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On the Rationality of Algebraic Tori of Norm Type
Journal of Algebra 235, 27᎐35 Ž2001. doi:10.1006rjabr.2000.8473, available online at http:rrwww.idealibrary.com on
On the Rationality of Algebraic Tori of Norm Type Shizuo Endo Teikyo Uni¨ ersity, Hachioji, Tokyo 192-0352, Japan Communicated by Leonard L. Scott, Jr. Received February 20, 1999
1 Let G be a finite group. A G-module is called a permutation module if r it is isomorphic to [is1 ZGrGi , where Gi , 1 O i O r, are subgroups of G. A G-module M is called a quasi-permutation module if there exists an exact sequence 0 ª M ª S ª S⬘ ª 0, where S and S⬘ are permutation G-modules. The dual module Hom Z Ž M, Z. of a G-module M is denoted by M⬚. Let k be a field and let F be an extension field of k. Then F is said to be rational over k if it is generated by a finite number of elements of F which are algebraically independent over k and to be stably rational over k if there exists an extension field of F which is rational over each of k and F. Further, F is said to be retract rational over k if it is the quotient field of an integral domain A = k such that there are k-algebra homomorphisms
: A ª k w X 1 , X 2 , . . . , X n xw 1rs x
and
: k w X 1 , X 2 , . . . , X n xw 1rs x ª A, where k w X 1 , X 2 , . . . , X n xw1rs x is the localization of a polynomial ring k w X 1 , X 2 , . . . , X n x with variables X 1 , X 2 , . . . , X n over k with respect to 0 / s g k w X 1 , X 2 , . . . , X n x and ⭈ is the identity on A. It is easy to see that rational « stably rational « retract rational. Let G be a finite group and let M be a G-module with a Z-free basis u1 , u 2 , . . . , u n . Let k be a field and let K be a Galois extension of k with group G. Define the action of G on the rational function field 27 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
28
SHIZUO ENDO
K Ž X 1 , X 2 , . . . , X n . with variables X 1 , X 2 , . . . , X n over K, as an extension of the action of G on K, as follows. For each g G, n
Ž Xi . s
Ł X jm
ij
1 O i O n,
,
js1
when u i s Ý njs1 m i j u j , m i j g Z, and denote K Ž X 1 , X 2 , . . . , X n . with this action of G by K Ž M .. As is well known, there is an algebraic torus T defined over k and split over K such that the character group of T is isomorphic to M as G-modules, and the invariant subfield K Ž M . G of K Ž M . can be identified with the function field of T over k. We now have the following fundamental PROPOSITION. Let G be a finite group and let k be a field. Let K be a Galois extension of k with group G and let M be a Z-free and Z-finitely generated G-module Ž i.e., a G-lattice.. Then: Ž1. Ž e. g., w2, 1.6x. M is a quasi-permutation G-module if and only if K Ž M . G is stably rational o¨ er k. Ž2. w7, 3.14x M is a direct summand of a quasi-permutation G-module if and only if K Ž M . G is retract rational o¨ er k. Let p be a prime, and let P be an elementary abelian p-group of order p m , m P 1. Let Pi , 1 O i O r, be distinct subgroups of index p in P, and let i : Z PrPi ª Z be the augmentation maps. Further, for h1 , h 2 , . . . , h r P 1, let r
⌽ s Ž 1h1 , 2h 2 , . . . , rh r . :
[ Ž Z PrPi . h
i
ªZ
is1
and set L s Ker ⌽. The main result of this note is the following THEOREM 1. Ž1. In case of p s 2, L⬚ is a Ž direct summand of a. quasi-permutation P-module if and only if r s 1, 2. Ž2. In case of p / 2, L⬚ is a Ž direct summand of a. quasi-permutation P-module if and only if r s 1. By the Proposition, Theorem 1 can be restated as follows: THEOREM 2. Let G, P, and L be as abo¨ e and let K be a Galois extension of a field k with group P. Then: Ž1. In case of p s 2, K Ž L⬚. G is stably Ž retract . rational o¨ er k if and only if r s 1, 2.
29
ALGEBRAIC TORI OF NORM TYPE
Ž2. In case of p / 2, K Ž L⬚. G is stably Ž retract . rational o¨ er k if and only if r s 1. It is noted that the algebraic torus corresponding to L⬚ defined over k and split over K is of norm type. Part Ž1. of Theorem 2 is an answer to the question asked the late professor T. Miyata by A. Merkurjev in 1982 w6x. It should be noted that Theorem 1 was obtained in 1982 and reported without proof in w4x.
2 We now begin to prove Theorem 1. In order to prove this, we need to consider a more general situation. Let P, Pi , and i , 1 O i O r, be as above, and define the homomorphism ␦ : Z ª Z by ␦ Ž1. s p. For h1 , h 2 , . . . , h r P 1 and h P 0, let r
[ Ž Z PrPi . h
⌽ s Ž 1h1 , 2h 2 , . . . , rh r , ␦ . :
i
[ Z h ª Z,
is1
and set L s Ker ⌽. Further, let r
⌽⬘ s Ž 1 , 2 , . . . , r . :
[ Z PrPi ª Z, is1
and set L⬘ s Ker ⌽⬘. Then the key lemma is given as follows: LEMMA.
