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Some Cases of Extensions of Twisted Group Schemes over a Discrete Valuation Ring
Journal of Algebra 227, 474᎐498 Ž2000. doi:10.1006rjabr.1999.8086, available online at http:rrwww.idealibrary.com on
Some Cases of Extensions of Twisted Group Schemes over a Discrete Valuation Ring Toshiaki Ohno Department of Mathematics, Faculty of Science and Engineering, Chuo Uni¨ ersity, Tokyo 112, Japan Communicated by Da¨ id Buchsbaum Received May 2, 1996
1. INTRODUCTION Let A be a discrete valuation ring with maximal ideal ᒊ, let K be the field of fractions of A, and let k s Arᒊ. Let G Ž . be the model of ⺗ m, A given by G Ž . s Spec A X , Ž X q 1 .
y1
for some Ž/ 0. g A Žcf. w4, 2.5x.. Then there is an exact sequence G Ž . ª ⺗ m, A ªi#⺗ m, A rŽ . ª0, 0 ªG x ¬ x q 1, where i is the closed immersion i: Spec ArŽ . ¨ Spec A Žcf. w6, 3.1x.. Sekiguchi and Suwa computed several extensions with respect to G Ž ., for example, Ext 1 Ž G Ž ., G Ž . ., etc. Žcf. w6᎐9x.. Weisfeiler computed Ext 1 Ž G Ž ., ⺗ a, A . Žcf. w5x.. Let BrA be an unramified quadratic extension. We denote its Galois group by ⌫ s e, 4 . We remark that B s a q b ¬ a, b g A4 where 2 y m q n s 0 for some m, n g A and m2 y 4 n g A=. We denote the norm map by Norm: B=ª A=. Let G be the nontrivial BrA-form of ⺗ m . Then by Waterhouse and Weisfeiler Žcf. w4, 2.6x., we get G s Spec A w , x r Ž n 2 y m q 2 q m2 y 4 n . . 474 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
475
TWISTED GROUP SCHEMES
Let G be the model of G. Then we get G s Spec A w ¨ , w x r Ž Ž n¨ 2 y m¨ w q w 2 . y Ž m2 y 4 n . ¨ . for some Ž/ 0. g A Žcf. Section 2.2.. Then we have an exact sequence Žcf. Section 2.3. 0ª G ª G ªi#GA rŽ . ª0. Ž x, y . ¬Ž x q 2, y q m.. Moreover, we have GB , GBŽ ., that is to say, G is the form of GBŽ . Žcf. Section 2.2.. Then there is the commutative diagram consisting of exact lines Žcf. Section 2.3., GBŽ . ª⺗ m, B ªj#⺗ m, BrŽ . ª 0 0 ªG x x 0 ª G ª G ª i#GA rŽ . ª0, x x 0 0, where j is the closed immersion j: Spec BrŽ . ¨ Spec B. Let Ł Br A⺗ m be the Weil restriction of ⺗ m from B to A. Then
Ł ⺗ m s Spec Aw X , Y , Z x r Ž Z Ž X 2 q mXY q nY 2 . y 1 . .
BrA
In this case, we are interested in the several extensions with respect to G and G . The purpose of this paper is to decide the following extension groups. We consider the fppf topology over Spec A and denote the corresponding site by Sch r A . Hence all cohomology groups area fppf cohomologies. In particular, we regard the first cohomology group H 1 Ž X, G . as the set of isomorphism classes of G-torsors over X. We consider only commutative extensions in Sch r A . Our results are as follows. 1. First cohomologies H 1 Ž G , ⺗ m , A . , ⺪r2⺪
Ž cf. 3.1.
H 1 Ž G Ž . , G . , A=rNormŽ B= . = ⺪r2⺪ =
=
H Ž G , G . , A rNormŽ B . 1
H 1 Ž ⺗ a, A , G . , A=rNormŽ B= . .
Ž cf. 3.2. Ž cf. 3.3. Ž cf. 3.4.
476
TOSHIAKI OHNO
2. Extensions Ext 1 Ž G , ⺗ m , A . , ⺪r2⺪
Ž cf. 4.1.
Ext 1 Ž G Ž . , G . , ⺪r2⺪
Ž cf. 4.2.
Ext 1 Ž G , G . s 0.
Ž cf. 4.3.
3. The model of the Ł Br A ⺗ m , which is the Neron blow-up of ⺗ m, k ´ in Ł Br A ⺗ m , is the nontrivial extension of G by ⺗ m, A Žcf. Section 4.1.. 4. The model of the Ł Br A ⺗ m , which is the Neron ´ blow-up of Gk in Ł Br A ⺗ m , is the nontrivial extension of G Ž . by G Žcf. Section 4.2.. 5. Any element of Ext 1 Ž⺗ a, A , G . corresponds to the polynomial F Ž a1 , a p , a p 2 , . . . ; X . given in Section 4.4 Žcf. Section 4.4.. The results of 3.1 and 4.1 are suggested by Suwa.
2. FORMS AND MODELS 2.1. The Form of ⺗ m We will review the form of ⺗ m from Waterhouse and Weisfeiler Žcf. w4, 2.6x.. THEOREM 2.1.1.
Let G be the nontri¨ ial BrA-form of ⺗ m . Then G s Spec A w , x ,
where 1.
Relation n 2 y m q 2 s 4 n y m2 .
2. Comultiplication
¬ ¬
1 m y 4n 2
1 m y 4n 2
Ž Ž m2 y 2 n . m q 2 m y m Ž m q m . . Ž mn m q m m y 2 n Ž m q m . . .
3. Counit
¬2 ¬ m.
TWISTED GROUP SCHEMES
477
4. Coin¨ erse
¬ ¬ m y . Proof. The nontrivial action of ⌫ over Bw⺗ m x s Bw T, Ty1 x is given by
Bw T, Ty1 x ª Bw T, Ty1 x. < < j j y1 T ¬ T . Then the ⌫-invariant subring Bw T, Ty1 x ⌫ is given by ⌫
B w T , Ty1 x s A w , x , where
s T q Ty1 s T q Ž m y . Ty1 . Hence Aw G x has the form Aw , x, and the computations of relation, comultiplication, counit, and coinverse are easy. 2.2. The Model of G We will compute the model of G by using Neron blow-ups Žcf. w4, 2.6x.. ´ Let G be the model of G. Then
THEOREM 2.2.1.
G s Spec A w ¨ , w x , where 1.
Relation
Ž n¨ 2 y m¨ w q w 2 . s Ž m2 y 4 n . ¨ , 2. Comultiplication ¨ ¬¨ m1 q1 m¨
q
m y 4n 2
Ž Ž m2 y 2 n . ¨ m ¨ y m¨ m w y mw m ¨ q 2 w m w .
w¬wm1q1mw q 2 Ž mn¨ m ¨ y 2 n¨ m w y 2 nw m ¨ q mw m w . , m y 4n
478
TOSHIAKI OHNO
3. Counit ¨ ¬0
w ¬ 0, 4. Coin¨ erse ¨ ¬¨ w ¬ m¨ y w,
for some Ž/ 0. g A. Proof. We can get the G by using Neron blow-ups Žcf. w4, Sect. 1x.. Let ´ G1 be the Neron ´ blow-up of the subgroup defined by Ž , . s Ž2, m. of the special fiber Gk . Then Aw G1 x has the form A w G1 x s A , , y1 Ž y 2 . , y1 Ž y m . r Ž n 2 y m q 2 qm2 y 4 n . s A y1 Ž y 2 . , y1 Ž y m . r Ž n 2 y m q 2 q m2 y 4 n . s A w ¨ , w x r Ž Ž n¨ 2 y m¨ w q w 2 . y Ž m2 y 4 n . ¨ . , where s ¨ q 2, s w q m. The special fiber of G1 is ⺗ a , and we have k w G1 x , k w ¨ , w xrŽ ¨ .. Let G2 be the Neron blow-up of the subgroup ´ defined by Ž ¨ , w . s Žy1 Ž y 2., y1 Ž y m.. s Ž0, 0. of the special fiber G1 k . Then Aw G2 x has the form A w G2 x s A y1 Ž y 2 . , y1 Ž y m . , y2 Ž y 2 . , y2 Ž y m . r Ž n 2 y m q 2 q m2 y 4 n . s A y2 Ž y 2 . , y2 Ž y m . r Ž n 2 y m q 2 q m2 y 4 n . s A w ¨ , w x r Ž 2 Ž n¨ 2 y m¨ w q w 2 . y Ž m2 y 4 n . ¨ . , where s 2 ¨ q 2, s 2 w q m. The special fiber of G2 is also ⺗ a . We continue the construction and we get G . The comultiplication is defined so that the morphism G ª G Ž x, y . ¬Ž x q 2, y q m. is homomorphism. The computation of counit and coinverse are easy.
479
TWISTED GROUP SCHEMES
PROPOSITION 2.2.2.
G is the form of GBŽ ..
Proof. We put ¨ sXqY
w s X q Ž m y .Y, and we have
Ž n¨ 2 y m¨ w q w 2 . y Ž m2 y 4 n . ¨ s y Ž m2 y 4 n . Ž XY q X q Y . . Then B w G x , B w X , Y x r Ž XY q X q Y . s B X , yX Ž X q 1 .
y1
, B X , Ž X q 1.
y1
s B w G Ž . x .
2.3. Exact Sequence We have an exact sequence 0 ª G ª G ª i#GA rŽ . ª 0, where i is a closed immersion i: Spec ArŽ . ¨ Spec A. Let R be any faithfully flat A-algebra. The maps are G Ž R. ª 0 ªG GŽ R . ª i#GA rŽ .Ž R . < < j j Ž x, y . ¬Ž x q 2, y q m. Ž x⬘, y⬘. ¬Ž x⬘ modŽ ., y⬘ modŽ ... We obtain a commutative diagram consisting of exact lines, GBŽ . ª⺗ m, B ªj#⺗ m, BrŽ . ª 0 0 ªG x x 0 ª G ª G ª i#GA rŽ . ª0, x x 0 0, where j is a closed immersion j: Spec BrŽ . ¨ Spec B.
480
TOSHIAKI OHNO
3. FIRST COHOMOLOGY 3.1. H 1 Ž G , ⺗ m, A . LEMMA 3.1.1. H 1 Ž Spec A, ⺗ m , A . s 0. Proof. Cf. w2, III, 4.10x. COROLLARY 3.1.2.
Let H 1 Ž ⌫, B=. be the group cohomology. Then H 1 Ž ⌫, B= . s 0.
Proof. We note that the extension BrA is unramified. Then we obtain immediately. THEOREM 3.1.3. H 1 Ž G , ⺗ m , A . , H 1 Ž ⌫, ⺗ m , A Ž GBŽ . . . . Proof. There is the finite Galois covering GBŽ . ª G with Galois group ⌫, and the ⌫-action on GBŽ . is given as follows:
Bw X, Ž X q 1.y1 x ª Bw X, Ž X q 1.y1 x < < j j Ž X ¬ yX X q 1.y1 . Then, by Hochschild᎐Serre spectral sequence Žcf. w2, III, 2.20x., we have the exact sequence ⌫
0 ª H 1 Ž ⌫, ⺗ m , A Ž GBŽ . . . ª H 1 Ž G , ⺗ m , A . ª H 1 Ž GBŽ . , ⺗ m , A . . While we obtain H 1 Ž GBŽ . , ⺗ m , A . , H 1 Ž GBŽ . , ⺗ m , B . s 0. Then we have H 1 Ž G , ⺗ m , A . , H 1 Ž ⌫, ⺗ m , A Ž GBŽ . . . .
