで二二言ご一二鸞ゴ言二九謡言二二三で

yl be a G-module, 召 an 有一module and j 谷召 a pair oI H- homomorphism s such thl t yM is the identity on 召 and y1° Σ が ( ム) , direct sum. T hen for any G-module 皿

we have an isomorphism g ・

K azuo A mano

186

˜ - -

the canonical (MI -injection which maps a non zero element of 尺,l, onto the idele having that element as 畢 -component and having l as components at all places other than 畢 .

Since(55, (55わ H λ y 47(resp. H lh), and 眉 (resp. 販 ) satisfy theconditions of semi-

local theory, we have

路(町 X ② H 肩) ⊇か((も , X ⑧ 肩), か ((55, X ⑧ H 叫 )⊇か ((S 5わ X ⑧ 叫 )

for any fixed prime 雫 dividing p . Since Uい s cohomologically trivial if 心 is unram-

ified over んp, X ⑧ 1h is also cohomologically trivial by the theory of local fields.

T herefore, if our set (B contains all places p of ん which ramify in 瓦 we have

≒リa(JX⑧肩) ×口 IJ錨丿(8)U皿 Fo re a c h雫o f瓦wed e n o teb y沁:鰐→お

⊇Σg(1(8) j) (M 区) £)

and the two maps

。 (1 ⑧ j)res 。

亙” (G, M ⑧ j ) 二二ご堵 (H, M 妬 B) cor( 1 ⑧ j)

are mutually inverse isomorphisms, where ” γeyʼ denotes restriction map and ”cor ” denotes corestriction map.

2.4 1f S is a finite set of places p in ん, w e also denote by the same symbo1 ら the set of the places 畢 of 尺 which divide some place p E ら . P utting

ベニEITり X HTり, theadelegroupTAxof7¯ ʼisdefirledastheirlduct沁eli°itof 几 relative

平Es 平唾 s

to G , w here 几ぶ= χ 必 U 、j s the unit gr oup of 尺平. S ince t H = χ ⑧ 尺喬 , w e hav e 7¯毎

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T herefore we have the following

P ROPOsITlON L か (町 7¯ʼ≒ ) ≧ Σ か回礼几い力 r 凹 eひむ洽即 rs ・・.

2. 5 P utting yヽ= y Z 畢, we have y = Σ n ( direct) as (S5-module. A ccordingly

心ゆ 11

we have X ⑧ y = Σ ( χ ⑧ n ) as (S5-module. F or each place ̀μ oI K , we define a

(S 5ヽ l;-homomorphism ら : Z → y by 沁 0 ) = 71雫. Since (S 5, (55、 r, 妁 , and, Z43 satisfythe

the conditions of semi-local theory, we have

か (@ , X ⑧ y) ⊇Σ か (@ , X ⑧ 瓦 ) 二Σ か (叫 , 7≒ )

T herefore we have the foUowing

⊇Σか (叫 , X).

二肌かl(賜, 7¯ ʼり ).

pE ご

P assing to the inductive limit over sufficiently large S , we have

堵 ((聚 T愧) ⊇ lim μ”(眠 7笑)

で

か(眠 7瓦) ⊇H か (り, r17≒)

l・∈ 2 劉 p

が 2(町 7臨) Σ

HO¯2( @ , 几g) か 2回礼几l )

知明 , X ⑧ y)

2。6 U sing these propositions, w e have the follow ing P RoPOsITIoN 3. The fotLom維g diagm m 18 commut(ltiue:

PRoposm ox 2. か ( @ , X ⑧ y) = Σ か ((恥 , X ) foT 四 e巧 integeT8 n.

p

耳か(吼, X)

LVUα ,

・X

Z

˜

方O¯2(叫 , 几い恚

P RoOR T he top horizontal isomorphism iʼ is induced by the maps

か回礼蜀犬か(町 X ⑧ 杓

and the bottom horizontal isomorphism i is induced by the m aps

COバ1 ⑧ ふ)

By the fundamental relation between corestriction and cup product, we have

Q U(C OT(1 ⑧ バ) ぐ) = c o代Te8 a2U( 1⑧ に) ぐ)

= cor G ヽ1, ・ Tes Q 2 U ぐ) = co ぺ沁叫 U ぐ )

= coべi ⑧ 晦 ) (偽 U ぐ),

whereぐ ∈ か凹も萄, andバ‰ theprojectionHO ・ (y, χ) → X defined by八( 九)

= μ雫) for 雫 ∈ (B. Therefore we have our proposition.