L ( L⬘ [ [is1 ŽZ PrPi . h iy1 [ Z h. r
Proof. First, define the homomorphism i : Z PrPi ª ŽZ PrPi . h i by i Ž a i . s Ž a i , 0, 0, . . . , 0. for each 1 O i O r, and set r
s Ž 1 , 2 , . . . , r . :
[ Z PrPi ª is1
r
r
[ Ž Z PrPi . h i : [ Ž Z PrPi . h i [ Z h . is1
is1
Next, define the homomorphism i : ŽZ PrPi ª Z PrPi by i ŽŽ a i1 , i a i2 , . . . , a i h i .. s Ý hjs1 a i j for each 1 O i O r and the homomorphism : r h Z ª [is1 Z PrPi by ŽŽ b1 , b 2 , . . . , bh .. s ŽŽÝ his1 bi s, 0, 0, . . . , 0. Ž s 0 when h s 0., where s denotes the sum of all elements of PrP1 in Z PrP1 , and set .hi
s Ž 1 , 2 , . . . , r , . :
r
r
is1
is1
[ Ž Z PrPi . h i [ Z h ª [ Z PrPi .
Then ⭈ is the identity on [ and it can be easily seen that Ž L⬘. : L and Ž L. : L⬘. Therefore, denoting the restrictions of , to L⬘, L by ˜, ˜ , respectively, we get homomorphisms ˜ : L⬘ ª L and r is1 Z PrPi
30
SHIZUO ENDO
˜ : L ª L⬘ such that ˜ ⭈ ˜ is the identity on L⬘. This shows that L ( L⬘ [ Coker ˜. However, as is easily seen, Coker ˜ ( Coker ( r Ž . h iy 1 [ Z h . Thus this concludes that L ( L⬘ [ [is 1 Z PrPi r h [is1 ŽZ PrPi . iy1 [ Z h. By this lemma we have only to consider the case where h1 s h 2 s ⭈⭈⭈ s h r s 1, that is, the exact sequence r
0ªLª
[ Z PrPi ª Z ª 0,
Ž ).
is1
where P1 , P2 , . . . , Pr are distinct subgroups of index p in P. It is known that L⬚ is a quasi-permutation P-module for each of the cases Ža. r s 1 and Žb. r s 2 and p s 2 w6x. However, for completeness, we give a proof of it. Assume first that r s 1. Then PrP1 is cyclic, and so L⬚ ( L. Hence L⬚ is clearly a quasi-permutation P-module. Next, assume that r s 2 and p s 2. Then PrP1 l P2 is the Klein four group, and therefore, regarding PrP1 l P2 as P, we can set P s ² , N 2 s 2 s 1, s : and P1 s ² :, P2 s ² :. Now we have Ker ⌽ s Z P Ž1, y1.. Define the epimorphism ⌿ s Ž , . : Z P [ Z ª L by Ž1. s Ž1, y1. and Ž1. s Ž1 q , 1 q .. Then, as is easily seen, Ker ⌿ s ŽŽ a q b .Ž1 q ., yŽ a q b .. N a, b g Z4 , and so Ker ⌿ ( Z Pr² :. Thus we have the exact sequence 0 ª Z Pr² : ª Z P [ Z ª L ª 0, which shows that L⬚ is a quasi-permutation P-module.
3 Consider here the case here p s 2 and r P 3. This case can be divided into the following: Ž1. w P : Pi l Pi l Pi x s 4 for some i1 - i 2 - i 3 . 1 2 3 Ž2. w P : Pi l Pi l Pi x s 8 for any i1 - i 2 - i 3 . 1 2 3 Set P⬘ s Pi1 l Pi 2 l Pi 3 in both cases and restrict the exact sequence Ž). to P⬘. Then we obtain the exact sequence 0 ª LP ⬘ ª Z PrPi1 [ Z PrPi 2 [ Z PrPi 3 [
[ Ž Z PrPi . P ⬘ ª Z ª 0.
i/i 1 , i 2 , i 3
It is noted that L⬚ is not Ža direct summand of. a quasi-permutation P-module if Ž LP ⬘ .⬚ is not Ža direct summand of. a quasi-permutation PrP⬘-module. Regarding PrP⬘ as P and applying the lemma to the above
ALGEBRAIC TORI OF NORM TYPE
31
exact sequence, the cases Ž1. and Ž2. reduce, respectively, to the following Ži.
P s ² , N 2 s 2 s 1, s :, the Klein four group, 0 ª L ª Z Pr² : [ Z Pr² : [ Z Pr² : ª Z ª 0.
Žii. P s ² , , N 2 s 2 s 2 s 1, s , s , s : Ža. 0 ª L ª Z Pr² , : [ Z Pr² , : [ Z Pr² , : ª Z ª 0. Žb. 0 ª L ª Z Pr² , : [ Z Pr² , : [ Z Pr² , : [ Z Pr² , : ª Z ª 0. We have only to show for these cases that L⬚ is not a direct summand of any quasi-permutation P-module. For brevity, a direct summand of a quasi-permutation module is called a d-quasi-permutation module. It should be noted that the case Ži. was examined by Miyata Žunpubw5x. We give here a proof of this case for lished, 1974. and Hurlimann ¨ completeness. Let P be as in the case Ži. and let I be the augmentation ideal of Z P. Define the homomorphism : Z Pr² : [ Z Pr² : [ Z Pr² : ª Z P as
Ž Ž 1, 0, 0 . . s 1 q ,
Ž Ž 0, 1, 0 . . s 1 q ,
Ž Ž 0, 0, 1 . . s 1 q .
