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422 T. GeisserProposition 4.3. The following statements are equivalent:a) Conjecture P.0/.b) For all schemes X, and all i, the map 'X i is an isomorphism.c) For all smooth and affine schemes U of dimension d, the groups CH 0 .U; i/ aretorsion for i 6D d, and the composition ! W CH 0 .U; d/ Ql ! H d .U et ; yQ l / !H d . xU et ; yQ l / OG is an isomorphism.Proof. a) ) b): First consider the case that X is smooth and proper. Then ConjectureP(0) is equivalent to the vanishing of the left-hand side of (6) for i 6D 0; 1, whereas theright-hand side of (6) vanishes by the Weil-conjectures. On the other hand, 'X 0 inducesan isomorphism CH 0 .X/ ˝ Q l Š H 2d .X et ; yQ l .d// and 'X1 induces an isomorphismCH 0 .X/ ˝ Q l Š .CH 0 . xX/ ˝ Q l /G O Š H 2d . xX; yQ l .d//G O . Indeed, both sides areisomorphic to Q l if X is connected. Using localization and alterations, the statementfor smooth and proper X implies the statement for all X.b) ) a): The right-hand side of (6) is zero for i 6D 0; 1 by weight reasons forsmooth and projective X, hence so is the left side.b) ) c): This follows because H i .U et ; yQ l / D 0 unless i D d;d 1 for smoothand affine U .c) ) b): We first assume that X is smooth and affine. By hypothesis and the affineLefschetz theorem, both sides of (6) vanish for i 6D d;d 1, and are isomorphic fori D d. Fori D d 1, the vertical maps in the following diagram are isomorphismsby semi-simplicity,CH 0 .X; d/ Ql Š CH 0 . xX;d/ O GQl Hd . xX; yQ l / O G.CH 0 . xX;d/ Ql / O G Hd . xX; yQ l / O G Hd 1 .X; yQ l /.Hence the lower map 'Xd 1 is an isomorphism because the upper map is. Using localization,the statement for smooth and affine X implies the statement for all X.Proposition 4.4. Under resolution of singularities, the following are equivalent:a) Conjecture P.0/.b) For every smooth affine U of dimension d over F q , we have CH 0 .U; i/ Q ŠHi W .U; Q/ for all i, and these group vanish for i 6D d.c) For every smooth affine U of dimension d over F q , the groups CH 0 .U; i/ Q vanishfor i>d, and CH 0 .U; d/ Q Š H W .U; Q/.dProof. a) ) b): It follows from the previous proposition that CH 0 .U; i/ Q D 0 fori 6D d. On the other hand, P.0/ for all X implies that CH 0 .X; i/ Š HiW .X; Q/ forall i and X.c) ) a): The statement implies Conjecture A.0/, and then Conjecture B.0/, versionb), for all X. By Theorems 3.5 and 4.2, P.0/ follows.