CHAPTER VI I
Cohomological invariant s o f etal e algebra s
24. Propertie s o f cohomologica l invariant s o f Et
n
As before , w e fix our functo r H t o b e k i— H(k,C), wher e C i s a finite Tk
0
-
module o f order prim e t o th e characteristic .
24.1.
UNRAMIFIE D ETAL E ALGEBRAS .
Le t K be & field complet e wit h respec t
to a discret e valuatio n v. W e kee p th e notatio n o f § 7 and §11 : R is the valuatio n
ring o f v (i.e. , the se t o f x G K wit h v(x) 0) ; k i s the residu e field.
Let E b e a n etal e algebr a o f ran k n ove r K, an d le t ifE : IV— Sn b e th e
corresponding homomorphis m (well-define d u p t o conjugation) , cf . 3.2 . I f we view
(fE a s a cocycle, it s clas s i n H
X(K,
S n) = Et n(if) wil l be denote d b y (E).
PROPOSITION
24.2 . The following properties are equivalent:
(1)
PE(IK)
= 1; where
IK
is the inertia subgroup
OJTK,
cf 7.1.
(2) E is a product of fields which are unramified extensions of K.
(3) The class (E) in H1^, S
n
) belongs to the image of the canonical injection
UH^SJ^H^KiSn), cf 11.1.
(4) There is a K-basis (ei,.. . ,en) of E such that the matrix (Tr(e^.ej) ) be-
longs to GL n(R).
PROOF. Th e equivalence s (1) = (2) ^ (3 ) ^= (4 ) ar e clear .
DEFINITION
24.3 . A n etal e algebr a E i s calle d unramified i f i t ha s propertie s
(1) throug h (4 ) o f 24.2.
EXAMPLE
24.4 . I f char(fc) ^ 2 , a quadratic algebra
K[X]/{X2-a)
i s unramifie d
if and onl y i f v(a) = 0 (mo d 2) .
Let a G Inv(Etn, C) b e a cohomological invariant , ove r a ground field ko which
is contained i n R (an d henc e als o i n k an d i n K).
PROPOSITION
24.5 . If E is unramified, then the residue of a{E) is 0.
PROOF.
Thi s follow s fro m 11.7 .
24.6. A VERSA L TORSOR . Le t K = A:
0
(ci,... ,cn) wher e th e Q ar e indeter -
minates, an d se t £
g e n
= K[t}/{t
n
+ cit
711+
' + c n) a s i n 5.6.2 . Fo r k infinite ,
every etale /c-algebr a ca n b e generated b y one element [Bou59 , V.7.7, Prop. 7 ] and
it i s easily see n tha t th e generator s ar e Zariski-dense , s o that E
gen
G Etn(K) cor -
responds t o a versal torso r i n H
l{K,
S n). Th e fac t tha t thi s i s versal ma y als o b e
viewed a s a specia l cas e o f 5.4 , applie d t o th e natura l embeddin g o f S
n
i n GL
n
.
(This wil l b e use d i n part (2 ) o f the proo f o f Prop. 24. 6 below. )
Let D b e an irreducible divisor o f
Affn
= Spe c ko [c\,..., c n]. Suc h a divisor de-
fines a discrete valuation vrj o f K =
/CQ(CI ,
..., c
n
); we write Ko fo r th e completio n
55
http://dx.doi.org/10.1090/ulect/028/09
Previous Page Next Page