8 I . THE NOTION O F "INVARIANT "
E X A M P L E 2. 1 (Etal e algebra s [Bou59 , V.6]) . Le t n b e a positiv e integer . W e
define th e functor A by
k i— Et
n
(k) = {isomorphis m classe s o f etale algebra s ove r k o f rank n) .
Recall tha t a n etale k-algebra E o f ran k n i s a commutativ e fc-algebra suc h tha t
E = Yii ki fo r hi finit e separabl e fiel d extension s o f k an d XU^ i : k] n. A n
equivalent definitio n i s that E i s commutative, [E : k] = n, and the /c-bilinear for m
defined o n E b y (x, y) ^ Tr^(xy ) i s nondegenerate .
The tw o mos t importan t case s ar e the one where E i s a field, an d the one where
E i s split, i.e. , isomorphic t o k x x k.
E X A M P L E 2. 2 (G-Galoi s algebra s [BS94 , 1.3], [KMRT98 , 18.15]) . Le t G b e
a finit e group . A G-Galois algebra ove r k i s an etal e fc-algebra E wit h a n actio n
of G suc h tha t G act s simpl y transitivel y o n Hom(i£, ks), wher e k
s
i s a separabl e
closure o f k. Suc h a n algebr a i s determine d u p t o isomorphis m b y a continuou s
homomorphism cp: Gal(fcs/fc)— » G, and two such homomorphism s giv e isomorphi c
G-Galois algebra s i f and only i f they ar e conjugate b y an element o f G. W e set :
G-Gal(k) = {isomorphis m classe s o f G-Galois algebra s ove r k} .
E X A M P L E 2. 3 (Central simpl e algebras) . Fo r n 1 we se t
Cent. Simpl
n
(&;) = {isomorphis m classe s o f central simpl e ^-algebra s o f rank n 2 } .
In th e next fou r examples , w e assume char(/co ) 7 ^ 2 .
E X A M P L E 2. 4 (Quadrati c forms) . Fo r n 1, w e se t
r\ A /i. \ _ f isomorphism classe s o f nondegenerat e quadrati c form s 1
n love r k o f rank n )
We writ e ( a i , . . . , a
n
) fo r the diagonal quadrati c for m define d b y the eVs.
E X A M P L E 2. 5 (Quadrati c form s wit h specifie d discriminant) . Fo r a fixe d
5 e kl/k^ 2 an d n 1, set
^ i /
7
x fisomorphis m classe s o f nondegenerat e quadrati c form s 1
Quad
n s
(k) =
7
r ,
A
A- . - x f
Lover / c of rank n an d discriminant d J
Recall tha t th e discriminant a d(q) o f a quadrati c for m q = ( a i , . . . , a
n
) i s the
product a i an viewe d a s an element o f /c*//c* 2.
E X A M P L E 2. 6 (Hermitia n forms) . Le t k\ = fco(v^) b e a quadrati c extensio n
of &o - Th e functor Herm
n?
/Cl/fco (abbreviate d Herm
n
) i s defined a s follows: I f k is
an extensio n o f /co , Herm
n
(/c) i s the set o f isomorphis m classe s o f nondegenerat e
hermitian form s o f rank n ove r th e quadratic /c-algebr a k ®k0 &i = k[X]/(X 2 (5) ,
cf. e.g . [Knus91, §1.3]. Whe n S is not a squar e i n k, w e have k 0/^ k\ = k(y/S)\
this i s th e usua l settin g o f hermitia n theory . Whe n 5 i s a squar e i n /c , we hav e
k®k0 ki kxk, an d one find s tha t al l the hermitian form s o f rank n are isomorphic:
the se t Herm
n
(/c) ha s only on e element.
For a i , ... , a
n
G /c*, we write ( a i , . . . , c\
n
)H fo r th e correspondin g diagona l
hermitian form .
a
In [KMRT98] , thi s is called the "determinant " o f q, and the (signed) discriminan t i s defined
to b e (-l)
n
(
n
-
1)/ 2
d(g) .
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