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

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§ 4.2 Computing unramified cohomology
In this section, we list cases where unramified cohomology has been computed, or
at least where methods have been devised to detect nontrivial elements in unramified
cohomology groups.
Let us first discuss unramified H 1 .
PROPOSITION 4.2.1. — Let X be a smooth, complete, connected variety over a field k.
Let n be a positive integer prime to char(k).
(a) For any j ∈ Z, the natural map H 1 (X, µ n ) ⊗j ) → H 1 (k(X), µ ⊗j
n ) induces an
isomorphism between H 1 (X, µ ⊗j
n ) and Hnr(k(X)/k, 1 µ ⊗j
n ).
(b) If k is algebraically closed, this map induces an isomorphism of finite groups
nPic(X) ≃ Hnr(k(X)/k, 1 µ n ).
(c) If k is algebraically closed and char(k) = 0, then there is a (non-canonical)
isomorphism Hnr(k(X)/k, 1 µ ⊗j
n ) ≃ (Z/n) 2q ⊕ n NS(X), where q denotes the dimension
of the Picard variety of X, also equal to the dimension of the coherent cohomology group
H 1 (X, O X ), and where NS(X) denotes the Néron-Severi group of X, which is a finitely
generated abelian group.
Proof : After Theorem 4.1.1, Statement (a) is just a reinterpretation of exact sequence
(3.7) (as a matter of fact, the special case of Purity required for (a) is easy to prove and
holds under quite general assumtions).
Statement (b) then follows from exact sequence (3.1), since for X/k as above,
H 0 (X, G m ) = k ∗ and k ∗ /k ∗n = 1.
For X/k as in (c), the Néron-Severi group NS(X) of classes of divisors modulo algebraic
equivalence is well-known to be a finitely generated group. The kernel of the natural
surjective map Pic(X) → NS(X) is the group A(k) of k-points of an abelian variety A,
the Picard variety of X. As such, it is a divisible group. Thus, as abelian groups, and in a
non-canonical way, we have Pic(X) ≃ A(k)⊕NS(X). Since char(k) = 0, the dimension of
the Picard variety is equal to q = dimH 1 (X, O X ). By the theory of abelian varieties, this
implies that n A(k) ≃ (Z/n) 2q for all n > 0 (here again, use is made of the char(k) = 0
hypothesis).
Remark 4.2.2 : Let k be a field of characteristic prime to the positive integer n. Let A
be a discrete valuation ring with k ⊂ A and let K be the fraction field of A. By means
of the Kummer sequence one may identify the natural map A ∗ /A ∗n → K ∗ /K ∗n with
the map H 1 (A, µ n ) → H 1 (K, µ n ). Thus for any smooth, complete, integral variety X/k,
the group H 1 nr(k(X)/k, µ n ) coincides with the group of unramified elements in k(X)
associated to the functor R → R ∗ /R ∗n , in the sense of Definition 2.1 For any A as above,
there is an obvious exact sequence
1 → A ∗ /A ∗n → K ∗ /K ∗n → Z/n → 0.
We may now interpret the computations of § 1 as a computation of unramified H 1 with
coefficients Z/2 for hyperelliptic curves over the complex field, that is as a computation
in the 2-torsion subgroup of the Picard group of such curves.