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

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be a central simple algebra of prime index p over a field K (char(K) ≠ p). Let X/K be
the affine (p 2 − 1)-dimensional affine variety given by the equation
Nrd D/k (Ξ) = c
for some c ∈ K ∗ . Then (Suslin, loc. cit.) the kernel of the map H 3 (K, µ ⊗2
p ) →
H 3 (K(X), µ ⊗2
p ) is spanned by the class (D) ∪ (c) where (c) ∈ H 1 (K, µ p ) is the class
of c in K ∗ /K ∗p (under the Kummer identification) and (D) ∈ H 2 (K, µ p ) ≃ p Br(K) is
the class of the central simple algebra D in the Brauer group of K.
On the basis of this result, and with some inspiration from Bogomolov’s extension
[Bo87] of Saltman’s results [Sa84], for arbitrary prime p, E. Peyre [Pe93] managed to
produce many examples of unirational varieties X over an algebraically closed field k for
which the unramified cohomology group H 3 (k(X), Z/p) does not vanish, hence which
are not rational, even though the whole unramified Brauer group of k(X) vanishes.
Let us come back to the case p = 2. In this case, Amitsur’s H 2 result and Arason’s
H 3 result have been extended by Jacob and Rost [Ja/Ro89] to H 4 . Let K be a field,
char(K) ≠ 2, let a, b, c, d be elements of K ∗ and let X be the smooth projective quadric
associated to the 4-fold Pfister form < 1, −a > ⊗ < 1, −b > ⊗ < 1, −c > ⊗ < 1, −d >
(a, b, c, d ∈ K ∗ ). Then, with notation as above, the kernel of the map H 4 (K, Z/2) →
H 4 (K(X), Z/2) is spanned by the class of the cup product (a)∪(b)∪(c)∪(d). Here again,
Peyre was able to use this result to produce unirational varieties whose non-rationality
is detected by unramified H 4 (with coefficients Z/2).
When the ground field is not algebraically closed, unramified cohomology may still
provide some information (such is already the case for the Brauer group). For instance,
it may detect whether some varieties over k, even though they are rational over the
algebraic closure of the ground field, are not rational over the ground field.
I shall here mention two general cases where some higher cohomology groups for
varieties over a non algebraically closed field have been computed.
Suppose that k is the field R of real numbers. Let X/R be a smooth variety over
R. Then for any integer n > dim(X), the group H 0 (X, H n (Z/2)) is isomorphic to
(Z/2) s , where s denotes the number of real components of the topological space X(R)
([CT/Pa90] ; that result has since been proved for arbitrary separated varieties). In
particular, if X/R is a smooth, proper, integral variety over the reals, for any n > dim(X),
the unramified cohomology group Hnr(R(X)/R, n Z/2) is isomorphic to (Z/2) s , where s
denotes the number of connected components of X(R) (well-known not to depend on
the particular smooth complete model). These results are proved in a joint paper with
Parimala [CT/Pa90].
Suppose that k = F is a finite field of characteristic p. Let X/F be a smooth, projective,
integral variety of dimension d. Let n be a positive integer prime to p. According to
a conjecture of Kato [Ka86], the unramified cohomology groups Hnr
d+1 (F(X)/F, µ ⊗d
n )
should be trivial. For a curve, this is just a reformulation of the classical result Br(X) = 0.
For surfaces, this was proved by Sansuc, Soulé and the author in 1983, and independently