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

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One may ask whether an element of H 1 (K, G) which at each prime p of height one of A
comes from an element of H 1 (A p , G) actually comes from an element of H 1 (A, G).
The injectivity question was already raised by Serre and Grothendieck in the early
days of étale cohomology (1958) (cf. [CT79]).
For arbitrary discrete valuation rings, injectivity was proved by Nisnevich [Ni84]. For
split reductive groups, Nisnevich [Ni89] also proved injectivity over a regular local ring
of dimension two. For local rings of smooth varieties over an infinite perfect field k, and
for a reductive group G defined over k, injectivity was established in [CT/Oj92] (see § 5
below). The case where k is infinite but not necessarily perfect has since been handled
by Raghunathan [Ra93].
Codimension one purity holds over a regular local ring of dimension two ([CT/Sa79]).
In higher dimension, it is an open question, even for A a local ring of a smooth k-variety
and G a reductive k-group scheme. One special case is known, namely codimension one
purity for H 1 (A, SL(D)) for A a local ring of a smooth variety over a field k and D/k
a central simple algebra. Various proofs are available ([CT/Pa/Sr89] in the square free
index case, [Ro90], [CT/Oj92] ; another proof will be given in § 5.3).
§ 2.2.7 One more general question
To conclude this section, let us mention a question related to the injectivity property.
Let A be a regular local ring, let K be its field of fractions. Let G/A be a reductive
group scheme and let X/A be an A-scheme which is a homogeneous space of G. If the set
X(K) of K-rational points is nonempty, does it follow that the set of A-points X(A) is
nonempty ? If A is a discrete valuation ring, and X/A is proper, this is trivially so. But
for dim(A) ≥ 2 and X/A proper, the answer is far from obvious. One known case is that
of Severi-Brauer schemes ([Gr68]).
As a special example of the question, suppose 2 ∈ A ∗ , let a i ∈ A ∗ (i = 1, . . . , n).
Suppose that the quadratic form ∑ i=1,...,n a iXi
2 has a nontrivial zero with coordinates
in K n . Does it have a zero (α 1 , . . . , α n ) ∈ A n with at least one of the α i ’s a unit ? For
more on this topic, see [CT79]. The case of a regular local ring of dimension two was
handled by Ojanguren [Oj82b].
§ 3. Étale cohomology
The aim of this section is to recall some basic facts from étale cohomology, including
cohomological purity, as proved in [SGA4], then to go on to the Gersten conjecture in
the étale cohomological context, as stated and proved by Bloch and Ogus [Bl/Og74].
We shall try to motivate the Gersten conjecture (§ 3.5). In § 3.8 we shall see that the
injectivity and codimension one purity theorems for local rings of smooth varieties over
a field are immediate consequences of the Gersten conjecture.
§ 3.1 A few basic properties of étale cohomology
The reader is referred to [SGA4], to Deligne’s introductory lectures ([SGA4 1/2], pp.
4-75) or to Milne’s book [Mi80] for the definition of étale cohomology. Given a scheme X