Restricting to L, we obtain the epimorphism ˜ : L ª I. It is easy to see that Ker ˜ ( Z 2 is a trivial P-module. Hence we have the exact sequence 0 ª Z 2 ª L ª I ª 0. However, it is known w1, 3x that I⬚ is not a d-quasi-permutation module. This concludes that L⬚ is not a d-quasi-permutation module. Next, consider the case Žii., Ža.. We restrict the exact sequence to the subgroup P⬘ s ² , : of P. Then we see that Z Pr² , : [ Z Pr² , : [ Z Pr² , : ( Z P⬘r² : [ Z P⬘r² : [ Z P⬘r² : as P⬘-modules. This shows by the case Ži. that L⬚ is not a d-quasi-permutation P⬘-module. Thus L⬚ is also not a d-quasi-permutation P-module. The case Žii., Žb. is slightly complicated. We set M s Z Pr² : [ Z Pr² : [ Z Pr² : [ Z Pr² : [ Z Pr² : [ Z Pr² : [ Z 3 and N s Z Pr² , : [ Z Pr² , : [ Z Pr² , : [ Z Pr² , :. Define the 9 homomorphisms as Z Pr² : ª N
by 1 ª Ž 1, y1, 0, 0 .
Z Pr² : ª N
by 1 ª Ž 1, 0, y1, 0 .
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SHIZUO ENDO
Z Pr² : ª N
by 1 ª Ž 0, 1, y1, 0 .
Z Pr² : ª N
by 1 ª Ž 1, 0, 0, y1 .
Z Pr² : ª N
by 1 ª Ž 0, 0, 1, y1 .
Z Pr² : ª N
by 1 ª Ž 0, 1, 0, y1 .
ZªN
by 1 ª Ž 1 q , y Ž 1 q . , 0, 0 .
ZªN
by 1 ª Ž 1 q , 0, y Ž 1 q . , 0 .
ZªN
by 1 ª Ž 1 q , 0, 0, y Ž 1 q . . .
By adding these homomorphisms, we obtain the homomorphism ⌿ : M ª N. It is easy to see that the image of ⌿ coinsides with L. Set H s Ker ⌿. Then we have the exact sequence 0 ª H ª M ª L ª 0. By direct computation we show that M P ⬘ ª LP ⬘ is surjective for each subgroup P⬘ of P. Therefore it follows that H 1 Ž P⬘, H . s 0 for each subgroup P⬘ of P, where H i Ž , . denotes the Tate cohomology group. Assume that L⬚ is a d-quasi-permutation. Then, by Žthe proof of. w3, 1.6x, there is an exact sequence 0 ª T ª D ª L ª 0, where T is a permutation module and D is a direct summand of permutation module. Taking the pullback of M ª L ¤ D, we obtain H [ D ( T [ M. Since P is a 2-group, H is locally isomorphic to a permutation module S. We denote the completion of a module V at 2 by V *. Then we have H * ( S*. Set s . Considering ² :-invariants we have Ž L*. ² : ( ŽZ*. 3 and
Ž H *.
² :
2
( Ž Z*Pr² , : [ Z*Pr² , : [ Z*Pr² , : . .
Since rank Z H s 20 and rank Z H P s 6, we have H * ( Z*PrP1 [ Z*PrP2 [ Z*PrP3 [ Z*PrP4 [ Z*PrP1X [ Z*PrP2X , where < P1 < s < P2 < s < P3 < s < P4 < s 2 and < P1X < s < P2X < s 4. Using the cohomology sequence of the exact sequence 0 ª L ª N ª Z ª 0, we see that H 1 Ž P, L. ( Zr2Z and H 0 Ž P, L. ( ŽZr4Z. 3. Further, taking the cohomology sequence of the exact sequence 0 ª H ª M ª L ª 0, we have the exact sequence H 0 Ž P, H . ª H 0 Ž P, M . ª H 0 Ž P, L. ª H 1 Ž P, H . s 0. By direct computation we also have H 0 Ž P, H . ( ŽZr2Z. 4 [ ŽZr4Z. 2 and H 0 Ž P, M . ( ŽZr2Z. 4 [ ŽZr8Z. 3. Therefore we obtain the exact sequence 4 2 6 3 3 Ž Zr2Z. [ Ž Zr4Z. ª Ž Zr2Z. [ Ž Zr8Z. ª Ž Zr4Z. ª 0.
However, it is easy to see that the kernel of ŽZr2Z. 6 [ ŽZr8Z. 3 ª ŽZr4Z. 3
ALGEBRAIC TORI OF NORM TYPE
33
is isomorphic to ŽZr2Z. 9. This is a contradiction, because ŽZr2Z. 4 [ ŽZr4Z. 2 never maps onto ŽZr2Z. 9. Thus L⬚ is not a d-quasi-permutation.
4 Finally consider the case where p / 2 and r P 2. We now prove that L⬚ is not a d-quasi-permutation. Restricting the exact sequence Ž). to the subgroup P⬘ s P1 l P2 , regarding PrP⬘ as P and using the lemma as in the case where p s 2, we can reduce this case to the one where < P < s p 2 . From now on, assume that P is of order p 2 . Under this assumption it is noted that there exist exactly p q 1 subgroups of order p in P. Let Pi , r 1 O i O r, be distinct subgroups of order p in P and set N s [is1 Z PrPi . Let i : Z PrPi ª Z, 1 O i O r, be the augmentation maps and let ⌽ s Ž 1 , 2 , . . . , r . : N ª Z. Set L s Ker ⌽. Then we have the exact sequence 0 ª L ª N ª Z ª 0. py 1 For 1 O i O r, set PrPi s ² i : and si s Ý js0 ij g Z PrPi . Define the homomorphisms as ZP ª N
by 1 ª Ž 1, y1, 0, . . . , 0 .
ZP ª N
by 1 ª Ž 1, 0, y1, 0, . . . , 0 . ⭈⭈⭈ ⭈⭈⭈
ZP ª N
by 1 ª Ž 1, 0, . . . , 0, y1 .
ZªN
by 1 ª Ž s1 , ys2 , 0, . . . , 0 .
