Birational invariants, purity and the Gersten conjecture Lectures at ...

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46 a k-rational point, one may then use a result of Saito ([Sa91], see also [CT93], § 7, Thm. 7.6) ensuring that the torsion subgroup of CH 2 (Y ) is finite. Versions of the following theorem, which combines many of the previous results, were brought to my attention by N. Suwa and L. Barbieri-Viale (independently). THEOREM 4.3.10. — Let X/k be a smooth, projective, geometrically integral variety of dimension d over a field k ⊂ C. Under any of the two sets of hypotheses : (i) the field k is a number field and CH 0 (X C ) is represented by a curve ; (ii) the field k is finitely generated over Q, there is a k-rational point on X and the degree map CH 0 (X C ) −→ Z is an isomorphism ; the group CH 2 (X) is a finitely generated abelian group. Remarks : (a) The geometric assumption in (i) is that there exists a curve C/C a C-morphism from C to X C such that the induced map CH 0 (C C ) −→ CH 0 (X C ) is surjective. Concrete examples are provides by varieties dominated by the product of a curve and and an affine space. A special case is that of quadric bundles (of relative dimension at least one) over a curve. (b) The geometric assumption in (ii) is satisfied by unirational varieties, by quadric bundles (of relative dimension at least one) over rational varieties, and also by Fano varieties (Miyaoka, Campana [Ca92]). Proof of the theorem : Let k ⊂ C be a fixed algebraic closure of k, let G = Gal(k/k), let X = X × k k and X C = X × k C. There is a natural filtration on the Chow group CH 2 (X) : CH 2 (X) alg ⊂ CH 2 (X) hom ⊂ CH 2 (X). The smallest subgroup is that of cycles algebraically equivalent to zero, the middle subgroup is that of cycles homologically equivalent to zero, that is those cycles which are in the kernel of the composite map CH 2 (X × k k) −→ CH 2 (X C ) −→ H 4 Betti(X(C), Z). A standard specialization argument shows that the map CH 2 (X × k k) −→ CH 2 (X C ) is injective. On the other hand, the group H 4 Betti (X(C), Z) is a finitely generated abelian group. Thus the quotient CH 2 (X)/CH 2 (X) hom is a finitely generated group. By the same specialization argument, the quotient CH 2 (X) hom /CH 2 (X) alg is a subgroup of the classical Griffiths group CH 2 (X C ) hom /CH 2 (X C ) alg . Under any of the two assumptions in the theorem, and even under the weaker assumption that CH 0 (X C ) is representable by a surface, the group CH 2 (X C ) hom /CH 2 (X C ) alg
vanishes : this is a result of Bloch and Srinivas [Bl/Sr83], whose proof relies on the Merkur’ev-Suslin theorem. Thus the group CH 2 (X)/CH 2 (X) hom is an abelian group of finite type. According to standard usage, let us write A 2 (X) = CH 2 (X) hom . Murre [Mu83], relying on work of H. Saito [Sa79] and Merkur’ev-Suslin [Me/Su82], has shown that there exists a universal regular map ρ : A 2 (X) −→ A(k) where A is a certain abelian variety defined over k. Under the assumption that CH 0 (X C ) is represented by a curve, Bloch and Srinivas (loc. cit.) show that the map ρ is an isomorphism. There exists an abelian variety B over k and a cycle Z ∈ CH 2 (B × X) which induces maps B(k) −→ A 2 (X) −→ A(k), the composite map being induced by a k-morphism of abelian varieties ϕ : B −→ A (see [Sa79], Prop. 1.2 (ii)). Now the varieties A, B, the morphism ϕ and the cycle