By Serge Lang

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**Extra resources for Introduction to algebraic geometry**

**Example text**

G . When S = Spec K , K = p-adic field, it is more convenient, according to Raynand [R2], to identify 1-motives which are quasi-isomorphic in the rigid analytic category; for instance, if A is isomorphic to the rigid quotient G / M , we consider A (or [0 ) A] ) and [M ) G] as two incarnations of the same rigid 1-motive. Indeed, the associated p--~visible groups, resp. filtered De Rham realizations, are isomorphic; furthermore this isomorphism is compatible with the Fontaine---Messing comparison isomorphism, which extends to the case of 1-motives (its semi-stable refinement also extends to this case (Fontaine-Raynand)).

By means of some compactification ~ of the semi-abelian group scheme A R over R extending A , there is the notion of strict neighborhood in Trig,,K of the formal completion A . x"~mJA. JA ~ ' where JA runs over all embeddings of strict neighborhoods of A inside Afig ; j+ is an exact functor, and there is a canonical epimorphism ~" the covanishing complex by ¢ := Ker(f~ rig ~ sequence , Rn(Arig,¢) * ~j+ F [B]. (~ ) 6 involving Berthelot's rigid cohomology of the special fiber ~ . M. 1 A Het( ~r) ®Qp BDR Proof: let us introduce the Raynaud realization [M ~G] of the (rigid) 1-motive A.

Extends to an Abelian scheme A R over 29 R , and let us denote the special fiber of A R by ~ . o. S. b) If contrawise A has bad reduction, let us use Grothendieck's theorem to reduce to the case , of semi-stable reduction. [Jannsen had the idea that there is still a fine structure on ttDR, , involving some 'monodromy operator", and such that Het could be recovered in a similar way as in the good reduction case [J]. Fontaine then formulated a precise conjecture and proved it in the ease of Abelian varieties].