ZªN
by 1 ª Ž s1 , 0, ys3 , 0, . . . , 0 . ⭈⭈⭈ ⭈⭈⭈
ZªN
by 1 ª Ž s1 , 0, 0, . . . , 0, ysr .
Z PrP1 ª N
by 1 ª Ž 1 y 1, 0, . . . , 0 .
Z PrP2 ª N
by 1 ª Ž 0, 2 y 1, 0, . . . , 0 . ⭈⭈⭈ ⭈⭈⭈ by 1 ª Ž 0, 0, . . . , 0, r y 1 . .
Z PrPr ª N
Set M s Z P ry1 [ Z ry1 [ [is1 Z PrPi and define the homomorphism ⌿ : M ª N as the sum of the above homomorphisms. Then it is easy to r
34
SHIZUO ENDO
see that Im ⌿ s L. Further set H s Ker ⌿. Then we have the exact sequence 0 ª H ª M ª L ª 0. As is easily seen, we have rank Z N s pr ,
rank Z N P s r ,
rank Z L s pr y 1,
rank Z LP s r y 1,
rank Z M s Ž p 2 q 1 . Ž r y 1 . q pr ,
rank Z M P s 3r y 2,
rank Z H s p 2 Ž r y 1 . q r ,
rank Z H P s 2 r y 1.
By direct computation we can show that M P ⬘ ª LP ⬘ is surjective for each subgroup P⬘ of P. Hence it follows that H 1 Ž P⬘, H . s 0 for each subgroup P⬘ of P. Assume that L⬚ is a d-quasi-permutation. Then, as in the proof of the case where p s 2, it follows that H is a direct summand of a permutation module. Since P is a p-group, H is locally isomorphic to a permutation module S. We denote the completion of a module V at p by V *. Then we can decompose H * ( S* ( ŽZ*P . s [ Ž S⬘.*, where S⬘ does not have Z P as a direct summand. It is easy to see that rank Z S⬘ s p 2 Ž r y s y 1. q r and rank Z Ž S⬘. P s 2 r y s y 1. Since each indecomposable component of S⬘ is of rank 1 or p over Z, we have rank Z S⬘ O pŽ2 r y s y 1.. If r y s y 3 P 0, it follows from the fact that r O p q 1 - 2 p that rank Z S⬘ ) pŽ2 r y s y 1., which is a contradiction. Hence we have r y s y 2 O 0, that is, r y 2 O s. Since s O r y 1, this shows that s s r y 2 or r y 1. Forming the pushout of Ž S⬘.* ¤ S* ª M*, we obtain the following commutative diagram with exact rows and columns: 0
0 6
6
ŽZ*P . s s ŽZ*P . s
0
0
6
6
6
6
6
6
6 0.
6
0
L*
6
Ž M⬘.*
6
Ž S⬘.*
L*
6
M*
6
S*
6
0
6
0
ALGEBRAIC TORI OF NORM TYPE
35
Then the columns are split, and so Ž M⬘.* s ŽZ*P . rysy1 [ ŽZ*. ry1 r [[is1 Z*PrPi . Set U* s Ý P ⬘Ž M*. P ⬘ where P⬘ runs over all subgroups r of order p in P. Since ŽZ*. ry1 [ [is1 Z*PrPi : U*, there exists an rysy1 epimorphism ŽZ*P . ª M*rU* ª L*r⌿*ŽU*.. It can be shown that ⌿ )ŽU *. s Z*P Ž 1 y 1, 0, . . . , 0. q Z*P Ž0, 2 y 1, 0, . . . , 0. q ⭈⭈⭈ q Z*P Ž0, . . . , 0, r y 1. q Z*P Ž p, yp, 0, . . . , 0. q Z*P Ž p, 0, yp, 0, . . . , 0. q ⭈⭈⭈ qZ*P Ž p, 0, . . . , 0, yp .. Hence it follows that L*r⌿*ŽU*. ( ZrpZŽ1, y1, 0, . . . , 0. q ZrpZŽ1, 0, y1, 0, . . . , 0. q ⭈⭈⭈ q ZrpZŽ1, 0, . . . , 0, y1. ( ŽZrpZ. ry1. Therefore we obtain an epimorphism ŽZ*P . rysy1 ª ŽZrpZ. ry1. However, s s r y 2 or r y 1, as seen above. This shows that r s 2 and s s 0. Then we have rank Z H s p 2 q 2 and rank Z H P s 3. This is a contradiction, because H *Ž( Ž S⬘.*. does not have Z*P as a direct summand. Thus the proof is complete.
REFERENCES 1. C. Chevalley, On algebraic group varieties, J. Math. Soc. Japan 6 Ž1954., 303᎐324. 2. S. Endo and T. Miyata, Invariants of finite abelian groups, J. Math. Soc. Japan 25 Ž1973., 7᎐26. 3. S. Endo and T. Miyata, On a classification of the function fields of algebraic tori, Nagoya Math. J. 56 Ž1975., 85᎐104; Corrigenda, 79 Ž1980., 187᎐190. 4. S. Endo, Non-rationality of algebraic tori of norm type and its application to generic division rings, in ‘‘Proc. 22nd Symp. Ring Theory, Okayama Univ., 1989,’’ pp. 59᎐64. 5. W. Hurlimann, On algebraic tori of norm type, Comment. Math. Hel¨ . 59 Ž1984., 539᎐549. ¨ 6. A. Merkurjev, letter to T. Miyata, October 25, 1982. 7. D. J. Saltman, Retract rational fields and cyclic Galois extensions, Israel J. Math. 47 Ž1984., 165᎐215. 8. V. E. Voskresenskiı, ˇ Algebraic groups and their birational invariants, in Transl. Math. Monogr., Vol. 179, Amer. Math. Soc., Providence, 1998.