THEOREM 3.1.4. H 1 Ž G , ⺗ m , A . , ⺪r2⺪.
481
TWISTED GROUP SCHEMES
Proof. In Theorem 3.1.3, we have ⺗ m , A Ž GBŽ . . sHom A A w T , Ty1 x , B X , Ž X q 1 .
ž
y1
/ ,B
X , Ž X q 1.
y1 =
Hence, using Corollary 3.1.2, we get H 1 Ž ⌫, ⺗ m , A Ž GBŽ . . . , H 1 ⌫, B X , Ž X q 1 .
ž
n
y1 =
/
s aŽ X q 1. ¬ Ž aŽ X q 1.
n
. aŽ X q 1.
r Ž b Ž X q 1.
m
n
s 1, a g B=, n g ⺪ 4
. rb Ž X q 1.
m
¬ b g B=, m g ⺪ 4
, ⺪r2⺪. Then we obtain H 1 Ž G , ⺗ m , A . , ⺪r2⺪.
3.2. H 1 Ž G Ž ., G . THEOREM 3.2.1. H 1 Ž G Ž . , G . , A=rNormŽ B= . = ⺪r2⺪. Proof. As in Theorem 3.1.3, we have H 1 Ž G Ž . , G . , H 1 Ž ⌫, G Ž GBŽ . . . , where GŽ GBŽ . . s Hom AŽ Aw , x, Bw X, Ž X q 1.y1 x.. Since m y 2 g B= Žwe remark that BrA is unramified., for an element of GŽ GBŽ . ., there is a commutative diagram, T ¬ aŽ X q 1. n < < l l y1 y1 y1 x Ž . w w Ž T q T , T q m y T gB T, T ªB X, X q 1.y1 x ᎐ ,
᎐
D p g Aw , x.
Hence we have Hom A A w , x , B X , Ž X q 1 .
ž
y1
n
/
, Ž a Ž X q 1 . q ay1 Ž X q 1 . q Ž m y . ay1 Ž X q 1 .
yn
yn
, aŽ X q 1.
. ¬ a g B=, n g ⺪ 4
n
.
482
TOSHIAKI OHNO
, Hom B B w T , Ty1 x , B X , Ž X q 1 .
ž
, B X , Ž X q 1.
y1 =
y1
/
;
however, we must define a new ⌫-action in place of on Bw X, Ž X q 1.y1 x= so that the following diagram is commutative:
Ž aŽ X q 1.
n
q ay1 Ž X q 1 .
yn
, yn
n
a Ž X q 1 . q Ž m y . ay1 Ž X q 1 . . ¬ a Ž X q 1 . ᎐ l ᎐ < < l Ž . . y1 x= Ž w Ž . G GB ªB X, X q 1
n
6
6
GŽ GBŽ . . ªBw X, Ž X q 1.y1 x= < < j j yn n yn y1 Ž Ž a. Ž X q 1. q Ž a. Ž X q 1. ,¬ Ž a. y1 Ž X q 1. 6
6
Ž a .
y1
Ž X q 1.
yn
q Ž m y . Ž a. Ž X q 1.
n
..
Hence, we get H 1 Ž ⌫, G Ž GBŽ . . . , H 1 ⌫, B X , Ž X q 1 .
ž
y1 =
n
s aŽ X q 1. ¬ Ž aŽ X q 1. r Ž b Ž X q 1. n
m
/ n
. aŽ X q 1.
. rb Ž X q 1.
s aŽ X q 1. ¬ Ž a.
y1
r 1r Ž b . b Ž X q 1 .
m
n
s 1, a g B=, n g ⺪ 4
¬ b g B=, m g ⺪ 4
a s 1, a g B=, n g ⺪ 4
2m
¬ b g B=, m g ⺪ 4
, A=rNormŽ B= . = ⺪r2⺪. Then we obtain H 1 Ž G Ž . , G . , A=rNormŽ B= . = ⺪r2⺪.
Remark 3.2.2. Let A be complete and let k be finite field. Then Žcf. w3, V, 2, Remarksx. we get A=rNormŽ B= . s 0.
483
TWISTED GROUP SCHEMES
3.3. H 1 Ž G , G . THEOREM 3.3.1. H 1 Ž G , G . , A=rNormŽ B= . . Proof. As in Theorem 3.1.3, we have H 1 Ž G , G . , H 1 Ž ⌫, G Ž GBŽ . . . , where GŽ GBŽ . . s Hom AŽ Aw , x, Bw X, Ž X q 1.y1 x.. Since m y 2 g B=, for an element of GŽ GBŽ . ., there is a commutative diagram, T ¬ aŽ X q 1. n < < l l T q Ty1 , T q Ž m y .Ty1 gBw T, Ty1 x ªBw X, Ž X q 1.y1 x ᎐ ,
᎐
j p g Aw , x.
Hence we have Hom A A w , x , B X , Ž X q 1 .
ž
y1
/
n
, Ž a Ž X q 1 . q ay1 Ž X q 1 .
yn
,
n
a Ž X q 1 . q Ž m y . ay1 Ž X q 1 . , Hom B B w T , Ty1 x , B X , Ž X q 1 .
ž
, B X , Ž X q 1.
y1 =
y1
yn
. ¬ a g B=, n g ⺪ 4
/
;
however, we must define the new ⌫-action in place of on Bw X, Ž X q 1.y1 x= so that the following diagram is commutative:
Ž aŽ X q 1.
n
q ay1 Ž X q 1 .
yn
, yn
n
a Ž X q 1 . q Ž m y . ay1 Ž X q 1 . . ¬ a Ž X q 1 . ᎐ l ᎐ < < l Ž . y1 = GŽ GB . ªBw X, Ž X q 1. x
n
6
6
GŽ GBŽ . . ªBw X, Ž X q 1.y1 x= < < j j n yn n y1 Ž Ž a. Ž X q 1. q Ž a. Ž X q 1. ,¬ Ž a. y1 Ž X q 1. 6
6
Ž a .
y1
n
Ž X q 1. q Ž m y . Ž a. Ž X q 1.
yn
..
484
TOSHIAKI OHNO
Hence, we get H 1 Ž ⌫, G Ž GBŽ . . . , H 1 ⌫, B X , Ž X q 1 .
ž
y1 =
n
s aŽ X q 1. ¬ Ž aŽ X q 1. r Ž b Ž X q 1. n
m
/ n
. aŽ X q 1.
. rb Ž X q 1.
s aŽ X q 1. ¬ Ž a.
y1
m
aŽ X q 1.
n
s 1, a g B=, n g ⺪ 4
¬ b g B=, m g ⺪ 4 2n
s 1, a g B=, n g ⺪ 4
r 1r Ž b . b ¬ b g B= 4 s a ¬ Ž a . s a, a g B= 4 r Ž b . b ¬ b g B= 4 s A=rNormŽ B= . . Then we obtain H 1 Ž G , G . , A=rNormŽ B= . .
3.4. H 1 Ž⺗ a, A , G . THEOREM 3.4.1. H 1 Ž ⺗ a, A , G . , A=rNormŽ B= . . Proof. As in Theorem 3.1.3, we have H 1 Ž ⺗ a, A , G . , H 1 Ž ⌫, G Ž ⺗ a, B . . , where GŽ⺗ a, B . s Hom AŽ Aw , x, Bw X x.. Since m y 2 g B=, for an element of GŽ⺗ a, B ., there is a commutative diagram T ¬ a < < l l T q Ty1 , T q Ž m y .Ty1 gBw T, Ty1 x ªBw X x ᎐ ,
᎐
D p g Aw , x,
where Bw T, Ty1 x s Aw , x mA B. Hence we have Hom A Ž A w , x , B w X x . , Ž a q ay1 , a q Ž m y . ay1 . ¬ a g B= 4 , Hom B Ž B w T , Ty1 x , B w X x . , Bw X x s B= ;
=
TWISTED GROUP SCHEMES
485
however, we must define the new ⌫-action in place of on B= so that the following diagram is commutative:
Ž a q ay1 , a q Ž m y . ay1 . ¬ a < < l l GŽ⺗ a, B . ª B=
6
GŽ⺗ a, B . ªB=. < < j j
6
᎐
6
᎐
6
Ž Ž a. q Ž a. y1 , Ž a. y1 q Ž m y . Ž a. . ¬ Ž a. y1 . Hence, we get H 1 Ž ⌫, G Ž ⺗ a, B . . , H 1 Ž ⌫, B= . s a ¬ Ž a . a s 1, a g B= 4 r Ž b . rb ¬ b g B= 4 s a ¬ ar Ž a . s 1, a g B= 4 r 1r Ž b . b ¬ b g B= 4 s A=rNormŽ B= . . Then we obtain H 1 Ž ⺗ a, A , G . , A=rNormŽ B= . .
COROLLARY 3.4.2. H 1 Ž Spec A, G . , A=rNormŽ B= . .
4. EXTENSION 4.1. Ext 1 Ž G , ⺗ m, A . THEOREM 4.1.1.
Let
Ł ⺗ m s Spec Aw X , Y , Z x r Ž Z Ž X 2 q m XY q n 2 Y 2 . y 1 . ,
BrA
486
TOSHIAKI OHNO
where Ž/ 0. g A. Then Ł Br A⺗ m is the nontri¨ ial extension of G by ⺗ m, A : ␣
0 ª ⺗m , A ª

Ł ⺗ m ª G ª 0.
BrA
The homomorphisms ␣ and  are defined by the following ␣ * and  *:
␣ *: A w X , Y , Z x r Ž Z Ž X 2 q m XY q n 2 Y 2 . y 1 . ª A w T , Ty1 x
Ž X , Y , Z . ¬ Ž T , 0, Ty2 .  *: A w w, ¨ x r Ž Ž n¨ 2 y m¨ w q w 2 . y Ž m2 y 4 n . ¨ . ª A w X , Y , Z x r Ž Z Ž X 2 q m XY q n 2 Y 2 . y 1 .
Ž w, y¨ . ¬ Ž XYZ, Y 2 Z . Ž cf. w4, 3.1x.. Proof. We know the nontrivial extension 0 ª ⺗m , A ª
Ł ⺗ m ª G ª 0.
BrA
Then we get Ł Br A⺗ m by the Neron blow-up of ⺗ m, k in Ł Br A ⺗ m . ´ THEOREM 4.1.2. Ext 1 Ž G , ⺗ m , A . , ⺪r2⺪. Proof. In general, there is an exact sequence Žcf. w1, III, 6.2.5x. 0 ª H02 Ž G , ⺗ m , A . ª Ext 1 Ž G , ⺗ m , A . ª H01 Ž G , H 1 Ž ⺗ m , A . . ª H03 Ž G , ⺗ m , A . ª Ext 2 Ž G , ⺗ m , A . , where H 1 Ž⺗ m, A . denotes the presheaf on Sch r A defined by X ¬ H 1 Ž X, ⺗ m, A ., and H02 Ž G , ⺗ m, A ., etc., are the Hochschild cohomology. First, we compute H02 Ž G , ⺗ m, A . and H03 Ž G , ⺗ m, A ..
