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It is a description of the underlying rules of algebraic geometry, a few of its very important advancements within the 20th century, and a few of the issues that occupy its practitioners this day. it's meant for the operating or the aspiring mathematician who's unexpected with algebraic geometry yet needs to realize an appreciation of its foundations and its ambitions with at the very least must haves. Few algebraic necessities are presumed past a uncomplicated path in linear algebra.

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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].

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