The ma p TK IV i i s an isomorphism, henc e the lemma.
Consider no w a subfield k\ o f k containin g k$. W e say that k\ i s K-liftable i f
there i s a /cg-morphism k\ R suc h tha t it s composite wit h R —• k is equal to the
injection k\ k. I n other words , it is a lifting (i n the usual sense) o f k\ i n R whic h
extends th e given liftin g k
LEMMA 11.5 . Suppose k\ is K-liftable, and x G i/1(A:,G?) is the image of an
element xi of H1(k\^G) (i.e. , x "comes fro m &i") . T/ie n (11.2 ) holds for x.
PROOF. Choos e a /ci-G-torso r 7 \ whos e clas s i n H l(k\,G) i s x\. A liftin g
k\ ^ R transforms T\ by base change into an iZ-C-torsor. Thi s is the TR mentioned
at th e beginning o f the section. Th e embedding R » K transform s TR into TK-
The clas s of TK is i(x). Bu t it is also the image of x\ vi a k\ if. Henc e a^(z(x) )
is equa l t o ji(ak 1{xi)), wher e a/
i s relative t o k\ an d ji come s fro m th e liftin g
fei - i? -• if .
The diagra m
H(kuC) —^— H(K,C)
i i i
#(/c,C) —^— H(K,C)
commutes, wher e the left-hand arro w come s fro m th e embedding k\ -+ k. Becaus e
a is functorial, w e have ji(ak
(xi)) = j(a/e(x)), whic h prove s the lemma.
LEMMA 11.6 . Let k\ be a finitely generated extension of k$ contained in k.
Then there exists a finite purely inseparable extension K\ of K such that k\ is
K\ -liftable.
(The statemen t o f the lemma make s sens e becaus e th e discrete valuatio n o f K
extends i n a unique wa y to a discrete valuatio n o f K\, whic h i s also complete. )
Case I: k\ is a finite purely inseparable extension of k$. Th e two
extensions ki/ko an d K/ko ca n be embedded i n some extensio n K 2. Defin e K\ t o
be the subfield o f K2 generated b y k\ an d K. I t has the required properties .
Case II: k\ is a finite extension of ko. Le t k' be the largest separabl e extensio n
of& o contained i n k\. Th e extension k\/k' i s purel y inseparable . B y Hensel' s
Lemma, k f i s X-liftable. On e then applie s Cas e I, with ko replaced b y k'.
Case III: The general case. Le t £1, ..., td be a transcendence basi s o f fci/fco,
and le t A?o(t ) b e the extensio n o f ko generated b y ti, ... , t&. Th e field k\ i s a
finite extensio n o f &o(t). Th e field &o(t) is if-liftable, a s one sees b y choosing R-
representatives o f the ti. On e then applie s Cas e II, with& o replace d b y ko(t).
END O F PROOF O F THEORE M 11.1Le . t x be an element of H^ik.G). B y 11.3,
we ma y choose k\ finitely generate d ove r fc0 such tha t x come s fro m k\. B y 11.5
and 11. 6 we may choose K\jK a s in 11.6 such tha t (11.2 ) hold s afte r replacin g K
by K\. Thi s mean s tha t th e two sides of (11.2) hav e the same imag e in H(Ki,C).
By the injectivity propert y 11.4 , these two sides are equal.
11.7. Suppos e agai n tha t w e have a discret e valuatio n rin g R containin g a
field /CQ , with quotien t field K an d residue field £;, and corresponding to a valuation
v. Tha t is , R i s as above excep t tha t w e do not assume tha t R i s complete. An y
i?-G-torsor TR gives torsors TK and T& .
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