5. VERSA L OBJECT S
11
II.1.1]; the y ar e denote d b y H
l(k,C),
an d thei r direc t su m i s writte n H(k,C).
These group s ar e functoria l i n /c , i.e., the y defin e functor s
Fields/^— » Abelian Groups .
These will be the functor s "JT ' tha t w e will consider mos t o f the time, an d fo r suc h
H we write
Inv2(^4,
G) or Inv(^4, G), instead of Inv^, H); similarly , if A = Torsorsc ,
where G is as in 3.1, we use the notatio n Inv
2(G,
G) o r Inv(G, G). Th e element s of
Inv(.A,G) an d Inv(G,G ) ar e calle d cohomological invariants. Whe n i t i s necessar y
to mention ko explicitly, we write ko as a subscript, a s in Invfco
(G,C).d
Mos t o f the
time we will assume that G is finite and o f order no t divisibl e by the characteristic ,
the principa l exampl e bein g G = Z/2Z .
EXAMPLE
4.2 . Le t G b e a finite grou p an d le t G b e a G-modul e wit h trivia l
action. Writ e H(G, C) fo r th e direc t su m o f the cohomolog y group s H l(G, G) . A n
element x o f i/
1(A:,
G) i s represented b y a continuous homomorphis m p
x
: Tk » G ;
such homomorphism s represen t th e sam e clas s x i f and onl y i f they ar e conjugate .
The ma p (p
x
give s a ma p cp x: H(G,C) —+ H(k,C) whic h depend s onl y o n x (bu t
not o n the choic e oiip x). I f u i s a given element o f H(G, G) , define a u{x) a s (f x(u).
One get s i n thi s wa y a cohomologica l invarian t a
u
whic h belong s t o Inv(G , G) i n
the notatio n o f 4.1 above .
For a concrete exampl e o f this constructio n whe n G = S n, se e 25.1.
Note tha t eve n whe n u i s no t 0 , th e correspondin g invarian t a
u
ma y b e zer o
for al l possible choice s o f ko. Suc h a class u i s called negligible, see §26.
4.3. Anothe r exampl e o f a functo r H i s H(k) = W(k), th e Wit t rin g o f
nondegenerate quadrati c form s o n k modul o hyperboli c forms , se e Chapte r VIII .
DEFINITION
4.4 . Ther e i s a natural embeddin g H(k
0
) —»Inv/
Co
(A,i7); namely ,
if h belong s t o H(ko), w e define th e invarian t ah by settin g
Qh(x) image o f h i n H{k)
for ever y x G A(k). Suc h a n invarian t i s called constant.
4.5. Suppos e we have fixed a base point fo r A. Fo r example, if A(*) = i^ 1(*, G)
as in §3, the base point i s the unit element "1". W e say that a is normalized (relativ e
to th e choic e o f bas e poin t fo r A) i f a vanishes o n th e bas e point . Ever y invarian t
can b e writte n i n a unique wa y a s (constant ) + (normalized) .
5. Versa l object s
Let G be a smoot h affin e algebrai c grou p schem e define d ove r ko.
DEFINITION
5.1. A versal G-torsor i s a G-torso r P ove r a finitely generate d
extension K o f & o such tha t ther e exist s a smooth , irreducibl e variet y X ove r ko
with functio n field K an d a G-torso r Q X, wit h bas e X , wit h th e followin g tw o
properties:
(1) Th e fiber o f Q a t th e generi c poin t o f X i s P.
Our genera l polic y i s to us e shortene d notatio n a s muc h a s possible .
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