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

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property to A = O X,P to get α ∈ Im F (O X,P ). This proves d). Statement e) gathers the
previous results. The k-birational invariance of F nr (k(X)/k) is clear, hence also that of
the other groups
PROPOSITION 2.1.9. — Let k be a field and let F be a functor from the category C
of commutative k-algebras to abelian groups. Assume that F satisfies the specialization
property for discrete valuation rings and the codimension one purity property for regular
local rings A of dimension 2. Assume that for all fields K ⊃ k, the natural map
F (K) → F nr (K(t)/K) is a bijection, where K(t) denotes the function field in one
variable over K. Then for all fields K ⊃ k and all positive integers n, the natural map
F (K) → F nr (K(t 1 , . . . , t n )/K) is a bijection.
Proof : First note that the assumption implies that for any field K ⊃ k, the map
F (K) → F (K(t)) is injective. Induction then shows that for any such field K, and any
positive integer n, the map F (K) → F (K(t 1 , . . . , t n )) is injective (injectivity also follows
from the specialization property : simply use a local ring at a K-rational point).
We shall prove the theorem by induction on n. The case n = 1 holds by assumption.
Suppose we have proved the theorem for n. Consider the projection p of P 1 K × K P n K onto
the second factor P n K . Note that these varieties are smooth over k, hence their local rings
are regular. On function fields, the map p induces an inclusion E = K(t 1 , · · · , t n ) ⊂ L =
K(t 1 , · · · , t n+1 ).
Let A ∈ P 1 K (K) be a fixed K-rational point. The map x → (A, x) defines a section σ
of the projection p (i.e. p ◦ σ = id P
n ). Let η denote the generic point of Pn
K K .
Let now α ∈ F (K(t 1 , · · · , t n+1 )) be an element of F nr (K(t 1 , · · · , t n+1 )/K). We may
view α as an element of F (L) = F (E(P 1 E )). It certainly belongs to F nr(E(P 1 E )/E). From
the hypothesis, we conclude that α ∈ F (L) is the image of a unique β ∈ F (E) under the
inclusion p ∗ : F (E) ↩→ F (L).
Let x ∈ P n K be an arbitrary codimension one point. Let y = σ(x) ∈ P1 K × K P n K . This
is a codimension 2 point on P 1 K × K P n K , which is a specialization of the codimension one
point ω = σ(η). Let A be the (dimension one) local ring at x, B the (dimension two)
local ring at y and C the (dimension one) local ring at ω. The local ring at η is the field
E = K(t 1 , · · · , t n ). We have inclusions A ⊂ E, B ⊂ C ⊂ L. We also have compatible
maps σ ∗ : F (B) → F (A) and σ ∗ : F (C) → F (E)
Since F satisfies the codimension one purity assumption for regular local rings of
dimension two, and α belongs to F nr (K(P 1 K × K P n K )/K), there exists an element
γ ∈ F (B) whose image in F (L) is α. Now the image γ 1 of γ in F (C) under F (B) → F (C)
and the image β 1 of β in F (C) under the map F (E) → F (C) both restrict to α in F (L).
The ring C is a discrete valuation ring, the natural map C → E from C to its residue
field E being given by σ ∗ .
By the specialization property, we conclude that σ ∗ (β 1 ) = σ ∗ (γ 1 ) ∈ F (E). But
σ ∗ (β 1 ) = σ ∗ (p ∗ (β)) = β and σ ∗ (γ 1 ) is the restriction to F (E) of σ ∗ (γ) ∈ F (A).
We therefore conclude that β ∈ F (E) lies in the image of F (A). Since A was the
local ring of P n K at an arbitrary codimension one point, we conclude that β lies in