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o → 几二→ T4 二 → cバ勁 → 0

P assing to cohomology groups, we have the eχact sequence

¨ ʼ→ 万¯1(凧几) 二戸¯1(勁, T柚)二万¯1(眠 Cバ7¯ ʼ))

→ 分〇(既几) 二方o(眠八い二か (町 Cバ7つ) → ¨ t

P RoPosITlON 配 T he foLLom錐g diagm m is commut(l tiue

叫

Σ 叩

188

K azuo A mano

§ 3. HAssEʼsNoR,y I Ti-{EoRE,v l of ToRI

W e consider the following eχact sequence of (S5-modules

Σ か (叫 , X)

p レ・

か (吼萄

↓ 万卜 ↓ Uα I

Σ

耳が¯2((551, 711)二 →HO2((55・八g)

レ

かO (町 CバT) )

Uα1

か+2((55゛・几いこ →が 2(@・八g) 二か¯ 1 ʼ2((S 5・Cバフ))

T his proves the proposltlon

Denote by yV げ ) 二几ら ( O yV‰ Z臨T ≒ ) the subF oup of 几 collsistillg of elellle趾 s which are local norm from 几 at every places ol k, where 几 is the group of ん- rational points of 7¯I T hen, as our nlain result, we have the generalization of H asseʼs norm theorem to the case of. an algebraic torus.

THEon回. Ld F 酸 X加 s晶gro叩げ方¯3(町 X) geneT(1ted by こヽ駱(鉦3((Mも X) ) μ『

eveび p. 7¯ʼhe71 uXe h(1tJe (1n 180・ oTphis・

斤げ )/y V卜 l哉Tχ三H- 3((9, 萄/R

P RooF. By virtue of proposition 4, we have

yV( 7¯ ʼ)ZNい TX= KeT( f *)

≧か 1(@, CバT) ) /j *(か I (眠八い )

レ・

P RooR By virtue of T ateʼs commutative diagram 2.2 and proposition 3, we have the following commutative diagram

Σ ら : Σ か (叫 , X) → か ((M, X (8) y) → か ((55, X)

T herefore we have Ke八 i *) = O by proposition 1

7¯ 八)

こ H-3 ((55, X) /E

ConoLL八nY( 〔 5〕 )・U K Zk 18 cydic eχten810n, lj e hE e l¥y( 7) = y Vχ 71TIい

P ROOF. By virtue of K neserʼs paper 〔3〕 , there is, for every integers 71, a ca nonical injection

か (町 7¯ y) → Σ か (叫 ,

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Hiコj Haip︹巴回〇く乱 9の{ 〇 } } 〇だFnroヨ〇r phi s ms せ ( 夕肆)㈲tぺ芦こ﹃゛せ ( 夕 jし gJjyD JO

where E and E are the idele grI ps of i and L respectivelyo p ooaatoコaFn {〇rus 7 which thの∃oa巴の