487
TWISTED GROUP SCHEMES
We use Hopf algebra: ␦2
Bw G Ž . x mB Bw G Ž . x ª Bw G Ž . x mB Bw G Ž . x mB Bw G Ž . x n j
Bw T, Ty1 x j Aw T, Ty1 x
p j
q o Aw G x mA Aw G x ª Aw G x mA Aw G x mA Aw G x. After tensoring B, the two cochain is C02 Ž GBŽ . , ⺗ m , B . s Hom B Ž B w T , Ty1 x , B w G Ž . x mB B w G Ž . x . m
n
, a Ž X q 1 . m Ž X q 1 . ¬ a g B=, m, n g ⺪ 4 . We compute ␦ 2 Ž aŽ X q 1. m m Ž X q 1. n .. Then Žwe may assume that the action of G on ⺗ m, A is trivial. m
␦ 2 Ž aŽ X q 1. m Ž X q 1.
n
.
m
s Ž a m Ž X q 1. m Ž X q 1.
n
.Ž aŽ X q 1.
m
m Ž X q 1.
m
m Ž X q 1.
Ž aŽ X q 1.
m
n
m Ž X q 1. m Ž X q 1.
n
.Ž aŽ X q 1.
n y1
.
m n
m Ž X q 1. m 1. s Ž X q 1.
ym
y1
n
m 1 m Ž X q 1. .
Hence the cocycle condition implies m s n s 0. Then we have Z02 Ž GBŽ . , ⺗ m , B . , B=. Hence Z02 Ž G , ⺗ m, A . ; A. The coboundary map is linear with respect to constants, so Z02 Ž G , ⺗ m, A . s B02 Ž G , ⺗ m, A .. Then we obtain H02 Ž G , ⺗ m , A . s 0. Similarly, the three cochain is C03 Ž GBŽ . , ⺗ m , B . s Hom B Ž B w T , Ty1 x , B w G Ž . x mB B w G Ž . x mB B w G Ž . x . l
m
n
, a Ž X q 1 . m Ž X q 1 . m Ž X q 1 . ¬ a g B=, l, m, n g ⺪ 4 .
488
TOSHIAKI OHNO
We compute ␦ 3 Ž aŽ X q 1. l m Ž X q 1. m m Ž X q 1. n ., then m
l
␦ 3 Ž aŽ X q 1. m Ž X q 1. m Ž X q 1.
n
.
m
l
s Ž a m Ž X q 1. m Ž X q 1. m Ž X q 1.
n
. n y1
m
Ž aŽ X q 1. l m Ž X q 1. l m Ž X q 1. m Ž X q 1. . m m n Ž aŽ X q 1. l m Ž X q 1. m Ž X q 1. m Ž X q 1. . m
n y1
n
Ž aŽ X q 1. l m Ž X q 1. m Ž X q 1. m Ž X q 1. . m n Ž aŽ X q 1. l m Ž X q 1. m Ž X q 1. m 1 . m
m
s a m Ž X q 1 . m Ž X q 1 . m 1. Hence the cocycle condition implies a s 1, m s 0. Then we have n
Z03 Ž GBŽ . , ⺗ m , B . , Ž X q 1 . m 1 m Ž X q 1 . ¬ l, n g ⺪ 4 . l
Next we compute the element of Z03 Ž G , ⺗ m, A . given by T ¬ Ž X q 1 . m 1 m Ž X q 1 . g A w G x mA A w G x mA A w G x . l
n
Then we can easily see that l s n s 0, and there is only a unit element: T ¬ 1. Hence we have ⺪ 30 Ž G , ⺗ m, A . s 0 and H03 Ž G , ⺗ m , A . s 0. Therefore Ext 1 Ž G , ⺗ m , A . , H01 Ž G , H 1 Ž ⺗ m , A . . . Next, we compute H01 Ž G , H 1 Ž⺗ m, A ... We have Žwe may assume that the action of G on H 1 Ž⺗ m, A . is trivial. H01 Ž G , H 1 Ž ⺗ m , A . . , Hom Ayg r Ž G , H 1 Ž ⺗ m , A . . . Moreover, we obtain Hom Ay g r Ž G , H 1 Ž ⺗ m , A . . ; MorA Ž G , H 1 Ž ⺗ m , A . . s H 1 Ž ⺗m , A . Ž G . s H 1 Ž G , ⺗m , A . , ⺪r2⺪.
TWISTED GROUP SCHEMES
489
Hence Ext 1 Ž G , ⺗ m , A . ; ⺪r2⺪. While we have the existence of nontrivial extension in Theorem 4.1.1, Ext 1 Ž G , ⺗ m , A . , ⺪r2⺪.
4.2. Ext 1 Ž G Ž ., G . THEOREM 4.2.1.
Let
Ł ⺗ m s Spec
A X , Y , Z, Ž Z q 1 .
y1
BrA
r Ž X 2 q mXY q nY 2 q Ž m2 y 4 n . Ž Z q 1 . . , where Ž/ 0. g A. Then Ł Br A ⺗ m is the nontri¨ ial extension of G Ž . by G: ␥
0 ª Gª
Ł
BrA
␦
⺗ m ª G Ž . ª 0.
The homomorphisms ␥ and ␦ are defined by the following ␥ * and ␦ *:
␥ *: A X , Y , Z, Ž Z q 1 .
y1
r Ž X 2 q mXY q nY 2 q Ž m2 y 4 n . Ž Z q 1 . . ª A w , x r Ž n 2 y m q 2 q m2 y 4 n .
Ž X , Y , Z . ¬ Ž , y , 0 . ␦ *: A X , Ž X q 1 .
y1
ª A X , Y , Z, Ž Z q 1 .
y1
r Ž X 2 q mXY q nY 2 q Ž m2 y 4 n . Ž Z q 1 . . X ¬ Z. Proof. We know the nontrivial extension 0ªGª
Ł ⺗ m ª ⺗ m , A ª 0.
BrA
490
TOSHIAKI OHNO
Then we get Ł Br A ⺗ m by the Neron blow-up of Gk in Ł Br A ⺗ m . ´ THEOREM 4.2.2. Ext 1 Ž G Ž . , G . , ⺪r2⺪. Proof. There is an exact sequence Žcf. w1, III, 6, 2.5x. 0 ª H02 Ž G Ž . , G . ª Ext 1 Ž G Ž . , G . ª H01 Ž G Ž . , H 1 Ž G . . ª H03 Ž G Ž . , G . ª Ext 2 Ž G Ž . , G . . First, we compute H02 Ž G Ž ., G . and H03 Ž G Ž ., G .. We use Hopf algebra: ␦2
Bw G Ž . x mB Bw G Ž . x ª Bw G Ž . x mB Bw G Ž . x mB Bw G Ž . x n j
Bw T, Ty1 x j Aw , x
p j
q o Aw G Ž . x mA Aw G Ž . x ª Aw G Ž . x mA Aw G Ž . x mA Aw G Ž . x. As in Theorem 4.1.2, we have H02 Ž G Ž . , G . s 0. Similarly, the three cocycle is n
Z03 Ž GBŽ . , ⺗ m , B . , Ž X q 1 . m 1 m Ž X q 1 . ¬ l, n g ⺪ 4 . l
We compute the element of ⺪ 30 Ž G Ž ., G . given by n
l
¬ Ž X q 1. m 1 m Ž X q 1. q Ž X q 1.
yl
m 1 m Ž X q 1.
yn
g A w G Ž . x mA A w G Ž . x mA A w G Ž . x l
¬ Ž X q 1. m 1 m Ž X q 1. q Ž m y . Ž X q 1.
yl
n
m 1 m Ž X q 1.
yn
g A w G Ž . x mA A w G Ž . x mA A w G Ž . x . Then the cocycle condition implies that l s n s 0, so there is only a unit element, ¬ 2, ¬ m. Hence we have Z03 Ž G Ž ., G . s 0 and H03 Ž G Ž . , G . s 0.
491
TWISTED GROUP SCHEMES
Therefore Ext 1 Ž G Ž . , G . , H01 Ž G Ž . , H 1 Ž G . . . Next, we compute H01 Ž G Ž ., H 1 Ž G ... We have H01 Ž G Ž . , H 1 Ž G . . , Hom Ayg r Ž G Ž . , H 1 Ž G . . . Moreover, we obtain Hom Ay g r Ž G Ž . , H 1 Ž G . . ; MorA Ž G Ž . , H 1 Ž G . . s H 1 Ž G . Ž G Ž . . s H 1 Ž G Ž . , G . , A=rNormŽ B= . = ⺪r2⺪. Hence Ext 1 Ž G Ž . , G . ; A=rNormŽ B= . = ⺪r2⺪. Let u: Spec A ª G Ž . be the neutral element of G Ž .. Then for any element in Hom Ay g r Ž G Ž ., H 1 Ž G .., ( u s 0 g H 1 Ž G .ŽSpec A.. On the other hand, we have Žwe remark that H 1 Ž G .ŽSpec A. s H 1 ŽSpec A, G . , A=rNormŽ B=. Žcf. 3.4.2.. Hom Ay g r Ž G Ž . , H 1 Ž G . .
;
H 1 Ž G . Ž G Ž . .
ª
H 1 Ž G . Ž Spec A .
< j
¬
< j
¬
< j (u
Then we get Ext 1 Ž G Ž . , G . , Hom Ay g r Ž G Ž . , H 1 Ž G . . ; ⺪r2⺪. While we have the existence of nontrivial extension in Theorem 4.2.1, Ext 1 Ž G Ž . , G . , ⺪r2⺪.
4.3. Ext 1 Ž G , G . THEOREM 4.3.1. Ext 1 Ž G , G . s 0. Proof. There is an exact sequence Žcf. w1, III, 6.2.5x. 0 ª H02 Ž G , G . ª Ext 1 Ž G , G . ª H01 Ž G , H 1 Ž G . . ª H03 Ž G , G . ª Ext 2 Ž G , G . . First, we compute H02 Ž G , G . and H03 Ž G , G ..
492
TOSHIAKI OHNO
We use Hopf algebra, ␦2
Bw G Ž . x mB Bw G Ž . x ª Bw G Ž . x mB Bw G Ž . x mB Bw G Ž . x n j
Bw T, Ty1 x j Aw , x
p j
q o Aw G x mA Aw G x ª Aw G x mA Aw G x mA Aw G x, where Aw G x s Aw ¨ , w x. As in Theorem 4.1.2, we have H02 Ž G , G . s 0. Similarly, the three cocycle is n
Z03 Ž GBŽ . , ⺗ m , B . , Ž X q 1 . m 1 m Ž X q 1 . ¬ l, n g ⺪ 4 . l
We compute the element of Z03 Ž G , G . given by n
l
¬ Ž X q 1. m 1 m Ž X q 1. q Ž X q 1.
yl
m 1 m Ž X q 1.
yn
g A w G x mA A w G x mA A w G x l
¬ Ž X q 1. m 1 m Ž X q 1. q Ž m y . Ž X q 1.
yl
n
m 1 m Ž X q 1.
yn
g A w G x mA A w G x mA A w G x .
Then the cocycle condition implies that l s n s 0, so there is only a unit element, ¬ 2, ¬ m. Hence we have Z03 Ž G , G . s 0 and H03 Ž G , G . s 0. Therefore Ext 1 Ž G , G . , H01 Ž G , H 1 Ž G . . . Next, we compute H01 Ž G , H 1 Ž G ... We have H01 Ž G , H 1 Ž G . . , Hom Ayg r Ž G , H 1 Ž G . . . Moreover, we obtain Hom Ay g r Ž G , H 1 Ž G . . ; MorA Ž G , H 1 Ž G . . s H 1 Ž G. Ž G . s H 1 Ž G , G. , A=rNormŽ B= . , H 1 Ž G . Ž Spec A . .
