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

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Proof : Suppose first that k is a perfect field. Fix i. Let us prove that the restriction map
is surjective for c > (i+1)/2. The proof is by descending induction on the codimension of
F in X, subject to the condition c > (i+1)/2. Assume it has been proved for c+1. There
exists a closed subset F 1 ⊂ F of codimension at least c+1 in X such that F −F 1 is regular,
hence smooth over the perfect field k, and of pure codimension c in X −F 1 (the closed set
F 1 is built out of the components of F of dimension strictly smaller than the dimension of
F and of the singular locus of F ). From i + 1 − 2c < 0 and the theorem, we conclude that
(X − F, µ⊗j n ) is surjective. The induction
(X, µ⊗j n ) −→ Het í (X − F, µ⊗j n ) is
surjective, hence Het í (X, µ⊗j n ) −→ Het í (X − F, µ⊗j n ) is surjective. That the restriction
map is injective for c > i/2 is proved in a similar manner. The proof will be left to the
reader. For the induction to work we need to be over a perfect field, to ensure that regular
schemes are smooth. But since étale cohomology is invariant under purely inseparable
extensions ([SGA4] VIII 1.1, [Mi80] Rem 3.17 p. 77), the corollary holds over all fields.
the restriction map Het í (X − F 1, µ ⊗j
n ) −→ Het í
assumption now implies that the restriction map Het í
Let X be a smooth integral k-variety. For c = 1, Corollary 3.4.2 says that for a
nonempty open set U ⊂ X, the restriction maps on sections Hét 0 (X, µ⊗j n ) −→ Hét 0 (U, µ⊗j n )
are isomorphisms, which is nearly obvious, and that the maps Hét 1 (X, µ⊗j n ) −→
Hét 1 (U, µ⊗j n ) are injective.
For c = 2, Corollary 3.4.2 says that if F is a closed subset of codimension at least 2, the
restriction maps Het í (X, µ⊗j n ) −→ Het í (X −F, µ⊗j n ) are injective for i ≤ 3 and bijective for
i ≤ 2. Using purity for discrete valuation rings, commutativity of étale cohomology with
direct limits ([Mi80], III 3.17 p. 119), and the Mayer-Vietoris sequence ([Mi80], III.2.24
p. 110), from the result just proved one may deduce the existence of exact sequences
(3.7) 0 −→ H 1 (X, µ ⊗j
n ) −→ H 1 (k(X), µ ⊗j
n ) −→ ⊕
H 0 (k(x), µ n ⊗(j−1) )
x∈X (1)
and
(3.8) H 2 (X, µ ⊗j
n ) −→ H 2 (k(X), µ ⊗j
n ) −→ ⊕
H 1 (k(x), µ n ⊗(j−1) ).
x∈X (1)
(A similar argument, with more details, will appear in the proof of Theorem 3.8.2 below.)
From the Kummer sequence we get a functorial surjection H 2 (X, µ n ) → n Br(X) (see
(3.2)). The map H 2 ét (X, µ n) −→ H 2 ét (X − F, µ n) is bijective by Corollary 3.4.2. Hence if
X/k is a smooth integral variety over a field k of characteristic zero, and F is a closed
subset of codimension at least 2, the restriction map Br(X) → Br(X −F ) is surjective. It
is therefore an isomorphism, since over any regular integral scheme X, for any nonempty
open set U, the restriction map Br(X) → Br(U) is injective ([GB II] 1.10). All in all, for
X a smooth variety over a field k and char(k) = 0, there is an exact sequence
(3.9) 0 −→ Br(X) −→ Br(k(X)) −→ ⊕
x∈X (1) H 1 (k(x), Q/Z).