# Cohomologie cristalline des schemas de caracteristique p O by P. Berthelot By P. Berthelot

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Pi 2 h). 1. Let g be an affine algebra of type X},:') where either r = r v = 1 or r > 1, let A E P~ and let u E N. ,o, jl E (b) If k E Q is a principal admissible rational number with the denominator u and 1 are such that y(AO - jlo - (u -l)Ao) - A E Q, then AE P:,Y' P:,t bA0 ).. o,/lo. PROOF: a) (resp. 1 (resp. 2. 1b was obtained in [12, Proposition 3]. 1. Let g be an affine algebra of type X N(r) and let A E P~- hV 29 ' ,J1 E P~ - hV (resp. E p~p'-h). One has the following asymptotics as T 1 0: 'P)....

We will construct inductively the classes of complements of MINj in GINj . It suffices to describe a typical step from GINj to GINj +!. , we may assume that ,m, F. CELLER ET AL. 64 N = N j is elementary abelian and that the results are known for G / N j . 1 Lemma: Let G be a group, let M

This in turn implies that the squares of X22, X23, X32, X33, X34 are conjugate to X12, X13, X31, X3I! X31, respectively, hence O(X12) = O(X23) = 4, O(X32) = O(X33) = O(X34) = 6. Thus the element orders and also the second powermap are found. The other powermaps are trivially obtained. In the above example it was very easy to see that a character (Xfl) was induced from a character of one of the inertia subgroups (1'3). PAHLINGS 54 Cl( i). These are defined by Now, if n X= 2: ('IjJ(m»)G m=l with 'IjJ(m) = 2:a~m)cpr), cp~m) E Irr(Tm' Am) k then (X, C(m,e») = 2: a~m)'ljJlm)(yfe' N) ('ljJlm) E Irr(Tm)) k Thus using these scalar products, which can be computed in CAS using the cldecompose command, one can decompose a character into its pieces belonging to the various inertia groups.