̀ 心 r

○

1 卜 3

ン

レト

X

レミ 1

0

Now we consider the exact sequence ofQ5 m odules2

94 ArrΓ一〇yH一〇z Ho z oz GAΓo一㎝ mxHmz∽一z

? 51s sectionswe give Hassets norm thのo9日 {〇 9の QSの o{ コ○コー(い巴or のx芯コー

器〇品回F叫 p roくのの〇mmut a t i vedi agram f orTasakag ss peci alt orH2 8︺゜

Ho 5e Z jee 6 modulej4 there ls

the ch aracter m odule H om ( リ Q し ʼ

sr諮 N ︹ ( ぶへむ)=M NF Q=り? Q 乱心=M P

4j ﹂et ﹂ be a separabl e ext ensi on 〇f i and K a fi ni t e ( い巴〇i s extensi on 〇f 1

containing L W e denote b y 6 a乱か 9 の 0 巴 or 9 0up of K 4 and K ZL respectiv e ly ﹂ et Q5 u g g b e a left coset decom p 〇sition W e consider the follow ing left 05 m 04已 a

y=N︹ 9ふ︺゛哨=ンヘ NF

}⁚)assing to coh om 〇10gy groupS w e obtain th e toUow lng ex act sequence2

where Rxzj Gし cIonotes the algebraic group detined over A70btained by restricting the

11eld ot delln ltIon A to t B y ︹吝︺o w e h av e j M R J O o a乱 hence w e obtain }にlassets n orm th eo rem fo r ﹂ Z k

REFERENCES

̀ ヨ

D e Rxa (Q して 7 ~ ︑= y 乙 = こ

-⌒tヽ、⊃← -- E o ARTIN AND I T ATEoClass leld theoly Benjam R 19670 (いʻ (2HEvA )﹂EY4 C lass field theoly N agoyao 195E 54

MoKNESERo﹂ecture on Galois cohomology ofclassicalgroupsoTata lnstoFundoRe4 Bombay

● ● S 士 _{せ (} _{夕 j )1い}

せ( 夕丿し士 ● ● ● や( ﹃ yj しレ﹄・ ( ( 洲の気き) 上

cub ic ex tension of M and Q5 = 一I = 飛

G al oi s extensi on X Zk W e consider a

く巴Ξ ︹81 ︺et ﹂ be a separable n〇n cyclic

= 1s r 1X r J二 9の (い巴oF 9 0にt o{6の

{ so︲Eヨのコ巴〇コ巴 { 〇回Q

By vi rt ue 〇{ 9 e com m utati ve di agram 〇{ 召〇で〇吃に〇コ 4 pコQ pご〇ve i som orphi sm S w e

h av e the 10110w m g

T 4EOREMO ﹂d y (0 6e a e n 4 ro9 j a e m zzM pacaG e gro9 1 ︒oZ i co71M M ag

oZ d em eM s zzjM d a re 70cd n xm j om ﹂ d g et p M ces j k 71 9 r e A aR a7z ja

moqMsm

司( ″ ) へ芦︑ y ︒旧H MR Xj Z E

where F de71des Ztze sg gru p oXH 2(R X 4 a erd d 4yG ﹃﹄︲ズ 9夕にべ))ybr eR 71yW t2 ﹂Jsing above thの0rem owe give an example such thatH assegsnorm theorem is

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K azuo A mano

190

196 9.

〔4〕 T. 0NO,Arithmetic of algebraic tori, Ann. of Math., 74(1961), 101-139.

〔5〕 T. 0χ0, 0n the Tamagawa number of algebraic tori, Ann. of Math., 78(1963), 47-74.

〔6〕 J.-P. SERRE, Corps locauχ, Hermann Paris, 1962.

〔7〕 T. TASAKか On the second cohomology groups of the fundamental groups of simple algebraic groups over perfect fields, J. M ath. Soc. Japan, 21( 1969) , 244-258.

〔8〕 T. TASAKA, Remarks on the validity of Hasseʼs norm theorem, J. Math. Soc. Japan, 22( 1970),

330- 34 1. f ,

〔9〕 J. TATE, The cohomology groups of tori in finite Galois eχtensions Qf number fields, Nagoya Math. J., 27( 1966), 709-719.

〔11〕 V. E. VOSKRESENSKII, 0n two-dimensional algebraic tori, Amer. Math. Soc. Transl., 73( 1968) ,

On the Galois cohomology groups of algebraic tori and Hasse's norm theorem

Title On the Galois cohomology groups of algebraic tori and Hasse's norm theorem

Author(s) AMANO, Kazuo

Citation [岐阜大学教養部研究報告] vol.[13] p.[185]-[190]

Issue Date 1977

Rights

Version 岐阜大学教養部 (Dep. of Math., Fac. of Gen. Educ., Gifu Univ.)