On the Rationality of Algebraic Tori of Norm Type Shizuo Endo Teikyo Uni¨ ersity, Hachioji, Tokyo 192-0352, Japan Communicated by Leonard L. Scott, Jr. Received February 20, 1999
1 Let G be a finite group. A G-module is called a permutation module if r it is isomorphic to [is1 ZGrGi , where Gi , 1 O i O r, are subgroups of G. A G-module M is called a quasi-permutation module if there exists an exact sequence 0 ª M ª S ª S⬘ ª 0, where S and S⬘ are permutation G-modules. The dual module Hom Z Ž M, Z. of a G-module M is denoted by M⬚. Let k be a field and let F be an extension field of k. Then F is said to be rational over k if it is generated by a finite number of elements of F which are algebraically independent over k and to be stably rational over k if there exists an extension field of F which is rational over each of k and F. Further, F is said to be retract rational over k if it is the quotient field of an integral domain A = k such that there are k-algebra homomorphisms
: A ª k w X 1 , X 2 , . . . , X n xw 1rs x
and
: k w X 1 , X 2 , . . . , X n xw 1rs x ª A, where k w X 1 , X 2 , . . . , X n xw1rs x is the localization of a polynomial ring k w X 1 , X 2 , . . . , X n x with variables X 1 , X 2 , . . . , X n over k with respect to 0 / s g k w X 1 , X 2 , . . . , X n x and ⭈ is the identity on A. It is easy to see that rational « stably rational « retract rational. Let G be a finite group and let M be a G-module with a Z-free basis u1 , u 2 , . . . , u n . Let k be a field and let K be a Galois extension of k with group G. Define the action of G on the rational function field 27 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
28
SHIZUO ENDO
K Ž X 1 , X 2 , . . . , X n . with variables X 1 , X 2 , . . . , X n over K, as an extension of the action of G on K, as follows. For each g G, n
Ž Xi . s
Ł X jm
ij
1 O i O n,
,
js1
when u i s Ý njs1 m i j u j , m i j g Z, and denote K Ž X 1 , X 2 , . . . , X n . with this action of G by K Ž M .. As is well known, there is an algebraic torus T defined over k and split over K such that the character group of T is isomorphic to M as G-modules, and the invariant subfield K Ž M . G of K Ž M . can be identified with the function field of T over k. We now have the following fundamental PROPOSITION. Let G be a finite group and let k be a field. Let K be a Galois extension of k with group G and let M be a Z-free and Z-finitely generated G-module Ž i.e., a G-lattice.. Then: Ž1. Ž e. g., w2, 1.6x. M is a quasi-permutation G-module if and only if K Ž M . G is stably rational o¨ er k. Ž2. w7, 3.14x M is a direct summand of a quasi-permutation G-module if and only if K Ž M . G is retract rational o¨ er k. Let p be a prime, and let P be an elementary abelian p-group of order p m , m P 1. Let Pi , 1 O i O r, be distinct subgroups of index p in P, and let i : Z PrPi ª Z be the augmentation maps. Further, for h1 , h 2 , . . . , h r P 1, let r
⌽ s Ž 1h1 , 2h 2 , . . . , rh r . :
[ Ž Z PrPi . h
i
ªZ
is1
and set L s Ker ⌽. The main result of this note is the following THEOREM 1. Ž1. In case of p s 2, L⬚ is a Ž direct summand of a. quasi-permutation P-module if and only if r s 1, 2. Ž2. In case of p / 2, L⬚ is a Ž direct summand of a. quasi-permutation P-module if and only if r s 1. By the Proposition, Theorem 1 can be restated as follows: THEOREM 2. Let G, P, and L be as abo¨ e and let K be a Galois extension of a field k with group P. Then: Ž1. In case of p s 2, K Ž L⬚. G is stably Ž retract . rational o¨ er k if and only if r s 1, 2.
29
ALGEBRAIC TORI OF NORM TYPE
Ž2. In case of p / 2, K Ž L⬚. G is stably Ž retract . rational o¨ er k if and only if r s 1. It is noted that the algebraic torus corresponding to L⬚ defined over k and split over K is of norm type. Part Ž1. of Theorem 2 is an answer to the question asked the late professor T. Miyata by A. Merkurjev in 1982 w6x. It should be noted that Theorem 1 was obtained in 1982 and reported without proof in w4x.
2 We now begin to prove Theorem 1. In order to prove this, we need to consider a more general situation. Let P, Pi , and i , 1 O i O r, be as above, and define the homomorphism ␦ : Z ª Z by ␦ Ž1. s p. For h1 , h 2 , . . . , h r P 1 and h P 0, let r
[ Ž Z PrPi . h
⌽ s Ž 1h1 , 2h 2 , . . . , rh r , ␦ . :
i
[ Z h ª Z,
is1
and set L s Ker ⌽. Further, let r
⌽⬘ s Ž 1 , 2 , . . . , r . :
[ Z PrPi ª Z, is1
and set L⬘ s Ker ⌽⬘. Then the key lemma is given as follows: LEMMA.
L ( L⬘ [ [is1 ŽZ PrPi . h iy1 [ Z h. r
Proof. First, define the homomorphism i : Z PrPi ª ŽZ PrPi . h i by i Ž a i . s Ž a i , 0, 0, . . . , 0. for each 1 O i O r, and set r
s Ž 1 , 2 , . . . , r . :
[ Z PrPi ª is1
r
r
[ Ž Z PrPi . h i : [ Ž Z PrPi . h i [ Z h . is1
is1
Next, define the homomorphism i : ŽZ PrPi ª Z PrPi by i ŽŽ a i1 , i a i2 , . . . , a i h i .. s Ý hjs1 a i j for each 1 O i O r and the homomorphism : r h Z ª [is1 Z PrPi by ŽŽ b1 , b 2 , . . . , bh .. s ŽŽÝ his1 bi s, 0, 0, . . . , 0. Ž s 0 when h s 0., where s denotes the sum of all elements of PrP1 in Z PrP1 , and set .hi
s Ž 1 , 2 , . . . , r , . :
r
r
is1
is1
[ Ž Z PrPi . h i [ Z h ª [ Z PrPi .
Then ⭈ is the identity on [ and it can be easily seen that Ž L⬘. : L and Ž L. : L⬘. Therefore, denoting the restrictions of , to L⬘, L by ˜, ˜ , respectively, we get homomorphisms ˜ : L⬘ ª L and r is1 Z PrPi
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SHIZUO ENDO
˜ : L ª L⬘ such that ˜ ⭈ ˜ is the identity on L⬘. This shows that L ( L⬘ [ Coker ˜. However, as is easily seen, Coker ˜ ( Coker ( r Ž . h iy 1 [ Z h . Thus this concludes that L ( L⬘ [ [is 1 Z PrPi r h [is1 ŽZ PrPi . iy1 [ Z h. By this lemma we have only to consider the case where h1 s h 2 s ⭈⭈⭈ s h r s 1, that is, the exact sequence r
0ªLª
[ Z PrPi ª Z ª 0,
Ž ).
is1
where P1 , P2 , . . . , Pr are distinct subgroups of index p in P. It is known that L⬚ is a quasi-permutation P-module for each of the cases Ža. r s 1 and Žb. r s 2 and p s 2 w6x. However, for completeness, we give a proof of it. Assume first that r s 1. Then PrP1 is cyclic, and so L⬚ ( L. Hence L⬚ is clearly a quasi-permutation P-module. Next, assume that r s 2 and p s 2. Then PrP1 l P2 is the Klein four group, and therefore, regarding PrP1 l P2 as P, we can set P s ² , N 2 s 2 s 1, s : and P1 s ² :, P2 s ² :. Now we have Ker ⌽ s Z P Ž1, y1.. Define the epimorphism ⌿ s Ž , . : Z P [ Z ª L by Ž1. s Ž1, y1. and Ž1. s Ž1 q , 1 q .. Then, as is easily seen, Ker ⌿ s ŽŽ a q b .Ž1 q ., yŽ a q b .. N a, b g Z4 , and so Ker ⌿ ( Z Pr² :. Thus we have the exact sequence 0 ª Z Pr² : ª Z P [ Z ª L ª 0, which shows that L⬚ is a quasi-permutation P-module.
3 Consider here the case here p s 2 and r P 3. This case can be divided into the following: Ž1. w P : Pi l Pi l Pi x s 4 for some i1 - i 2 - i 3 . 1 2 3 Ž2. w P : Pi l Pi l Pi x s 8 for any i1 - i 2 - i 3 . 1 2 3 Set P⬘ s Pi1 l Pi 2 l Pi 3 in both cases and restrict the exact sequence Ž). to P⬘. Then we obtain the exact sequence 0 ª LP ⬘ ª Z PrPi1 [ Z PrPi 2 [ Z PrPi 3 [
[ Ž Z PrPi . P ⬘ ª Z ª 0.
i/i 1 , i 2 , i 3
It is noted that L⬚ is not Ža direct summand of. a quasi-permutation P-module if Ž LP ⬘ .⬚ is not Ža direct summand of. a quasi-permutation PrP⬘-module. Regarding PrP⬘ as P and applying the lemma to the above
ALGEBRAIC TORI OF NORM TYPE
31
exact sequence, the cases Ž1. and Ž2. reduce, respectively, to the following Ži.
P s ² , N 2 s 2 s 1, s :, the Klein four group, 0 ª L ª Z Pr² : [ Z Pr² : [ Z Pr² : ª Z ª 0.
Žii. P s ² , , N 2 s 2 s 2 s 1, s , s , s : Ža. 0 ª L ª Z Pr² , : [ Z Pr² , : [ Z Pr² , : ª Z ª 0. Žb. 0 ª L ª Z Pr² , : [ Z Pr² , : [ Z Pr² , : [ Z Pr² , : ª Z ª 0. We have only to show for these cases that L⬚ is not a direct summand of any quasi-permutation P-module. For brevity, a direct summand of a quasi-permutation module is called a d-quasi-permutation module. It should be noted that the case Ži. was examined by Miyata Žunpubw5x. We give here a proof of this case for lished, 1974. and Hurlimann ¨ completeness. Let P be as in the case Ži. and let I be the augmentation ideal of Z P. Define the homomorphism : Z Pr² : [ Z Pr² : [ Z Pr² : ª Z P as
Ž Ž 1, 0, 0 . . s 1 q ,
Ž Ž 0, 1, 0 . . s 1 q ,
Ž Ž 0, 0, 1 . . s 1 q .
Restricting to L, we obtain the epimorphism ˜ : L ª I. It is easy to see that Ker ˜ ( Z 2 is a trivial P-module. Hence we have the exact sequence 0 ª Z 2 ª L ª I ª 0. However, it is known w1, 3x that I⬚ is not a d-quasi-permutation module. This concludes that L⬚ is not a d-quasi-permutation module. Next, consider the case Žii., Ža.. We restrict the exact sequence to the subgroup P⬘ s ² , : of P. Then we see that Z Pr² , : [ Z Pr² , : [ Z Pr² , : ( Z P⬘r² : [ Z P⬘r² : [ Z P⬘r² : as P⬘-modules. This shows by the case Ži. that L⬚ is not a d-quasi-permutation P⬘-module. Thus L⬚ is also not a d-quasi-permutation P-module. The case Žii., Žb. is slightly complicated. We set M s Z Pr² : [ Z Pr² : [ Z Pr² : [ Z Pr² : [ Z Pr² : [ Z Pr² : [ Z 3 and N s Z Pr² , : [ Z Pr² , : [ Z Pr² , : [ Z Pr² , :. Define the 9 homomorphisms as Z Pr² : ª N
by 1 ª Ž 1, y1, 0, 0 .
Z Pr² : ª N
by 1 ª Ž 1, 0, y1, 0 .
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SHIZUO ENDO
Z Pr² : ª N
by 1 ª Ž 0, 1, y1, 0 .
Z Pr² : ª N
by 1 ª Ž 1, 0, 0, y1 .
Z Pr² : ª N
by 1 ª Ž 0, 0, 1, y1 .
Z Pr² : ª N
by 1 ª Ž 0, 1, 0, y1 .
ZªN
by 1 ª Ž 1 q , y Ž 1 q . , 0, 0 .
ZªN
by 1 ª Ž 1 q , 0, y Ž 1 q . , 0 .
ZªN
by 1 ª Ž 1 q , 0, 0, y Ž 1 q . . .
By adding these homomorphisms, we obtain the homomorphism ⌿ : M ª N. It is easy to see that the image of ⌿ coinsides with L. Set H s Ker ⌿. Then we have the exact sequence 0 ª H ª M ª L ª 0. By direct computation we show that M P ⬘ ª LP ⬘ is surjective for each subgroup P⬘ of P. Therefore it follows that H 1 Ž P⬘, H . s 0 for each subgroup P⬘ of P, where H i Ž , . denotes the Tate cohomology group. Assume that L⬚ is a d-quasi-permutation. Then, by Žthe proof of. w3, 1.6x, there is an exact sequence 0 ª T ª D ª L ª 0, where T is a permutation module and D is a direct summand of permutation module. Taking the pullback of M ª L ¤ D, we obtain H [ D ( T [ M. Since P is a 2-group, H is locally isomorphic to a permutation module S. We denote the completion of a module V at 2 by V *. Then we have H * ( S*. Set s . Considering ² :-invariants we have Ž L*. ² : ( ŽZ*. 3 and
Ž H *.
² :
2
( Ž Z*Pr² , : [ Z*Pr² , : [ Z*Pr² , : . .
Since rank Z H s 20 and rank Z H P s 6, we have H * ( Z*PrP1 [ Z*PrP2 [ Z*PrP3 [ Z*PrP4 [ Z*PrP1X [ Z*PrP2X , where < P1 < s < P2 < s < P3 < s < P4 < s 2 and < P1X < s < P2X < s 4. Using the cohomology sequence of the exact sequence 0 ª L ª N ª Z ª 0, we see that H 1 Ž P, L. ( Zr2Z and H 0 Ž P, L. ( ŽZr4Z. 3. Further, taking the cohomology sequence of the exact sequence 0 ª H ª M ª L ª 0, we have the exact sequence H 0 Ž P, H . ª H 0 Ž P, M . ª H 0 Ž P, L. ª H 1 Ž P, H . s 0. By direct computation we also have H 0 Ž P, H . ( ŽZr2Z. 4 [ ŽZr4Z. 2 and H 0 Ž P, M . ( ŽZr2Z. 4 [ ŽZr8Z. 3. Therefore we obtain the exact sequence 4 2 6 3 3 Ž Zr2Z. [ Ž Zr4Z. ª Ž Zr2Z. [ Ž Zr8Z. ª Ž Zr4Z. ª 0.
However, it is easy to see that the kernel of ŽZr2Z. 6 [ ŽZr8Z. 3 ª ŽZr4Z. 3
ALGEBRAIC TORI OF NORM TYPE
33
is isomorphic to ŽZr2Z. 9. This is a contradiction, because ŽZr2Z. 4 [ ŽZr4Z. 2 never maps onto ŽZr2Z. 9. Thus L⬚ is not a d-quasi-permutation.
4 Finally consider the case where p / 2 and r P 2. We now prove that L⬚ is not a d-quasi-permutation. Restricting the exact sequence Ž). to the subgroup P⬘ s P1 l P2 , regarding PrP⬘ as P and using the lemma as in the case where p s 2, we can reduce this case to the one where < P < s p 2 . From now on, assume that P is of order p 2 . Under this assumption it is noted that there exist exactly p q 1 subgroups of order p in P. Let Pi , r 1 O i O r, be distinct subgroups of order p in P and set N s [is1 Z PrPi . Let i : Z PrPi ª Z, 1 O i O r, be the augmentation maps and let ⌽ s Ž 1 , 2 , . . . , r . : N ª Z. Set L s Ker ⌽. Then we have the exact sequence 0 ª L ª N ª Z ª 0. py 1 For 1 O i O r, set PrPi s ² i : and si s Ý js0 ij g Z PrPi . Define the homomorphisms as ZP ª N
by 1 ª Ž 1, y1, 0, . . . , 0 .
ZP ª N
by 1 ª Ž 1, 0, y1, 0, . . . , 0 . ⭈⭈⭈ ⭈⭈⭈
ZP ª N
by 1 ª Ž 1, 0, . . . , 0, y1 .
ZªN
by 1 ª Ž s1 , ys2 , 0, . . . , 0 .
ZªN
by 1 ª Ž s1 , 0, ys3 , 0, . . . , 0 . ⭈⭈⭈ ⭈⭈⭈
ZªN
by 1 ª Ž s1 , 0, 0, . . . , 0, ysr .
Z PrP1 ª N
by 1 ª Ž 1 y 1, 0, . . . , 0 .
Z PrP2 ª N
by 1 ª Ž 0, 2 y 1, 0, . . . , 0 . ⭈⭈⭈ ⭈⭈⭈ by 1 ª Ž 0, 0, . . . , 0, r y 1 . .
Z PrPr ª N
Set M s Z P ry1 [ Z ry1 [ [is1 Z PrPi and define the homomorphism ⌿ : M ª N as the sum of the above homomorphisms. Then it is easy to r
34
SHIZUO ENDO
see that Im ⌿ s L. Further set H s Ker ⌿. Then we have the exact sequence 0 ª H ª M ª L ª 0. As is easily seen, we have rank Z N s pr ,
rank Z N P s r ,
rank Z L s pr y 1,
rank Z LP s r y 1,
rank Z M s Ž p 2 q 1 . Ž r y 1 . q pr ,
rank Z M P s 3r y 2,
rank Z H s p 2 Ž r y 1 . q r ,
rank Z H P s 2 r y 1.
By direct computation we can show that M P ⬘ ª LP ⬘ is surjective for each subgroup P⬘ of P. Hence it follows that H 1 Ž P⬘, H . s 0 for each subgroup P⬘ of P. Assume that L⬚ is a d-quasi-permutation. Then, as in the proof of the case where p s 2, it follows that H is a direct summand of a permutation module. Since P is a p-group, H is locally isomorphic to a permutation module S. We denote the completion of a module V at p by V *. Then we can decompose H * ( S* ( ŽZ*P . s [ Ž S⬘.*, where S⬘ does not have Z P as a direct summand. It is easy to see that rank Z S⬘ s p 2 Ž r y s y 1. q r and rank Z Ž S⬘. P s 2 r y s y 1. Since each indecomposable component of S⬘ is of rank 1 or p over Z, we have rank Z S⬘ O pŽ2 r y s y 1.. If r y s y 3 P 0, it follows from the fact that r O p q 1 - 2 p that rank Z S⬘ ) pŽ2 r y s y 1., which is a contradiction. Hence we have r y s y 2 O 0, that is, r y 2 O s. Since s O r y 1, this shows that s s r y 2 or r y 1. Forming the pushout of Ž S⬘.* ¤ S* ª M*, we obtain the following commutative diagram with exact rows and columns: 0
0 6
6
ŽZ*P . s s ŽZ*P . s
0
0
6
6
6
6
6
6
6 0.
6
0
L*
6
Ž M⬘.*
6
Ž S⬘.*
L*
6
M*
6
S*
6
0
6
0
ALGEBRAIC TORI OF NORM TYPE
35
Then the columns are split, and so Ž M⬘.* s ŽZ*P . rysy1 [ ŽZ*. ry1 r [[is1 Z*PrPi . Set U* s Ý P ⬘Ž M*. P ⬘ where P⬘ runs over all subgroups r of order p in P. Since ŽZ*. ry1 [ [is1 Z*PrPi : U*, there exists an rysy1 epimorphism ŽZ*P . ª M*rU* ª L*r⌿*ŽU*.. It can be shown that ⌿ )ŽU *. s Z*P Ž 1 y 1, 0, . . . , 0. q Z*P Ž0, 2 y 1, 0, . . . , 0. q ⭈⭈⭈ q Z*P Ž0, . . . , 0, r y 1. q Z*P Ž p, yp, 0, . . . , 0. q Z*P Ž p, 0, yp, 0, . . . , 0. q ⭈⭈⭈ qZ*P Ž p, 0, . . . , 0, yp .. Hence it follows that L*r⌿*ŽU*. ( ZrpZŽ1, y1, 0, . . . , 0. q ZrpZŽ1, 0, y1, 0, . . . , 0. q ⭈⭈⭈ q ZrpZŽ1, 0, . . . , 0, y1. ( ŽZrpZ. ry1. Therefore we obtain an epimorphism ŽZ*P . rysy1 ª ŽZrpZ. ry1. However, s s r y 2 or r y 1, as seen above. This shows that r s 2 and s s 0. Then we have rank Z H s p 2 q 2 and rank Z H P s 3. This is a contradiction, because H *Ž( Ž S⬘.*. does not have Z*P as a direct summand. Thus the proof is complete.
REFERENCES 1. C. Chevalley, On algebraic group varieties, J. Math. Soc. Japan 6 Ž1954., 303᎐324. 2. S. Endo and T. Miyata, Invariants of finite abelian groups, J. Math. Soc. Japan 25 Ž1973., 7᎐26. 3. S. Endo and T. Miyata, On a classification of the function fields of algebraic tori, Nagoya Math. J. 56 Ž1975., 85᎐104; Corrigenda, 79 Ž1980., 187᎐190. 4. S. Endo, Non-rationality of algebraic tori of norm type and its application to generic division rings, in ‘‘Proc. 22nd Symp. Ring Theory, Okayama Univ., 1989,’’ pp. 59᎐64. 5. W. Hurlimann, On algebraic tori of norm type, Comment. Math. Hel¨ . 59 Ž1984., 539᎐549. ¨ 6. A. Merkurjev, letter to T. Miyata, October 25, 1982. 7. D. J. Saltman, Retract rational fields and cyclic Galois extensions, Israel J. Math. 47 Ž1984., 165᎐215. 8. V. E. Voskresenskiı, ˇ Algebraic groups and their birational invariants, in Transl. Math. Monogr., Vol. 179, Amer. Math. Soc., Providence, 1998.