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

48
Proof : 1) Since étale cohomology does not change under inseparable extensions, we
may assume that k and K are algebraically closed.
2) The functor V → H 0 (V, H i (µ ⊗j
n )) is functorial contravariant on the category of all k-
schemes. Indeed, étale cohomology is functorial contravariant under arbitrary morphisms.
If f : V → W is any k-morphism of k-varieties, we have an induced map of sheaves on
W : HW i (µ⊗j n ) → f ∗ (HV i (µ⊗j n )) hence a map
f ∗ : H 0 (W, HW i (µ ⊗j
n )) → H 0 (V, HV i (µ ⊗j
n ))
and one easily checks that this map is functorial, i.e. respects composition of morphisms.
3) Given a finite field extension K ⊂ L with k ⊂ K, an excellent discrete valuation
ring A with k ⊂ A, with residue field κ and with field of fractions K, let B be the integral
closure of A in L, which we assume to be of finite type over A. Let q α be the finitely
many maximal ideals of B, and let κ α be the corresponding residue field. There is a
commutative diagram
H i (L, µ ⊗j
n )
(∂ α )
−−−−−→
⊕ α Hi−1 (κ α , µ ⊗j−1
n )
↓ Cores L,K
H i (K, µ ⊗j
n )
∂ A
−−−−−→
↓ ∑ α Cores κα,κ
H i−1 (κ, µ ⊗j−1
n ).
This is most easily proved by means of the Galois cohomological description of the residue
map.
If now f : V → W is a finite flat morphism of smooth integral k-varieties, we deduce
that there is an induced norm map
f ∗ : H 0 (V, HV i (µ ⊗j
n )) → H 0 (W, HW i (µ ⊗j
n )).
In particular given a smooth, proper, integral k-variety X and f : V → W a finite flat
morphism of smooth integral k-curves, there is an induced norm map
f ∗ : H 0 (X × V, HV i (µ ⊗j
n )) → H 0 (X × W, HW i (µ ⊗j
n )).
Suppose that V is a smooth affine curve and that W = A 1 k . Given any point x ∈ A1 (k),
we have a commutative diagram
H 0 (X × C, H i ) −−−−−→ H 0 (X, H i )
Cores ↓
id ↓
H 0 (X × A 1 k , Hi ) −−−−−→ H 0 (X, H i ).