, (C) sm /S) which is by definition given by -for U/S a smooth morphism
, (C)/S) is by definition a natural extension of
, Denote h 12 : U 12 := U 1 × S U 2 ? S and p 112 : U 1 × S U 2 ? U 1
, ?)(?, ?) are usu local equivalence by proposition 9, ? since h : U ? S is a smooth morphism, the inclusion ?
, Z UT ) is a quasi-isomorphism
, AnSp(C) 2,smpr /S) is ?-filtered D 1 S local for the usual or etale topology (see definition 16) and admits transferts. an open affine covering and denote, for I ? [1, · · · l], S I = ? i?I S i and j I : S I ?? S the open embedding. Let i i : S i ??S i closed embeddings, The complex of presheaves
,
,
,
, Let i i : S i ??S i closed embeddings, withS i ? SmVar(C), ) sm /S)
,
,
,
,
,
,
,
, AnSp(C) sm /S an )
, T an )) ? D D(1,0)f il,? (T an /(Y an × S an I )), see definition 121 and definition 142 commutes g * mod Hdg, the following diagram in ? T (D(M HM
, we define the Betti realization functor as
, S cw ) * Cw * S M = e(S cw ) * sing I * Cw et)(F )
, For the Corti-Hanamura weight structure on DA ? (S), we have by functoriality of (i) the functor
, Note that, by considering the explicit D 1 S local model for presheaves on AnSp(C) sm /S an
, S an ) ; by considering the explicit I 1 S local model for presheaves on CW sm /S cw
,
, Definition 152, We have, for M ? DA(S), (F, W ) ? C f il (Var(C) sm /S) such that (M, W ) = D(A 1 , et)(F, W ), and an equivalence (A 1 , et) local e : f * (F, W ) ? (F ? , W ) with (F ? , W ) ? C f il (Var(C) sm /S) such that (f * M, W ) = D
,
,
,
,
,
,
,
, IJ
, IJ
, IJ
, IJ
,
,
,
, C f il (Var(C) sm /S) is such that (M, W ) = D(A 1 , et)(F, W )
, S )) is the isomorphism of theorem 23, ? the map T (e, hom)(?, ?) is a filtered equivalence usu local by proposition 33 and proposition 121
, M ) : (F DR S (M )) an ? ? ? F DR S,an (M ) is an isomorphism by theorem 41
, We now give the functoriality with respect to the five operation using the De Rahm realization case and the Betti realization case : given in proposition 133(i0)
, the maps of definition 122 and of definition 153 induce an isomorphism in D(M HM (S)
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