493
TWISTED GROUP SCHEMES
Let u: Spec A ª G be the neutral element of G . Then for any element in Hom Ay g r Ž G , H 1 Ž G .., ( u s 0 g H 1 Ž G .ŽSpec A.. On the other hand, we have Hom Ay g r Ž G , H 1 Ž G .. ; H 1 Ž G .Ž G . , H 1 Ž G .ŽSpec A.. < j
< j
¬
< j (u
¬
Hence, we get Ext 1 Ž G , G . , Hom Ay g r Ž G , H 1 Ž G . . s 0.
4.4. Ext 1 Ž⺗ a, A , G . THEOREM 4.4.1. Suppose char Ž K . s 0 and char Ž k . s p ) 0. Then an element of Ext 1 Ž⺗ a, A , G . corresponds to F Ž a1 , a p , a p 2 , . . . ; X . s 1 q a1 X q q ap X p q q ⭈⭈⭈ q q
a12 2!
X 2 q ⭈⭈⭈ q
a1 a p pq1
Ž p y 1. !
X pq1 q
Ž p q 2. Ž p q 1.
Ž 2 p y 1 . ⭈⭈⭈ Ž p q 1 .
Ž 2 p . ⭈⭈⭈ Ž p q 1 .
X
2p
q
X py1
a12 a p
a1py 1 a p p!a2p
q ⭈⭈⭈ q
a1py 1
X pq2
X 2 py1 p!a1 a2p
Ž 2 p q 1 . ⭈⭈⭈ Ž p q 1 .
p!a1py 1 a2p
Ž 3 p y 1 . ⭈⭈⭈ Ž p q 1 .
X 2 pq1
X 3 py1
2
2
Ž p! . a3p Ž p! . a1 a3p q X 3p q X 3 pq1 Ž 3 p . ⭈⭈⭈ Ž p q 1 . Ž 3 p q 1 . ⭈⭈⭈ Ž p q 1 . 2
q ⭈⭈⭈ q
Ž p! . a1py1a3p X 4 py1 Ž 4 p y 1 . ⭈⭈⭈ Ž p q 1 . ⭈⭈⭈
494
TOSHIAKI OHNO
a1 a p 2
2
q ap2 X p q
p2 q 1
Xp
2
q1
q ⭈⭈⭈ ,
where pa p s
a1p
Ž p y 1. !
,
p2ap2 s
Ž p! .
py 2
a1p a ppy1
Ž p 2 y 1 . ⭈⭈⭈ Ž p q 1.
,...,
a1 , a p , a p 2 , . . . are nilpotents in BrŽ ., and, moreo¨ er,
Ž 2 y m . a1 , Ž 2 y m . a p , Ž 2 y m . a p 2 , . . . g Ar Ž . . Proof. We obtain the exact sequence 0 ª G ª G ª i#GA rŽ . ª 0 in 2.3, so there is a long exact sequence 0 ª Hom Ay g r Ž ⺗ a, A , G . ª Hom Ayg r Ž ⺗ a, A , G . r
ª Hom Ay g r Ž ⺗ a, A , i#GA rŽ . . ª Ext 1 Ž ⺗ a, A , G . ª Ext 1 Ž ⺗ a, A , G . ª ⭈⭈⭈ . We know that Ext 1 Ž⺗ a, A , G . s 0, so Ext 1 Ž ⺗ a, A , G . , Hom Ayg r Ž ⺗ a, A , i#GA rŽ . . rr Ž Hom Ayg r Ž ⺗ a, A , G . . . Since Hom Ay g r Ž⺗ a, A , G . s 0, Ext 1 Ž ⺗ a, A , G . , Hom Ayg r Ž ⺗ a, A , i#GA rŽ . . s Hom A rŽ .yH o p f Ž Ar Ž . w , x , A w X x mA Ar Ž . . , Hom A rŽ .yH o p f Ž Ar Ž . w , x , Ar Ž . w X x . , where Hom A rŽ .yH o p f Ž ArŽ .w , x, ArŽ .w X x. denotes the set of ArŽ .Hopf algebra homomorphisms. After tensoring B, there is a commutative diagram, T ¬ fŽ X . < < l l T q Ty1 , T q Ž m y .Ty1 gBrŽ .w T, Ty1 x ª BrŽ .w X x ᎐ ,
᎐
D D Ž .w x Ž g Ar , ªAr .w X x.
495
TWISTED GROUP SCHEMES
Hence we have Hom A rŽ .- H o p f Ž Ar Ž . w , x , Ar Ž . w X x . =
, f Ž X . ¬ f Ž X . g Br Ž . w X x , fŽ X. q fŽ X.
y1
,fŽ X. q Žm y . fŽ X.
y1
g Ar Ž . w X x ,
fŽ X m 1 q 1 m X . s fŽ X . m fŽ X .4. Let f Ž X . s Ý iG 0 a i X i g BrŽ .w X x=. Then the condition f Ž X m 1 q 1 m X . s f Ž X . m f Ž X . implies that f Ž X . has the following style: f Ž X . s 1 q a1 X q
a12 2!
a1 a p
q ap X p q q q q
X 2 q ⭈⭈⭈ q
pq1
a1py 1
X pq1 q
a1py 1 a p
Ž 2 p . ⭈⭈⭈ Ž p q 1 .
a12 a p
Ž p q 2. Ž p q 1.
p!a1 a2p
X2p q
p!a1py 1a2p
Ž 3 p y 1 . ⭈⭈⭈ Ž p q 1 .
X pq2 q ⭈⭈⭈
X 2 py1
Ž 2 p y 1 . ⭈⭈⭈ Ž p q 1 . p!a2p
X py1
Ž p y 1. !
Ž 2 p q 1 . ⭈⭈⭈ Ž p q 1 .
X 2 pq1 q ⭈⭈⭈
X 3 py1
2
2
Ž p! . a3p Ž p! . a1 a3p 3p q X q X 3 pq1 q ⭈⭈⭈ Ž 3 p . ⭈⭈⭈ Ž p q 1 . Ž 3 p q 1 . ⭈⭈⭈ Ž p q 1 . 2
Ž p! . a1py1a3p q X 4 py1 Ž 4 p y 1 . ⭈⭈⭈ Ž p q 1 . ⭈⭈⭈ a1 a p 2
2
q ap2 X p q
p q1 2
Xp
2
q1
q ⭈⭈⭈ ,
where pa p s
a1p
Ž p y 1. !
,
p2ap2 s
Ž p! .
py 2
a1p a ppy1
Ž p 2 y 1 . ⭈⭈⭈ Ž p q 1.
,...,
496
TOSHIAKI OHNO
a1 , a p , a p 2 , . . . are nilpotent in BrŽ .. Hence
fŽ X. q fŽ X.
y1
a12
s2q2 q2 q2 q2
2!
X q ⭈⭈⭈ q2
a1py 1
2
a1 a p pq1
Ž p y 1. !
X py1
X pq1 q ⭈⭈⭈ p!a2p
Ž 2 p . ⭈⭈⭈ Ž p q 1 .
X 2 p q ⭈⭈⭈
p!a1py 1 a2p
Ž 3 p y 1 . ⭈⭈⭈ Ž p q 1 .
X 3 py1
2
Ž p! . a1 a3p q2 X 3 pq1 q ⭈⭈⭈ Ž 3 p q 1 . ⭈⭈⭈ Ž p q 1 . ⭈⭈⭈ q2
a1 a p 2 p2 q 1
fŽ X. q Žm y . fŽ X.
Xp
qŽ 2 y m. ap X p q m
qŽ 2 y m. qm
qm
a12
q ⭈⭈⭈ ,
2!
X q ⭈⭈⭈ qm
a1 a p pq1
Ž p q 2. Ž p q 1.
Ž p y 1. !
X pq1 X pq2 q ⭈⭈⭈
a1py 1 a p
Ž 2 p y 1 . ⭈⭈⭈ Ž p q 1 .
Ž 2 p . ⭈⭈⭈ Ž p q 1 .
a1py 1
2
a12 a p
p!a2p
qŽ 2 y m.
q1
y1
s m q Ž 2 y m . a1 X q m
qŽ 2 y m.
2
X 2 py1
X2p
p!a1 a2p
Ž 2 p q 1 . ⭈⭈⭈ Ž p q 1 .
p!a1py 1a2p
Ž 3 p y 1 . ⭈⭈⭈ Ž p q 1 .
X 3 py1
X 2 pq1 q ⭈⭈⭈
X py1
497
TWISTED GROUP SCHEMES 2
Ž p! . a3p qŽ 2 y m. X 3p Ž 3 p . ⭈⭈⭈ Ž p q 1 . 2
qm
Ž p! . a1 a3p X 3 pq1 q ⭈⭈⭈ Ž 3 p q 1 . ⭈⭈⭈ Ž p q 1 . 2
Ž p! . a1py1a3p qŽ 2 y m. X 4 py1 q ⭈⭈⭈ Ž 4 p y 1 . ⭈⭈⭈ Ž p q 1 . ⭈⭈⭈ 2
qŽ 2 y m. ap2 X p q m
a1 a p 2 p2 q 1
Xp
2
q1
q ⭈⭈⭈ .
Then we obtain a12 , a1 a p , a2p ,a1 a3p , a4p , a1 a5p , . . . , a1 a ppy 2 , a ppy 1 , a1 a p 2 , a2p 2 , a1 a3p 2 , . . .g ArŽ . ,
Ž 2 y m . a1 , Ž 2 y m . a p , Ž 2 y m . a3p , Ž 2 y m . a5p , . . . , Ž 2 y m . a ppy 2 , Ž 2 y m . a p 2 , Ž 2 y m . a3p 2 , . . . g Ar Ž . . These conditions are equivalent to
Ž 2 y m . a1 , Ž 2 y m . a p , Ž 2 y m . a p 2 , . . . g Ar Ž . .
ACKNOWLEDGMENTS Finally, I am grateful to Prof. T. Sekiguchi and Prof. N. Suwa for their many suggestions.
REFERENCES 1. M. Demazure and P. Gabriel, ‘‘Groupes Algebriques,’’ Tome I, North-Holland, Amster´ dam, 1970. ´ 2. J. S. Milne, ‘‘Etale Cohomology,’’ Princeton Univ. Press, Princeton, NJ, 1980. 3. J.-P. Serre, ‘‘Local Fields,’’ Graduate Texts in Mathematics 67, Springer-Verlag, BerlinrNew York, 1979. 4. W. C. Waterhouse and B. Weisfeiler, One-dimensional affine group schemes, J. Algebra 66 Ž1980., 550᎐568. 5. B. Weisfeiler, On a case of extensions of group schemes, Trans. Amer. Math. Soc. 248, No. 1 Ž1979., 171᎐189.
498
TOSHIAKI OHNO
6. T. Sekiguchi, On the deformations of Witt groups to tori II, J. Algebra 138 Ž1991., 273᎐297. 7. T. Sekiguchi and N. Suwa, A case of extensions of group schemes over a discrete valuation ring, Tsukuba J. Math. 14, No. 2 Ž1990., 459᎐487. 8. T. Sekiguchi and N. Suwa, Some cases of extensions of group schemes over a discrete valuation ring I, J. Fac. Sci. Uni¨ . Tokyo Sect. IA Math. 38 Ž1991., 1᎐45. 9. T. Sekiguchi and N. Suwa, Some cases of extensions of group schemes over a discrete valuation ring II, Bull. Fac. Sci. Engrg. Chuo Uni¨ . Ser. I Math. 32 Ž1989., 17᎐35.