URL http://hdl.handle.net/20.500.12099/47402

※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。

D叩 , of M ath・, F ae. j Gen. E dl, e. , G ㈲ U nil, .

On the Galois cohomology groups of algeblʻaic tori and H asseʼs norm theor em

§2. GALoIs CoHoM01.0GY oF ToR1

s E @ , y E X, and l E 7¯ I Thenitiswenknown that 几 弓 X ⑧ 尺̀ and 不帰ミ X ⑧ 収

oごr瓦 The nthec upm ultiplic atio nbythecano nical c lassof 万 2(@, (λ) induc e s an

isomorphism か 明, C) ⊇か ・2((55, G ( 勁) for every integers 7z〔5〕.

Tり) for every integersI・71 〔3〕゜ / ・ʼ

j(ミ|%雫)=回り(邱). yed e n o teb yWtりeke rn e lo fs u rje c tive叫h o m o m o rp h is m

§|. INTRoDucTIoN

group凧ん )/倣y ぶ帽sisomorphictoafactor groupof 万一 3凹, Z).

ΣM 妬 gi(召)

戸“((S 5, X (8) .Z) ≧HO¯2( (M・ X (8) Cg), か (@ , X ⑧ 杓⊇刄o-2(@ , X ⑧ み), か(@ , X ⑧ W) ⊇が 2(@ , X ⑧ 7ぐ勺;

・ ・ ・→ か (@ , X ⑧ W) → か (肌 X ⑧ y) → か (吼 X ⑧ Z) → ‥

↓ ↓ ↓

・ ・ ・→ 方o¯2((55, X ⑧ 尺勺→ H“ 2((S 5, X ⑧ み)→ HO2((S 5, X ⑧ Cχ )→ ‥

よで二二言ご一 二鸞ゴ 言二九謡言二二三で

K azuo A mano

Since(55, (55わ H λ y 47(resp. H lh), and 眉 (resp. 販 ) satisfy theconditions of semi-

路(町 X ② H 肩) ⊇か((も , X ⑧ 肩), か ((55, X ⑧ H 叫 )⊇か ((S 5わ X ⑧ 叫 )

ified over んp, X ⑧ 1h is also cohomologically trivial by the theory of local fields.

≒リa(JX⑧肩) ×口 IJ錨丿(8)U皿 Fo re a c h雫o f瓦wed e n o teb y沁:鰐→お

⊇Σg(1(8) j) (M 区) £)

亙” (G, M ⑧ j ) 二 二 ご 堵 (H, M 妬 B) cor( 1 ⑧ j)

ベニEITり X HTり, theadelegroupTAxof7¯ ʼisdefirledastheirlduct沁eli°itof 几 relative

to G , w here 几 ぶ= χ 必 U 、j s the unit gr oup of 尺 平. S ince t H = χ ⑧ 尺喬 , w e hav e 7¯毎

(S 5ヽ l;-homomorphism ら : Z → y by 沁 0 ) = 71雫. Since (S 5, (55、 r, 妁 , and, Z43 satisfythe

か (@ , X ⑧ y) ⊇Σ か (@ , X ⑧ 瓦 ) 二Σ か (叫 , 7≒ )

⊇Σか (叫 , X).

二肌 かl(賜, 7¯ ʼり ).