Z are all defined over a finite field extension L of k. Let H = Gal(k/L). It then follows that the isomorphism ρ : A 2 (X) −→ A(k) is H-equivariant. Thus the group (A 2 (X)) G ⊂ (A 2 (X)) H is a subgroup of A(L), and for k, hence L, finitely generated over Q, this last abelian group is finitely generated by the Mordell-Weil-Néron theorem. Hence (A 2 (X)) G is a finitely generated abelian group. To conclude the proof, it only remains to show that under the assumptions of (i) and (ii), the group Ker[CH 2 (X) −→ CH 2 (X] is a finite group. In case (i), observe that the geometric assumption on 0-cycles implies the vanishing of H 2 (X, O X ) (see [Ja90] p.157). The required finiteness now follows from [CT/Ra91] and [Sal93]. In case (ii), the geometric assumption on 0-cycles implies the vanishing of both H 2 (X, O X ) and H 1 (X, O X ). The required finiteness result is then due to S. Saito ([Sa91],[CT93]). § 4.4 Rigidity for unramified cohomology The following theorem is an adaptation to étale cohomology of Suslin’s celebrated rigidity theorem for K-theory with coefficients ([Su83] [Su88], see also Lecomte’s paper [Le86] for interesting variants in the Chow group context) : THEOREM 4.4.1. — Let k ⊂ K be separably closed fields, let X/k be a smooth, integral, proper k-variety. Let k(X) be the function field of X and K(X) the function field of X K = X × k K. Let n > 0 be an integer prime to char(k). Let i > 0 and j be integers.Then the natural map H i (k(X), µ ⊗j n i.e. an isomorphism ) → H i (K(X), µ ⊗j n ) induces an isomorphism Hnr(k(X)/k, i µ ⊗j n ) ≃ Hnr(K(X)/K, i µ ⊗j n ) H 0 (X, H i (µ ⊗j n )) ≃ H 0 (X K , H i (µ ⊗j n )). In other words, unramified cohomology is rigid : it does not change by extension of separably closed field. 47
Page 1 and 2: 1 Birational invariants, purity and
Page 3 and 4: cohomology with the scheme-theoreti
Page 5 and 6: § 1. An exercise in elementary alg
Page 7 and 8: LEMMA 1.2. — Let L/K be a finite
Page 9 and 10: 9 § 2. Unramified elements § 2.1
Page 11 and 12: 11 Remark 2.1.7 : Note that there i
Page 13 and 14: 13 F nr (E/K) = F nr (K(t 1 , · ·
Page 15 and 16: Now (g ◦ f) ∗ (α) ∈ F loc (X
Page 17 and 18: local rings in dimension at least 5
Page 19 and 20: and an étale sheaf F on X, the ét
Page 21 and 22: COHOMOLOGICAL PURITY CONJECTURE 3.2
Page 23 and 24: Remark 3.3.2 : Let A be a discrete
Page 25 and 26: If char(k) = p ≠ 0, then for any
Page 27 and 28: The Gersten conjecture in that case
Page 29 and 30: Proof : Let us first assume that k
Page 31 and 32: Remark 3.8.4 : Except for the injec
Page 33 and 34: THEOREM 4.1.5. — Let k be a field
Page 35 and 36: Let us now discuss unramified H 2 .
Page 37 and 38: This is precisely what happens in t
Page 39 and 40: y K. Kato (see [Ka86]). For threefo
Page 41 and 42: Finally, we shall also use the foll
Page 43 and 44: LEMMA 4.3.3. — Let Z/k be a varie
Page 45: from exact sequence (4.2) together
Page 49 and 50: Here the top row is the sum ∑ i
Page 51 and 52: 1. One starts from the presentation
Page 53 and 54: K be the fraction field of A. Assum
Page 55 and 56: In this diagram, the map CH n−1 (
Page 57 and 58: groups 1 −→ SL(D) −→ GL(D)
Page 59 and 60: COROLLARY 5.3.2. — Let Z/k be a p
Page 61 and 62: [CT79] J.-L. COLLIOT-THÉLÈNE. —
Page 63 and 64: 63 [Ni84] Ye. A. NISNEVICH. — Esp
Page 65: [Sh89] C. SHERMAN. — K-theory of

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