Some Cases of Extensions of Twisted Group Schemes over a Discrete Valuation Ring Toshiaki Ohno Department of Mathematics, Faculty of Science and Engineering, Chuo Uni¨ ersity, Tokyo 112, Japan Communicated by Da¨ id Buchsbaum Received May 2, 1996
1. INTRODUCTION Let A be a discrete valuation ring with maximal ideal ᒊ, let K be the field of fractions of A, and let k s Arᒊ. Let G Ž . be the model of ⺗ m, A given by G Ž . s Spec A X , Ž X q 1 .
y1
for some Ž/ 0. g A Žcf. w4, 2.5x.. Then there is an exact sequence G Ž . ª ⺗ m, A ªi#⺗ m, A rŽ . ª0, 0 ªG x ¬ x q 1, where i is the closed immersion i: Spec ArŽ . ¨ Spec A Žcf. w6, 3.1x.. Sekiguchi and Suwa computed several extensions with respect to G Ž ., for example, Ext 1 Ž G Ž ., G Ž . ., etc. Žcf. w6᎐9x.. Weisfeiler computed Ext 1 Ž G Ž ., ⺗ a, A . Žcf. w5x.. Let BrA be an unramified quadratic extension. We denote its Galois group by ⌫ s e, 4 . We remark that B s a q b ¬ a, b g A4 where 2 y m q n s 0 for some m, n g A and m2 y 4 n g A=. We denote the norm map by Norm: B=ª A=. Let G be the nontrivial BrA-form of ⺗ m . Then by Waterhouse and Weisfeiler Žcf. w4, 2.6x., we get G s Spec A w , x r Ž n 2 y m q 2 q m2 y 4 n . . 474 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
475
TWISTED GROUP SCHEMES
Let G be the model of G. Then we get G s Spec A w ¨ , w x r Ž Ž n¨ 2 y m¨ w q w 2 . y Ž m2 y 4 n . ¨ . for some Ž/ 0. g A Žcf. Section 2.2.. Then we have an exact sequence Žcf. Section 2.3. 0ª G ª G ªi#GA rŽ . ª0. Ž x, y . ¬Ž x q 2, y q m.. Moreover, we have GB , GBŽ ., that is to say, G is the form of GBŽ . Žcf. Section 2.2.. Then there is the commutative diagram consisting of exact lines Žcf. Section 2.3., GBŽ . ª⺗ m, B ªj#⺗ m, BrŽ . ª 0 0 ªG x x 0 ª G ª G ª i#GA rŽ . ª0, x x 0 0, where j is the closed immersion j: Spec BrŽ . ¨ Spec B. Let Ł Br A⺗ m be the Weil restriction of ⺗ m from B to A. Then
Ł ⺗ m s Spec Aw X , Y , Z x r Ž Z Ž X 2 q mXY q nY 2 . y 1 . .
BrA
In this case, we are interested in the several extensions with respect to G and G . The purpose of this paper is to decide the following extension groups. We consider the fppf topology over Spec A and denote the corresponding site by Sch r A . Hence all cohomology groups area fppf cohomologies. In particular, we regard the first cohomology group H 1 Ž X, G . as the set of isomorphism classes of G-torsors over X. We consider only commutative extensions in Sch r A . Our results are as follows. 1. First cohomologies H 1 Ž G , ⺗ m , A . , ⺪r2⺪
Ž cf. 3.1.
H 1 Ž G Ž . , G . , A=rNormŽ B= . = ⺪r2⺪ =
=
H Ž G , G . , A rNormŽ B . 1
H 1 Ž ⺗ a, A , G . , A=rNormŽ B= . .
Ž cf. 3.2. Ž cf. 3.3. Ž cf. 3.4.
476
TOSHIAKI OHNO
2. Extensions Ext 1 Ž G , ⺗ m , A . , ⺪r2⺪
Ž cf. 4.1.
Ext 1 Ž G Ž . , G . , ⺪r2⺪
Ž cf. 4.2.
Ext 1 Ž G , G . s 0.
Ž cf. 4.3.
3. The model of the Ł Br A ⺗ m , which is the Neron blow-up of ⺗ m, k ´ in Ł Br A ⺗ m , is the nontrivial extension of G by ⺗ m, A Žcf. Section 4.1.. 4. The model of the Ł Br A ⺗ m , which is the Neron ´ blow-up of Gk in Ł Br A ⺗ m , is the nontrivial extension of G Ž . by G Žcf. Section 4.2.. 5. Any element of Ext 1 Ž⺗ a, A , G . corresponds to the polynomial F Ž a1 , a p , a p 2 , . . . ; X . given in Section 4.4 Žcf. Section 4.4.. The results of 3.1 and 4.1 are suggested by Suwa.
2. FORMS AND MODELS 2.1. The Form of ⺗ m We will review the form of ⺗ m from Waterhouse and Weisfeiler Žcf. w4, 2.6x.. THEOREM 2.1.1.
Let G be the nontri¨ ial BrA-form of ⺗ m . Then G s Spec A w , x ,
where 1.
Relation n 2 y m q 2 s 4 n y m2 .
2. Comultiplication
¬ ¬
1 m y 4n 2
1 m y 4n 2
Ž Ž m2 y 2 n . m q 2 m y m Ž m q m . . Ž mn m q m m y 2 n Ž m q m . . .
3. Counit
¬2 ¬ m.
TWISTED GROUP SCHEMES
477
4. Coin¨ erse
¬ ¬ m y . Proof. The nontrivial action of ⌫ over Bw⺗ m x s Bw T, Ty1 x is given by
Bw T, Ty1 x ª Bw T, Ty1 x. < < j j y1 T ¬ T . Then the ⌫-invariant subring Bw T, Ty1 x ⌫ is given by ⌫
B w T , Ty1 x s A w , x , where
s T q Ty1 s T q Ž m y . Ty1 . Hence Aw G x has the form Aw , x, and the computations of relation, comultiplication, counit, and coinverse are easy. 2.2. The Model of G We will compute the model of G by using Neron blow-ups Žcf. w4, 2.6x.. ´ Let G be the model of G. Then
THEOREM 2.2.1.
G s Spec A w ¨ , w x , where 1.
Relation
Ž n¨ 2 y m¨ w q w 2 . s Ž m2 y 4 n . ¨ , 2. Comultiplication ¨ ¬¨ m1 q1 m¨
q
m y 4n 2
Ž Ž m2 y 2 n . ¨ m ¨ y m¨ m w y mw m ¨ q 2 w m w .
w¬wm1q1mw q 2 Ž mn¨ m ¨ y 2 n¨ m w y 2 nw m ¨ q mw m w . , m y 4n
478
TOSHIAKI OHNO
3. Counit ¨ ¬0
w ¬ 0, 4. Coin¨ erse ¨ ¬¨ w ¬ m¨ y w,
for some Ž/ 0. g A. Proof. We can get the G by using Neron blow-ups Žcf. w4, Sect. 1x.. Let ´ G1 be the Neron ´ blow-up of the subgroup defined by Ž , . s Ž2, m. of the special fiber Gk . Then Aw G1 x has the form A w G1 x s A , , y1 Ž y 2 . , y1 Ž y m . r Ž n 2 y m q 2 qm2 y 4 n . s A y1 Ž y 2 . , y1 Ž y m . r Ž n 2 y m q 2 q m2 y 4 n . s A w ¨ , w x r Ž Ž n¨ 2 y m¨ w q w 2 . y Ž m2 y 4 n . ¨ . , where s ¨ q 2, s w q m. The special fiber of G1 is ⺗ a , and we have k w G1 x , k w ¨ , w xrŽ ¨ .. Let G2 be the Neron blow-up of the subgroup ´ defined by Ž ¨ , w . s Žy1 Ž y 2., y1 Ž y m.. s Ž0, 0. of the special fiber G1 k . Then Aw G2 x has the form A w G2 x s A y1 Ž y 2 . , y1 Ž y m . , y2 Ž y 2 . , y2 Ž y m . r Ž n 2 y m q 2 q m2 y 4 n . s A y2 Ž y 2 . , y2 Ž y m . r Ž n 2 y m q 2 q m2 y 4 n . s A w ¨ , w x r Ž 2 Ž n¨ 2 y m¨ w q w 2 . y Ž m2 y 4 n . ¨ . , where s 2 ¨ q 2, s 2 w q m. The special fiber of G2 is also ⺗ a . We continue the construction and we get G . The comultiplication is defined so that the morphism G ª G Ž x, y . ¬Ž x q 2, y q m. is homomorphism. The computation of counit and coinverse are easy.
479
TWISTED GROUP SCHEMES
PROPOSITION 2.2.2.
G is the form of GBŽ ..
Proof. We put ¨ sXqY
w s X q Ž m y .Y, and we have
Ž n¨ 2 y m¨ w q w 2 . y Ž m2 y 4 n . ¨ s y Ž m2 y 4 n . Ž XY q X q Y . . Then B w G x , B w X , Y x r Ž XY q X q Y . s B X , yX Ž X q 1 .
y1
, B X , Ž X q 1.
y1
s B w G Ž . x .
2.3. Exact Sequence We have an exact sequence 0 ª G ª G ª i#GA rŽ . ª 0, where i is a closed immersion i: Spec ArŽ . ¨ Spec A. Let R be any faithfully flat A-algebra. The maps are G Ž R. ª 0 ªG GŽ R . ª i#GA rŽ .Ž R . < < j j Ž x, y . ¬Ž x q 2, y q m. Ž x⬘, y⬘. ¬Ž x⬘ modŽ ., y⬘ modŽ ... We obtain a commutative diagram consisting of exact lines, GBŽ . ª⺗ m, B ªj#⺗ m, BrŽ . ª 0 0 ªG x x 0 ª G ª G ª i#GA rŽ . ª0, x x 0 0, where j is a closed immersion j: Spec BrŽ . ¨ Spec B.
480
TOSHIAKI OHNO
3. FIRST COHOMOLOGY 3.1. H 1 Ž G , ⺗ m, A . LEMMA 3.1.1. H 1 Ž Spec A, ⺗ m , A . s 0. Proof. Cf. w2, III, 4.10x. COROLLARY 3.1.2.
Let H 1 Ž ⌫, B=. be the group cohomology. Then H 1 Ž ⌫, B= . s 0.
Proof. We note that the extension BrA is unramified. Then we obtain immediately. THEOREM 3.1.3. H 1 Ž G , ⺗ m , A . , H 1 Ž ⌫, ⺗ m , A Ž GBŽ . . . . Proof. There is the finite Galois covering GBŽ . ª G with Galois group ⌫, and the ⌫-action on GBŽ . is given as follows:
Bw X, Ž X q 1.y1 x ª Bw X, Ž X q 1.y1 x < < j j Ž X ¬ yX X q 1.y1 . Then, by Hochschild᎐Serre spectral sequence Žcf. w2, III, 2.20x., we have the exact sequence ⌫
0 ª H 1 Ž ⌫, ⺗ m , A Ž GBŽ . . . ª H 1 Ž G , ⺗ m , A . ª H 1 Ž GBŽ . , ⺗ m , A . . While we obtain H 1 Ž GBŽ . , ⺗ m , A . , H 1 Ž GBŽ . , ⺗ m , B . s 0. Then we have H 1 Ž G , ⺗ m , A . , H 1 Ž ⌫, ⺗ m , A Ž GBŽ . . . .
