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

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of the integral cohomology group H 2 (G, Q/Z) ≃ H 3 (G, Z) consisting of classes whose
restrictions to all abelian subgroups of G vanish. This condition is equivalent to the
vanishing of restrictions to all abelian subgroups with at most two generators.
Bogomolov’s formula in turn is subsumed in Saltman’s formula [Sa90] for the unramified
Brauer group of twisted multiplicative field invariants.
Let G be a connected, reductive group over an algebraically closed field k of characteristic
zero. Let G ⊂ GL n be an injective homomorphism. One may then consider the
quotient variety X = GL n /G, which is an affine variety since G is reductive. It is an
open question whether the (obviously unirational) variety X is a rational variety. Saltman
([Sa85], [Sa88]) and Bogomolov [Bo89] have studied this problem. For G = P GL r ,
Saltman [Sa85] (see [CT/Sa87] for a different proof) and Bogomolov for G arbitrary
([Bo89], see also [CT/Sa88]) showed that the unramified Brauer group of k(GL n /G) is
always zero.
Taking the point of view of unramified cohomology, one may handle the Artin-
Mumford example quoted above, as well as other examples, in a manner totally parallel
to that of § 1. Indeed, in § 1, the function field L = C(X) is a quadratic field extension
of the rational function field in one variable K = C(t), that is it is the function field
of a 0-dimensional conic over C(t). A key point in the proof we gave in § 1 is the exact
sequence
1 → Z/2 → K ∗ /K ∗2 → L ∗ /L ∗2
that is
1 → Z/2 → H 1 (K, Z/2) → H 1 (L, Z/2)
associated to the quadratic field extension L/K., which gives control on the kernel of
the map K ∗ /K ∗2 → L ∗ /L ∗2 . Let α ∈ H 1 (K, Z/2) be the generator of the above kernel.
Note that α is a ramified element in K ∗ /K ∗2 . Nontrivial unramified elements in L ∗ /L ∗2
are then produced in the following way. Let β ≠ 1, α be a ramified element in K ∗ /K ∗2 .
Suppose that all ramification of β is “contained” in the ramification of α. Since α dies in
L ∗ /L ∗2 , so does its ramification, hence so does the ramification of β when going over to
L. However, because β does not lie in the the kernel of K ∗ /K ∗2 → L ∗ /L ∗2 , β itself does
not die when going up to L : this produces a nontrivial unramified element in L ∗ /L ∗2 .
Now let L be the function field of a 1-dimensional conic over the rational function
field in two variables K = C(t 1 , t 2 ). If L is the function field of a smooth projective
conic over a field K, it is a special case, already known to Witt ([Wi35] p.465), of a
result of Amitsur that the kernel of the map H 2 (K, Z/2) → H 2 (L, Z/2) is either 0 or
Z/2, depending on whether the conic has a K-rational point or not. In the second case,
the nontrivial element in H 2 (K, Z/2) ≃ 2 Br(K) is the class of the quaternion algebra
α associated to the conic. Now this is an exact analogue of the situation above. To
produce an unramified element in H 2 (L, Z/2) one should therefore look for an element
β ∈ H 2 (K, Z/2) which is neither 0 nor α, but whose ramification (as an element of
H 2 (K, Z/2)) is “somehow” dominated by the ramification of α, so that when going over
to L, all ramification of β L ∈ H 2 (L, Z/2) vanishes – although β itself does not vanish.