堵 ((聚 T愧) ⊇ lim μ”(眠 7笑)

か(眠 7瓦) ⊇H か (り, r17≒)

が 2(町 7臨) Σ

HO¯2( @ , 几g) か 2回礼 几l )

知 明 , X ⑧ y)

耳か(吼, X)

LVUα ,

方O¯2(叫 , 几い 恚

か回礼蜀 犬 か(町 X ⑧ 杓

COバ1 ⑧ ふ)

Q U(C OT(1 ⑧ バ) ぐ) = c o代Te8 a2U( 1⑧ に) ぐ)

= coべi ⑧ 晦 ) (偽 U ぐ),

whereぐ ∈ か 凹も 萄, andバ‰ theprojectionHO ・ (y, χ) → X defined by八( 九)

= μ雫) for 雫 ∈ (B. Therefore we have our proposition.

o → 几 二→ T4 二 → cバ勁 → 0

¨ ʼ→ 万¯1(凧 几) 二 戸¯1(勁, T柚)二 万¯1(眠 Cバ7¯ ʼ))

→ 分〇(既 几) 二 方o(眠 八い二 か (町 Cバ7つ) → ¨ t

Σ 叩

K azuo A mano

§ 3. HAssEʼsNoR,y I Ti-{EoRE,v l of ToRI

Σ か (叫 , X)

か (吼 萄

↓ 万 卜 ↓ Uα I

Σ

耳が¯2((551, 711)二 →HO2((55・八g)

レ

かO (町 CバT) )

か+2((55゛・几いこ →が 2(@・八g) 二 か¯ 1 ʼ2((S 5・Cバフ))

THEon回. Ld F 酸 X加 s晶gro叩 げ 方¯3(町 X) geneT(1ted by こヽ 駱(鉦3((Mも X) ) μ『

斤げ )/y V卜 l哉Tχ三H- 3((9, 萄/R

yV( 7¯ ʼ)ZNい TX= KeT( f *)

≧か 1(@, CバT) ) /j *(か I (眠 八い )

レ・

Σ ら : Σ か (叫 , X) → か ((M, X (8) y) → か ((55, X)

7¯ 八)

こ H-3 ((55, X) /E

ConoLL八nY( 〔 5〕 )・U K Zk 18 cydic eχten810n, lj e hE e l¥y( 7) = y Vχ 71TIい

か (町 7¯ y) → Σ か (叫 ,

Hiコj Haip︹ 巴 回〇く乱 9の{ 〇 } } 〇だFnroヨ〇r phi s ms せ ( 夕 肆)㈲tぺ 芦こ﹃ ゛ せ ( 夕 jし gJjyD JO

̀ 心 r

1 卜 3

レ ト

s E @ , y E X, and l E 7¯ I Thenitiswenknown that 几弓 X ⑧ 尺̀ and 不帰ミ X ⑧ 収

isomorphism か明, C) ⊇か・2((55, G ( 勁) for every integers 7z〔5〕.

・・・→ か (@ , X ⑧ W) → か (肌 X ⑧ y) → か (吼 X ⑧ Z) → ‥

・・・→ 方o¯2((55, X ⑧ 尺勺→ H“ 2((S 5, X ⑧ み)→ HO2((S 5, X ⑧ Cχ )→ ‥

よで二二言ご一二鸞ゴ言二九謡言二二三で

亙” (G, M ⑧ j ) 二二ご堵 (H, M 妬 B) cor( 1 ⑧ j)

to G , w here 几ぶ= χ 必 U 、j s the unit gr oup of 尺平. S ince t H = χ ⑧ 尺喬 , w e hav e 7¯毎

二肌かl(賜, 7¯ ʼり ).

HO¯2( @ , 几g) か 2回礼几l )

知明 , X ⑧ y)

方O¯2(叫 , 几い恚

か回礼蜀犬か(町 X ⑧ 杓

whereぐ ∈ か凹も萄, andバ‰ theprojectionHO ・ (y, χ) → X defined by八( 九)

o → 几二→ T4 二 → cバ勁 → 0

¨ ʼ→ 万¯1(凧几) 二戸¯1(勁, T柚)二万¯1(眠 Cバ7¯ ʼ))

→ 分〇(既几) 二方o(眠八い二か (町 Cバ7つ) → ¨ t

か (吼萄

↓ 万卜 ↓ Uα I

か+2((55゛・几いこ →が 2(@・八g) 二か¯ 1 ʼ2((S 5・Cバフ))

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