THEOREM 3.1.4. H 1 Ž G , ⺗ m , A . , ⺪r2⺪.
481
TWISTED GROUP SCHEMES
Proof. In Theorem 3.1.3, we have ⺗ m , A Ž GBŽ . . sHom A A w T , Ty1 x , B X , Ž X q 1 .
ž
y1
/ ,B
X , Ž X q 1.
y1 =
Hence, using Corollary 3.1.2, we get H 1 Ž ⌫, ⺗ m , A Ž GBŽ . . . , H 1 ⌫, B X , Ž X q 1 .
ž
n
y1 =
/
s aŽ X q 1. ¬ Ž aŽ X q 1.
n
. aŽ X q 1.
r Ž b Ž X q 1.
m
n
s 1, a g B=, n g ⺪ 4
. rb Ž X q 1.
m
¬ b g B=, m g ⺪ 4
, ⺪r2⺪. Then we obtain H 1 Ž G , ⺗ m , A . , ⺪r2⺪.
3.2. H 1 Ž G Ž ., G . THEOREM 3.2.1. H 1 Ž G Ž . , G . , A=rNormŽ B= . = ⺪r2⺪. Proof. As in Theorem 3.1.3, we have H 1 Ž G Ž . , G . , H 1 Ž ⌫, G Ž GBŽ . . . , where GŽ GBŽ . . s Hom AŽ Aw , x, Bw X, Ž X q 1.y1 x.. Since m y 2 g B= Žwe remark that BrA is unramified., for an element of GŽ GBŽ . ., there is a commutative diagram, T ¬ aŽ X q 1. n < < l l y1 y1 y1 x Ž . w w Ž T q T , T q m y T gB T, T ªB X, X q 1.y1 x ᎐ ,
᎐
D p g Aw , x.
Hence we have Hom A A w , x , B X , Ž X q 1 .
ž
y1
n
/
, Ž a Ž X q 1 . q ay1 Ž X q 1 . q Ž m y . ay1 Ž X q 1 .
yn
yn
, aŽ X q 1.
. ¬ a g B=, n g ⺪ 4
n
.
482
TOSHIAKI OHNO
, Hom B B w T , Ty1 x , B X , Ž X q 1 .
ž
, B X , Ž X q 1.
y1 =
y1
/
;
however, we must define a new ⌫-action in place of on Bw X, Ž X q 1.y1 x= so that the following diagram is commutative:
Ž aŽ X q 1.
n
q ay1 Ž X q 1 .
yn
, yn
n
a Ž X q 1 . q Ž m y . ay1 Ž X q 1 . . ¬ a Ž X q 1 . ᎐ l ᎐ < < l Ž . . y1 x= Ž w Ž . G GB ªB X, X q 1
n
6
6
GŽ GBŽ . . ªBw X, Ž X q 1.y1 x= < < j j yn n yn y1 Ž Ž a. Ž X q 1. q Ž a. Ž X q 1. ,¬ Ž a. y1 Ž X q 1. 6
6
Ž a .
y1
Ž X q 1.
yn
q Ž m y . Ž a. Ž X q 1.
n
..
Hence, we get H 1 Ž ⌫, G Ž GBŽ . . . , H 1 ⌫, B X , Ž X q 1 .
ž
y1 =
n
s aŽ X q 1. ¬ Ž aŽ X q 1. r Ž b Ž X q 1. n
m
/ n
. aŽ X q 1.
. rb Ž X q 1.
s aŽ X q 1. ¬ Ž a.
y1
r 1r Ž b . b Ž X q 1 .
m
n
s 1, a g B=, n g ⺪ 4
¬ b g B=, m g ⺪ 4
a s 1, a g B=, n g ⺪ 4
2m
¬ b g B=, m g ⺪ 4
, A=rNormŽ B= . = ⺪r2⺪. Then we obtain H 1 Ž G Ž . , G . , A=rNormŽ B= . = ⺪r2⺪.
Remark 3.2.2. Let A be complete and let k be finite field. Then Žcf. w3, V, 2, Remarksx. we get A=rNormŽ B= . s 0.
483
TWISTED GROUP SCHEMES
3.3. H 1 Ž G , G . THEOREM 3.3.1. H 1 Ž G , G . , A=rNormŽ B= . . Proof. As in Theorem 3.1.3, we have H 1 Ž G , G . , H 1 Ž ⌫, G Ž GBŽ . . . , where GŽ GBŽ . . s Hom AŽ Aw , x, Bw X, Ž X q 1.y1 x.. Since m y 2 g B=, for an element of GŽ GBŽ . ., there is a commutative diagram, T ¬ aŽ X q 1. n < < l l T q Ty1 , T q Ž m y .Ty1 gBw T, Ty1 x ªBw X, Ž X q 1.y1 x ᎐ ,
᎐
j p g Aw , x.
Hence we have Hom A A w , x , B X , Ž X q 1 .
ž
y1
/
n
, Ž a Ž X q 1 . q ay1 Ž X q 1 .
yn
,
n
a Ž X q 1 . q Ž m y . ay1 Ž X q 1 . , Hom B B w T , Ty1 x , B X , Ž X q 1 .
ž
, B X , Ž X q 1.
y1 =
y1
yn
. ¬ a g B=, n g ⺪ 4
/
;
however, we must define the new ⌫-action in place of on Bw X, Ž X q 1.y1 x= so that the following diagram is commutative:
Ž aŽ X q 1.
n
q ay1 Ž X q 1 .
yn
, yn
n
a Ž X q 1 . q Ž m y . ay1 Ž X q 1 . . ¬ a Ž X q 1 . ᎐ l ᎐ < < l Ž . y1 = GŽ GB . ªBw X, Ž X q 1. x
n
6
6
GŽ GBŽ . . ªBw X, Ž X q 1.y1 x= < < j j n yn n y1 Ž Ž a. Ž X q 1. q Ž a. Ž X q 1. ,¬ Ž a. y1 Ž X q 1. 6
6
Ž a .
y1
n
Ž X q 1. q Ž m y . Ž a. Ž X q 1.
yn
..
484
TOSHIAKI OHNO
Hence, we get H 1 Ž ⌫, G Ž GBŽ . . . , H 1 ⌫, B X , Ž X q 1 .
ž
y1 =
n
s aŽ X q 1. ¬ Ž aŽ X q 1. r Ž b Ž X q 1. n
m
/ n
. aŽ X q 1.
. rb Ž X q 1.
s aŽ X q 1. ¬ Ž a.
y1
m
aŽ X q 1.
n
s 1, a g B=, n g ⺪ 4
¬ b g B=, m g ⺪ 4 2n
s 1, a g B=, n g ⺪ 4
r 1r Ž b . b ¬ b g B= 4 s a ¬ Ž a . s a, a g B= 4 r Ž b . b ¬ b g B= 4 s A=rNormŽ B= . . Then we obtain H 1 Ž G , G . , A=rNormŽ B= . .
3.4. H 1 Ž⺗ a, A , G . THEOREM 3.4.1. H 1 Ž ⺗ a, A , G . , A=rNormŽ B= . . Proof. As in Theorem 3.1.3, we have H 1 Ž ⺗ a, A , G . , H 1 Ž ⌫, G Ž ⺗ a, B . . , where GŽ⺗ a, B . s Hom AŽ Aw , x, Bw X x.. Since m y 2 g B=, for an element of GŽ⺗ a, B ., there is a commutative diagram T ¬ a < < l l T q Ty1 , T q Ž m y .Ty1 gBw T, Ty1 x ªBw X x ᎐ ,
᎐
D p g Aw , x,
where Bw T, Ty1 x s Aw , x mA B. Hence we have Hom A Ž A w , x , B w X x . , Ž a q ay1 , a q Ž m y . ay1 . ¬ a g B= 4 , Hom B Ž B w T , Ty1 x , B w X x . , Bw X x s B= ;
=
TWISTED GROUP SCHEMES
485
however, we must define the new ⌫-action in place of on B= so that the following diagram is commutative:
Ž a q ay1 , a q Ž m y . ay1 . ¬ a < < l l GŽ⺗ a, B . ª B=
6
GŽ⺗ a, B . ªB=. < < j j
6
᎐
6
᎐
6
Ž Ž a. q Ž a. y1 , Ž a. y1 q Ž m y . Ž a. . ¬ Ž a. y1 . Hence, we get H 1 Ž ⌫, G Ž ⺗ a, B . . , H 1 Ž ⌫, B= . s a ¬ Ž a . a s 1, a g B= 4 r Ž b . rb ¬ b g B= 4 s a ¬ ar Ž a . s 1, a g B= 4 r 1r Ž b . b ¬ b g B= 4 s A=rNormŽ B= . . Then we obtain H 1 Ž ⺗ a, A , G . , A=rNormŽ B= . .
COROLLARY 3.4.2. H 1 Ž Spec A, G . , A=rNormŽ B= . .
4. EXTENSION 4.1. Ext 1 Ž G , ⺗ m, A . THEOREM 4.1.1.
Let
Ł ⺗ m s Spec Aw X , Y , Z x r Ž Z Ž X 2 q m XY q n 2 Y 2 . y 1 . ,
BrA
486
TOSHIAKI OHNO
where Ž/ 0. g A. Then Ł Br A⺗ m is the nontri¨ ial extension of G by ⺗ m, A : ␣
0 ª ⺗m , A ª

Ł ⺗ m ª G ª 0.
BrA
The homomorphisms ␣ and  are defined by the following ␣ * and  *:
␣ *: A w X , Y , Z x r Ž Z Ž X 2 q m XY q n 2 Y 2 . y 1 . ª A w T , Ty1 x
Ž X , Y , Z . ¬ Ž T , 0, Ty2 .  *: A w w, ¨ x r Ž Ž n¨ 2 y m¨ w q w 2 . y Ž m2 y 4 n . ¨ . ª A w X , Y , Z x r Ž Z Ž X 2 q m XY q n 2 Y 2 . y 1 .
Ž w, y¨ . ¬ Ž XYZ, Y 2 Z . Ž cf. w4, 3.1x.. Proof. We know the nontrivial extension 0 ª ⺗m , A ª
Ł ⺗ m ª G ª 0.
BrA
Then we get Ł Br A⺗ m by the Neron blow-up of ⺗ m, k in Ł Br A ⺗ m . ´ THEOREM 4.1.2. Ext 1 Ž G , ⺗ m , A . , ⺪r2⺪. Proof. In general, there is an exact sequence Žcf. w1, III, 6.2.5x. 0 ª H02 Ž G , ⺗ m , A . ª Ext 1 Ž G , ⺗ m , A . ª H01 Ž G , H 1 Ž ⺗ m , A . . ª H03 Ž G , ⺗ m , A . ª Ext 2 Ž G , ⺗ m , A . , where H 1 Ž⺗ m, A . denotes the presheaf on Sch r A defined by X ¬ H 1 Ž X, ⺗ m, A ., and H02 Ž G , ⺗ m, A ., etc., are the Hochschild cohomology. First, we compute H02 Ž G , ⺗ m, A . and H03 Ž G , ⺗ m, A ..
