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

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separable closure of K and let K nr ⊂ K be the maximal unramified extension of K inside
K. Let G = Gal(K/K) be the absolute Galois group, I = Gal(K/K nr ) be the inertia
group and G = Gal(K nr /K) = Gal(κ/κ). We then have the Hochschild-Serre spectral
sequence for Galois cohomology
E pq
2 = Hp (G, H q (I, µ ⊗j
n )) =⇒ H p+q (G, µ ⊗j
n ).
Let p be the characteristic of κ. Now we have an exact sequence
1 −→ I p −→ I −→ ∏ l≠p
Z l (1) −→ 1
where Z l (1) is the projective limit over m of the G-modules µ l
m = µ l m(K) = µ l m(K nr )
(Here I p = 0 if p = 0). For this, see [Se68]. Also, we have H q (I, µ ⊗j
n ) = 0 for q > 2, and
H 1 (I, µ ⊗j
n ) = µ ⊗j−1
n , as a G-module. The above spectral sequence thus gives rise to a
long exact sequence
. . . → H i (G, µ ⊗j
n
hence in particular to a map
) → H i (G, µ ⊗j
n
) → H i−1 (G, µ n
⊗j−1 ) → H i+1 (G, µ ⊗j
n ) → . . .
∂ : H i (K, µ ⊗j
n ) −→ H i−1 (κ, µ n ⊗(j−1) ) .
and one may check that this map agrees (up to a sign) with the map ∂ A in (3.6) .
Equipped with the previous description of the map ∂ A , one proves :
PROPOSITION 3.3.1. — Let A ