CHAPTER I V
Specialization propertie s o f cohomologica l
invariants
11. Compatibilit y wit h goo d reductio n
Let G b e a smoot h algebrai c grou p ove r /co , and le t G b e a finite T
ko
-modul e
of order no t divisibl e b y the characteristic.
a
We conside r a n extensio n K o f /co , wit h a discret e valuatio n v, w e cal l it s
valuation rin g R an d it s residu e field k. W e assum e tha t R contain s /co , s o tha t
R an d K ar e /c 0-algebras; k i s als o a /co-algebr a vi a th e compositio n fco R * k.
Until 11.7, we assume als o that K i s complete. (Thi s assumptio n coul d b e replace d
by "henselian". )
Let Tk b e a /c-G-torsor . I t i s well known tha t ther e i s an i?-G-torso r TR whic h
has "specia l fiber" T
k
an d tha t TR i s unique u p to isomorphism, cf . [SGA3 , p. 401,
Prop. 8.1]. Sinc e an y i?-G-torso r TR give s b y bas e extensio n a i^-G-torsor , thi s
defines a ma p i: H l{k,G) - ^(K.G).
The followin g theore m an d it s proo f ar e du e t o M . Ros t (lette r t o Serr e date d
2 May 1994). W e reproduce the m her e wit h hi s permission :
COMPATIBILITY THEORE M
11.1For . a e lnv ko(G,C), the diagram
Hl{k,G) l - ^ H^K.G)
ak
\a
K
H(k,C) —^— H{K,C)
commutes.
The ma p j i s the natura l embeddin g define d i n 7. 6 an d 7.7 .
PROOF O F THEORE M
11.1W . e have t o sho w tha t th e equatio n
(11.2) j(a
k
(x)) = a
K
(i(x))
holds fo r al l xeH 1(k,G).
We need a fe w lemmas :
LEMMA
11.3 . iJ
1(/c,G)
is the inductive limit of the if
1(fci,G)
where k\ runs
through the finitely generated extensions of ko contained in k.
This i s clear .
LEMMA 11.4 . If'Ki/K is purely inseparable, the map H(K,C) H(Ki,C) is
bijective.
a As Ros t ha s pointe d ou t t o us , n o hypothesi s o n th e T ^ -modul e C i s neede d unti l 11.7.
29
http://dx.doi.org/10.1090/ulect/028/06
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