487
TWISTED GROUP SCHEMES
We use Hopf algebra: ␦2
Bw G Ž . x mB Bw G Ž . x ª Bw G Ž . x mB Bw G Ž . x mB Bw G Ž . x n j
Bw T, Ty1 x j Aw T, Ty1 x
p j
q o Aw G x mA Aw G x ª Aw G x mA Aw G x mA Aw G x. After tensoring B, the two cochain is C02 Ž GBŽ . , ⺗ m , B . s Hom B Ž B w T , Ty1 x , B w G Ž . x mB B w G Ž . x . m
n
, a Ž X q 1 . m Ž X q 1 . ¬ a g B=, m, n g ⺪ 4 . We compute ␦ 2 Ž aŽ X q 1. m m Ž X q 1. n .. Then Žwe may assume that the action of G on ⺗ m, A is trivial. m
␦ 2 Ž aŽ X q 1. m Ž X q 1.
n
.
m
s Ž a m Ž X q 1. m Ž X q 1.
n
.Ž aŽ X q 1.
m
m Ž X q 1.
m
m Ž X q 1.
Ž aŽ X q 1.
m
n
m Ž X q 1. m Ž X q 1.
n
.Ž aŽ X q 1.
n y1
.
m n
m Ž X q 1. m 1. s Ž X q 1.
ym
y1
n
m 1 m Ž X q 1. .
Hence the cocycle condition implies m s n s 0. Then we have Z02 Ž GBŽ . , ⺗ m , B . , B=. Hence Z02 Ž G , ⺗ m, A . ; A. The coboundary map is linear with respect to constants, so Z02 Ž G , ⺗ m, A . s B02 Ž G , ⺗ m, A .. Then we obtain H02 Ž G , ⺗ m , A . s 0. Similarly, the three cochain is C03 Ž GBŽ . , ⺗ m , B . s Hom B Ž B w T , Ty1 x , B w G Ž . x mB B w G Ž . x mB B w G Ž . x . l
m
n
, a Ž X q 1 . m Ž X q 1 . m Ž X q 1 . ¬ a g B=, l, m, n g ⺪ 4 .
488
TOSHIAKI OHNO
We compute ␦ 3 Ž aŽ X q 1. l m Ž X q 1. m m Ž X q 1. n ., then m
l
␦ 3 Ž aŽ X q 1. m Ž X q 1. m Ž X q 1.
n
.
m
l
s Ž a m Ž X q 1. m Ž X q 1. m Ž X q 1.
n
. n y1
m
Ž aŽ X q 1. l m Ž X q 1. l m Ž X q 1. m Ž X q 1. . m m n Ž aŽ X q 1. l m Ž X q 1. m Ž X q 1. m Ž X q 1. . m
n y1
n
Ž aŽ X q 1. l m Ž X q 1. m Ž X q 1. m Ž X q 1. . m n Ž aŽ X q 1. l m Ž X q 1. m Ž X q 1. m 1 . m
m
s a m Ž X q 1 . m Ž X q 1 . m 1. Hence the cocycle condition implies a s 1, m s 0. Then we have n
Z03 Ž GBŽ . , ⺗ m , B . , Ž X q 1 . m 1 m Ž X q 1 . ¬ l, n g ⺪ 4 . l
Next we compute the element of Z03 Ž G , ⺗ m, A . given by T ¬ Ž X q 1 . m 1 m Ž X q 1 . g A w G x mA A w G x mA A w G x . l
n
Then we can easily see that l s n s 0, and there is only a unit element: T ¬ 1. Hence we have ⺪ 30 Ž G , ⺗ m, A . s 0 and H03 Ž G , ⺗ m , A . s 0. Therefore Ext 1 Ž G , ⺗ m , A . , H01 Ž G , H 1 Ž ⺗ m , A . . . Next, we compute H01 Ž G , H 1 Ž⺗ m, A ... We have Žwe may assume that the action of G on H 1 Ž⺗ m, A . is trivial. H01 Ž G , H 1 Ž ⺗ m , A . . , Hom Ayg r Ž G , H 1 Ž ⺗ m , A . . . Moreover, we obtain Hom Ay g r Ž G , H 1 Ž ⺗ m , A . . ; MorA Ž G , H 1 Ž ⺗ m , A . . s H 1 Ž ⺗m , A . Ž G . s H 1 Ž G , ⺗m , A . , ⺪r2⺪.
TWISTED GROUP SCHEMES
489
Hence Ext 1 Ž G , ⺗ m , A . ; ⺪r2⺪. While we have the existence of nontrivial extension in Theorem 4.1.1, Ext 1 Ž G , ⺗ m , A . , ⺪r2⺪.
4.2. Ext 1 Ž G Ž ., G . THEOREM 4.2.1.
Let
Ł ⺗ m s Spec
A X , Y , Z, Ž Z q 1 .
y1
BrA
r Ž X 2 q mXY q nY 2 q Ž m2 y 4 n . Ž Z q 1 . . , where Ž/ 0. g A. Then Ł Br A ⺗ m is the nontri¨ ial extension of G Ž . by G: ␥
0 ª Gª
Ł
BrA
␦
⺗ m ª G Ž . ª 0.
The homomorphisms ␥ and ␦ are defined by the following ␥ * and ␦ *:
␥ *: A X , Y , Z, Ž Z q 1 .
y1
r Ž X 2 q mXY q nY 2 q Ž m2 y 4 n . Ž Z q 1 . . ª A w , x r Ž n 2 y m q 2 q m2 y 4 n .
Ž X , Y , Z . ¬ Ž , y , 0 . ␦ *: A X , Ž X q 1 .
y1
ª A X , Y , Z, Ž Z q 1 .
y1
r Ž X 2 q mXY q nY 2 q Ž m2 y 4 n . Ž Z q 1 . . X ¬ Z. Proof. We know the nontrivial extension 0ªGª
Ł ⺗ m ª ⺗ m , A ª 0.
BrA
490
TOSHIAKI OHNO
Then we get Ł Br A ⺗ m by the Neron blow-up of Gk in Ł Br A ⺗ m . ´ THEOREM 4.2.2. Ext 1 Ž G Ž . , G . , ⺪r2⺪. Proof. There is an exact sequence Žcf. w1, III, 6, 2.5x. 0 ª H02 Ž G Ž . , G . ª Ext 1 Ž G Ž . , G . ª H01 Ž G Ž . , H 1 Ž G . . ª H03 Ž G Ž . , G . ª Ext 2 Ž G Ž . , G . . First, we compute H02 Ž G Ž ., G . and H03 Ž G Ž ., G .. We use Hopf algebra: ␦2
Bw G Ž . x mB Bw G Ž . x ª Bw G Ž . x mB Bw G Ž . x mB Bw G Ž . x n j
Bw T, Ty1 x j Aw , x
p j
q o Aw G Ž . x mA Aw G Ž . x ª Aw G Ž . x mA Aw G Ž . x mA Aw G Ž . x. As in Theorem 4.1.2, we have H02 Ž G Ž . , G . s 0. Similarly, the three cocycle is n
Z03 Ž GBŽ . , ⺗ m , B . , Ž X q 1 . m 1 m Ž X q 1 . ¬ l, n g ⺪ 4 . l
We compute the element of ⺪ 30 Ž G Ž ., G . given by n
l
¬ Ž X q 1. m 1 m Ž X q 1. q Ž X q 1.
yl
m 1 m Ž X q 1.
yn
g A w G Ž . x mA A w G Ž . x mA A w G Ž . x l
¬ Ž X q 1. m 1 m Ž X q 1. q Ž m y . Ž X q 1.
yl
n
m 1 m Ž X q 1.
yn
g A w G Ž . x mA A w G Ž . x mA A w G Ž . x . Then the cocycle condition implies that l s n s 0, so there is only a unit element, ¬ 2, ¬ m. Hence we have Z03 Ž G Ž ., G . s 0 and H03 Ž G Ž . , G . s 0.
491
TWISTED GROUP SCHEMES
Therefore Ext 1 Ž G Ž . , G . , H01 Ž G Ž . , H 1 Ž G . . . Next, we compute H01 Ž G Ž ., H 1 Ž G ... We have H01 Ž G Ž . , H 1 Ž G . . , Hom Ayg r Ž G Ž . , H 1 Ž G . . . Moreover, we obtain Hom Ay g r Ž G Ž . , H 1 Ž G . . ; MorA Ž G Ž . , H 1 Ž G . . s H 1 Ž G . Ž G Ž . . s H 1 Ž G Ž . , G . , A=rNormŽ B= . = ⺪r2⺪. Hence Ext 1 Ž G Ž . , G . ; A=rNormŽ B= . = ⺪r2⺪. Let u: Spec A ª G Ž . be the neutral element of G Ž .. Then for any element in Hom Ay g r Ž G Ž ., H 1 Ž G .., ( u s 0 g H 1 Ž G .ŽSpec A.. On the other hand, we have Žwe remark that H 1 Ž G .ŽSpec A. s H 1 ŽSpec A, G . , A=rNormŽ B=. Žcf. 3.4.2.. Hom Ay g r Ž G Ž . , H 1 Ž G . .
;
H 1 Ž G . Ž G Ž . .
ª
H 1 Ž G . Ž Spec A .
< j
¬
< j
¬
< j (u
Then we get Ext 1 Ž G Ž . , G . , Hom Ay g r Ž G Ž . , H 1 Ž G . . ; ⺪r2⺪. While we have the existence of nontrivial extension in Theorem 4.2.1, Ext 1 Ž G Ž . , G . , ⺪r2⺪.
4.3. Ext 1 Ž G , G . THEOREM 4.3.1. Ext 1 Ž G , G . s 0. Proof. There is an exact sequence Žcf. w1, III, 6.2.5x. 0 ª H02 Ž G , G . ª Ext 1 Ž G , G . ª H01 Ž G , H 1 Ž G . . ª H03 Ž G , G . ª Ext 2 Ž G , G . . First, we compute H02 Ž G , G . and H03 Ž G , G ..
492
TOSHIAKI OHNO
We use Hopf algebra, ␦2
Bw G Ž . x mB Bw G Ž . x ª Bw G Ž . x mB Bw G Ž . x mB Bw G Ž . x n j
Bw T, Ty1 x j Aw , x
p j
q o Aw G x mA Aw G x ª Aw G x mA Aw G x mA Aw G x, where Aw G x s Aw ¨ , w x. As in Theorem 4.1.2, we have H02 Ž G , G . s 0. Similarly, the three cocycle is n
Z03 Ž GBŽ . , ⺗ m , B . , Ž X q 1 . m 1 m Ž X q 1 . ¬ l, n g ⺪ 4 . l
We compute the element of Z03 Ž G , G . given by n
l
¬ Ž X q 1. m 1 m Ž X q 1. q Ž X q 1.
yl
m 1 m Ž X q 1.
yn
g A w G x mA A w G x mA A w G x l
¬ Ž X q 1. m 1 m Ž X q 1. q Ž m y . Ž X q 1.
yl
n
m 1 m Ž X q 1.
yn
g A w G x mA A w G x mA A w G x .
Then the cocycle condition implies that l s n s 0, so there is only a unit element, ¬ 2, ¬ m. Hence we have Z03 Ž G , G . s 0 and H03 Ž G , G . s 0. Therefore Ext 1 Ž G , G . , H01 Ž G , H 1 Ž G . . . Next, we compute H01 Ž G , H 1 Ž G ... We have H01 Ž G , H 1 Ž G . . , Hom Ayg r Ž G , H 1 Ž G . . . Moreover, we obtain Hom Ay g r Ž G , H 1 Ž G . . ; MorA Ž G , H 1 Ž G . . s H 1 Ž G. Ž G . s H 1 Ž G , G. , A=rNormŽ B= . , H 1 Ž G . Ž Spec A . .
