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

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THEOREM 4.3.6. — Let k be a real closed field, and let X be a smooth, geometrically
integral variety of dimension d over k. Assume that there exists a dominant rational map
of degree N from A d−2 × k
S to X, for some integral surface S/k. Let n > 0 be an integer.
Then :
a) The group Hnr(k(X)/k, 3 µ ⊗2
n ) is finite, and it is zero for all integers n prime to 2N.
b) If X/k is smooth, the group CH 2 (X)/n is finite.
Proof : One may assume that S/k is smooth and affine and find a finite and flat map
of constant degree 2N from a nonempty open set V of A d−2 × k
S to an open set U of
X. The proof of the theorem is then essentially identical to the proof of Theorem 4.3.1,
in view of the following facts :
(i) For any variety U/k with k real closed, the étale cohomology groups H i (U, µ ⊗j
n )
are finite. This follows from the Hochschild-Serre spectral sequence for the extension k/k
together with the finiteness of étale cohomology of varieties over an algebraically closed
field and finiteness of the Galois cohomology of finite Gal(k/k)-modules.
(ii) For any smooth variety X over a real closed field k, and any positive integer n,
the group n CH 2 (X) is finite. Indeed, it is a subquotient of H 3 (X, µ ⊗2
n ) which we have
just seen is finite.
(iii) H 3 (k, µ ⊗2
n ) = 0 for n odd.
(iv) Lemma 4.3.3 still holds, with the same proof, when k is real closed.
THEOREM 4.3.7. — Let k be a p-adic field (finite extension of Q p ), let X a smooth
geometrically integral variety of dimension d over k.
a) Assume that X = X × k k is rationally dominated by the product of an integral
curve and an affine space of dimension d − 1 ; then for any integer n > 0, the groups
) are finite.
b) If n is prime to p, then the same finiteness results hold if X is rationally dominated
by the product of an integral surface and an affine space of dimension d − 2.
CH 2 (X)/n and H 3 nr(k(X)/k, µ ⊗2
n
Proof : The proof of the first statement is essentially identical to the proof of Theorems
4.3.1 and 4.3.6. One chooses a finite extension K/k of fields, a smooth integral curve C/K,
resp. surface S/K, and a finite, flat morphism of constant degree N from a nonempty
open set of C × K A d−1
K
to an open set of X. One then combines the following facts :
(i) For any variety U/k with k p-adic, the étale cohomology groups H i (U, µ ⊗j
n ) are
finite. This follows from the Hochschild-Serre spectral sequence for the extension k/k
together with the finiteness of étale cohomology of varieties over an algebraically closed
field and finiteness of the Galois cohomology of finite Gal(k/k)-modules ([Se65]). This
implies that for any smooth variety U over k p-adic, and any positive integer n, the group
nCH 2 (U) is finite.
(ii) For the curve C, we obviously have CH 2 (C × K A d−1
K
) = CH2 (C) = 0. For the
smooth surface S, if n is prime to p, the group CH 2 (S)/n is finite. It is enough to prove
this for a smooth projective surface S over the p-adic field K. In that case, Saito and
Sujatha [Sa/Su93] have proved that H 0 (X, H 3 (S, µ ⊗2
n )) is finite . The result then follows