Groupe de Brauer: Séminaire, Les Plans-sur-Bex, Suisse 1980 by J. -P. Tignol (auth.), Michel Kervaire, Manuel Ojanguren

By J. -P. Tignol (auth.), Michel Kervaire, Manuel Ojanguren (eds.)

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Is KF' 3 is a c o m p l e x of free free K G m o d u l e s . free for all j. Hence x(M) 43 2. I n d u c t i o n Let seen H be a s u b g r o u p that H is VP too. , Then 4. The Let H' Euler x(M) Let Qo o N xF(M) poly-~ We use (F 1: F') with K over i = to u n d e r s t a n d (F/F' : H/H'). KH module, Obviously it is i s o m o r p h i c is c o m p a t i b l e subgroup the kF' with of F and let H' = H n F'. (K,M) of N over kF, kH. Then so c e r t a i n l y over kF' ~ K[F/F']®kHQ . ring K[F/F'] as an H module.

M ~ 0 b e a f i n i t e can compute t h e r a n k of a free kF' free module resolution that k over F by dimD,D'®kF,F. flatness Z(-l)3dimD,j (D'®kF,Fj) = d i m D , (D'®kF,M) . Thus 1 x(M) = ~ d i m D ' Identifying Clearly (D'®kp'M) = 1 U as a m i n i m a l K(F) • I = U so, dimD' (uS) left non-zero following Corollar[ are easy i. T h e G o l d i e denominator of Euler ideal let I = k F n U . d. 1 = -r consequences. rank is t h e characteristics, least common in r e d u c e d multiple form, of t h e of finitely 42 generated kF m o d u l e s .

3): ^ I ~ ,) ®'M -X 1 -~ X(G) ,,, eF @'M .... × X(G)® F ~ ®'Z/2Z--~ 0 X(G)---~ 1 ..... 4) ^ , M ) 1 "-~'~ ®'Mg_ - - ~ g ] ~ de F, qui de M, ainsi que de M et de M g pour X • X(G) X l'extension M/F n'est pas ramifi6e. Pour achever la d6monstration, droite de ce diagramme scind~es, O ' B r 2 ( M g) g , 1o, ~ Br2 (M) F par le choix d'une uniformisante est aussi une uniformisante et g • G, puisque > il suffit de prouver que la colonne est une suite exacte; la suite exacte d'homologie ¢omme les lignes de sont donne alors les isomorphismes cherch6s.

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