CHAPTER I
The notio n o f "invariant "
1. Definition s
We fix a ground field fco, and conside r th e categor y Fields// ^ o f field extensions
k o f fco and tw o functor s
A: Fields// ^ Set s
and
H: Fields/
ko
Abelian Group s
In th e applications , A wil l be , fo r instance , th e se t o f isomorphis m classe s o f etal e
fc-algebras of a given rank, an d H wil l be the mo d 2 Galois cohomology o f the field
k. (Fo r mor e examples , se e Section s 2 and 4. )
DEFINITION 1.1 . A n H-invariant of A i s a morphism o f functor s
a: A- H.
Here, w e vie w i f a s a functo r wit h value s i n Sets . Hence , givin g a : A H
means giving, for ever y k G Fields/^, a map a& : E - a{E) o f A(k) int o H(k) suc h
that, i f (j) : k k' i s a morphism i n Fields/ fco, th e diagra m
A{k) —^-^ H(k)
i i
A(fc') -^-^ H(k')
is commutative .
Our ai m wil l be t o determin e explicitl y i n som e case s (e.g. , th e on e mentione d
above) th e grou p Inv(A , H) o f al l suc h invariants . I f i t i s necessar y t o emphasiz e
the rol e of the bas e field ko, we will write Invfc
0
(A, H).
REMARK
1.2. I t i s well-known tha t considerin g th e "set " o f all field extensions
of ko may lead to logical difficulties. Bu t i n all cases we will consider, th e functo r A
commutes wit h direc t limits , s o that a n invarian t a is determined b y its restrictio n
to the category o f finitely generated extension s of ko (which one can further assum e
to be contained i n a large enough algebraically closed extension of fco). Moreover, i n
most cases, there is a finitely generated extension k\ o f fco an d an element E i n A(k\)
such tha t th e knowledg e o f a on E determine s a, i.e. , th e ma p Inv(A , H) H(k\)
given b y a ^ a (E) i s injective, se e Section 12.3.
2. Example s o f functor s A
In the examples below, we only give the value of the functor A fo r a n extensio n
k of ko. Th e arrow A(k) A{k') associate d wit h a fco-embedding k k' i s obvious
("extension o f scalars") .
7
http://dx.doi.org/10.1090/ulect/028/03
Previous Page Next Page