493
TWISTED GROUP SCHEMES
Let u: Spec A ª G be the neutral element of G . Then for any element in Hom Ay g r Ž G , H 1 Ž G .., ( u s 0 g H 1 Ž G .ŽSpec A.. On the other hand, we have Hom Ay g r Ž G , H 1 Ž G .. ; H 1 Ž G .Ž G . , H 1 Ž G .ŽSpec A.. < j
< j
¬
< j (u
¬
Hence, we get Ext 1 Ž G , G . , Hom Ay g r Ž G , H 1 Ž G . . s 0.
4.4. Ext 1 Ž⺗ a, A , G . THEOREM 4.4.1. Suppose char Ž K . s 0 and char Ž k . s p ) 0. Then an element of Ext 1 Ž⺗ a, A , G . corresponds to F Ž a1 , a p , a p 2 , . . . ; X . s 1 q a1 X q q ap X p q q ⭈⭈⭈ q q
a12 2!
X 2 q ⭈⭈⭈ q
a1 a p pq1
Ž p y 1. !
X pq1 q
Ž p q 2. Ž p q 1.
Ž 2 p y 1 . ⭈⭈⭈ Ž p q 1 .
Ž 2 p . ⭈⭈⭈ Ž p q 1 .
X
2p
q
X py1
a12 a p
a1py 1 a p p!a2p
q ⭈⭈⭈ q
a1py 1
X pq2
X 2 py1 p!a1 a2p
Ž 2 p q 1 . ⭈⭈⭈ Ž p q 1 .
p!a1py 1 a2p
Ž 3 p y 1 . ⭈⭈⭈ Ž p q 1 .
X 2 pq1
X 3 py1
2
2
Ž p! . a3p Ž p! . a1 a3p q X 3p q X 3 pq1 Ž 3 p . ⭈⭈⭈ Ž p q 1 . Ž 3 p q 1 . ⭈⭈⭈ Ž p q 1 . 2
q ⭈⭈⭈ q
Ž p! . a1py1a3p X 4 py1 Ž 4 p y 1 . ⭈⭈⭈ Ž p q 1 . ⭈⭈⭈
494
TOSHIAKI OHNO
a1 a p 2
2
q ap2 X p q
p2 q 1
Xp
2
q1
q ⭈⭈⭈ ,
where pa p s
a1p
Ž p y 1. !
,
p2ap2 s
Ž p! .
py 2
a1p a ppy1
Ž p 2 y 1 . ⭈⭈⭈ Ž p q 1.
,...,
a1 , a p , a p 2 , . . . are nilpotents in BrŽ ., and, moreo¨ er,
Ž 2 y m . a1 , Ž 2 y m . a p , Ž 2 y m . a p 2 , . . . g Ar Ž . . Proof. We obtain the exact sequence 0 ª G ª G ª i#GA rŽ . ª 0 in 2.3, so there is a long exact sequence 0 ª Hom Ay g r Ž ⺗ a, A , G . ª Hom Ayg r Ž ⺗ a, A , G . r
ª Hom Ay g r Ž ⺗ a, A , i#GA rŽ . . ª Ext 1 Ž ⺗ a, A , G . ª Ext 1 Ž ⺗ a, A , G . ª ⭈⭈⭈ . We know that Ext 1 Ž⺗ a, A , G . s 0, so Ext 1 Ž ⺗ a, A , G . , Hom Ayg r Ž ⺗ a, A , i#GA rŽ . . rr Ž Hom Ayg r Ž ⺗ a, A , G . . . Since Hom Ay g r Ž⺗ a, A , G . s 0, Ext 1 Ž ⺗ a, A , G . , Hom Ayg r Ž ⺗ a, A , i#GA rŽ . . s Hom A rŽ .yH o p f Ž Ar Ž . w , x , A w X x mA Ar Ž . . , Hom A rŽ .yH o p f Ž Ar Ž . w , x , Ar Ž . w X x . , where Hom A rŽ .yH o p f Ž ArŽ .w , x, ArŽ .w X x. denotes the set of ArŽ .Hopf algebra homomorphisms. After tensoring B, there is a commutative diagram, T ¬ fŽ X . < < l l T q Ty1 , T q Ž m y .Ty1 gBrŽ .w T, Ty1 x ª BrŽ .w X x ᎐ ,
᎐
D D Ž .w x Ž g Ar , ªAr .w X x.
495
TWISTED GROUP SCHEMES
Hence we have Hom A rŽ .- H o p f Ž Ar Ž . w , x , Ar Ž . w X x . =
, f Ž X . ¬ f Ž X . g Br Ž . w X x , fŽ X. q fŽ X.
y1
,fŽ X. q Žm y . fŽ X.
y1
g Ar Ž . w X x ,
fŽ X m 1 q 1 m X . s fŽ X . m fŽ X .4. Let f Ž X . s Ý iG 0 a i X i g BrŽ .w X x=. Then the condition f Ž X m 1 q 1 m X . s f Ž X . m f Ž X . implies that f Ž X . has the following style: f Ž X . s 1 q a1 X q
a12 2!
a1 a p
q ap X p q q q q
X 2 q ⭈⭈⭈ q
pq1
a1py 1
X pq1 q
a1py 1 a p
Ž 2 p . ⭈⭈⭈ Ž p q 1 .
a12 a p
Ž p q 2. Ž p q 1.
p!a1 a2p
X2p q
p!a1py 1a2p
Ž 3 p y 1 . ⭈⭈⭈ Ž p q 1 .
X pq2 q ⭈⭈⭈
X 2 py1
Ž 2 p y 1 . ⭈⭈⭈ Ž p q 1 . p!a2p
X py1
Ž p y 1. !
Ž 2 p q 1 . ⭈⭈⭈ Ž p q 1 .
X 2 pq1 q ⭈⭈⭈
X 3 py1
2
2
Ž p! . a3p Ž p! . a1 a3p 3p q X q X 3 pq1 q ⭈⭈⭈ Ž 3 p . ⭈⭈⭈ Ž p q 1 . Ž 3 p q 1 . ⭈⭈⭈ Ž p q 1 . 2
Ž p! . a1py1a3p q X 4 py1 Ž 4 p y 1 . ⭈⭈⭈ Ž p q 1 . ⭈⭈⭈ a1 a p 2
2
q ap2 X p q
p q1 2
Xp
2
q1
q ⭈⭈⭈ ,
where pa p s
a1p
Ž p y 1. !
,
p2ap2 s
Ž p! .
py 2
a1p a ppy1
Ž p 2 y 1 . ⭈⭈⭈ Ž p q 1.
,...,
496
TOSHIAKI OHNO
a1 , a p , a p 2 , . . . are nilpotent in BrŽ .. Hence
fŽ X. q fŽ X.
y1
a12
s2q2 q2 q2 q2
2!
X q ⭈⭈⭈ q2
a1py 1
2
a1 a p pq1
Ž p y 1. !
X py1
X pq1 q ⭈⭈⭈ p!a2p
Ž 2 p . ⭈⭈⭈ Ž p q 1 .
X 2 p q ⭈⭈⭈
p!a1py 1 a2p
Ž 3 p y 1 . ⭈⭈⭈ Ž p q 1 .
X 3 py1
2
Ž p! . a1 a3p q2 X 3 pq1 q ⭈⭈⭈ Ž 3 p q 1 . ⭈⭈⭈ Ž p q 1 . ⭈⭈⭈ q2
a1 a p 2 p2 q 1
fŽ X. q Žm y . fŽ X.
Xp
qŽ 2 y m. ap X p q m
qŽ 2 y m. qm
qm
a12
q ⭈⭈⭈ ,
2!
X q ⭈⭈⭈ qm
a1 a p pq1
Ž p q 2. Ž p q 1.
Ž p y 1. !
X pq1 X pq2 q ⭈⭈⭈
a1py 1 a p
Ž 2 p y 1 . ⭈⭈⭈ Ž p q 1 .
Ž 2 p . ⭈⭈⭈ Ž p q 1 .
a1py 1
2
a12 a p
p!a2p
qŽ 2 y m.
q1
y1
s m q Ž 2 y m . a1 X q m
qŽ 2 y m.
2
X 2 py1
X2p
p!a1 a2p
Ž 2 p q 1 . ⭈⭈⭈ Ž p q 1 .
p!a1py 1a2p
Ž 3 p y 1 . ⭈⭈⭈ Ž p q 1 .
X 3 py1
X 2 pq1 q ⭈⭈⭈
X py1
497
TWISTED GROUP SCHEMES 2
Ž p! . a3p qŽ 2 y m. X 3p Ž 3 p . ⭈⭈⭈ Ž p q 1 . 2
qm
Ž p! . a1 a3p X 3 pq1 q ⭈⭈⭈ Ž 3 p q 1 . ⭈⭈⭈ Ž p q 1 . 2
Ž p! . a1py1a3p qŽ 2 y m. X 4 py1 q ⭈⭈⭈ Ž 4 p y 1 . ⭈⭈⭈ Ž p q 1 . ⭈⭈⭈ 2
qŽ 2 y m. ap2 X p q m
a1 a p 2 p2 q 1
Xp
2
q1
q ⭈⭈⭈ .
Then we obtain a12 , a1 a p , a2p ,a1 a3p , a4p , a1 a5p , . . . , a1 a ppy 2 , a ppy 1 , a1 a p 2 , a2p 2 , a1 a3p 2 , . . .g ArŽ . ,
Ž 2 y m . a1 , Ž 2 y m . a p , Ž 2 y m . a3p , Ž 2 y m . a5p , . . . , Ž 2 y m . a ppy 2 , Ž 2 y m . a p 2 , Ž 2 y m . a3p 2 , . . . g Ar Ž . . These conditions are equivalent to
Ž 2 y m . a1 , Ž 2 y m . a p , Ž 2 y m . a p 2 , . . . g Ar Ž . .
ACKNOWLEDGMENTS Finally, I am grateful to Prof. T. Sekiguchi and Prof. N. Suwa for their many suggestions.
REFERENCES 1. M. Demazure and P. Gabriel, ‘‘Groupes Algebriques,’’ Tome I, North-Holland, Amster´ dam, 1970. ´ 2. J. S. Milne, ‘‘Etale Cohomology,’’ Princeton Univ. Press, Princeton, NJ, 1980. 3. J.-P. Serre, ‘‘Local Fields,’’ Graduate Texts in Mathematics 67, Springer-Verlag, BerlinrNew York, 1979. 4. W. C. Waterhouse and B. Weisfeiler, One-dimensional affine group schemes, J. Algebra 66 Ž1980., 550᎐568. 5. B. Weisfeiler, On a case of extensions of group schemes, Trans. Amer. Math. Soc. 248, No. 1 Ž1979., 171᎐189.
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TOSHIAKI OHNO
6. T. Sekiguchi, On the deformations of Witt groups to tori II, J. Algebra 138 Ž1991., 273᎐297. 7. T. Sekiguchi and N. Suwa, A case of extensions of group schemes over a discrete valuation ring, Tsukuba J. Math. 14, No. 2 Ž1990., 459᎐487. 8. T. Sekiguchi and N. Suwa, Some cases of extensions of group schemes over a discrete valuation ring I, J. Fac. Sci. Uni¨ . Tokyo Sect. IA Math. 38 Ž1991., 1᎐45. 9. T. Sekiguchi and N. Suwa, Some cases of extensions of group schemes over a discrete valuation ring II, Bull. Fac. Sci. Engrg. Chuo Uni¨ . Ser. I Math. 32 Ž1989., 17᎐35.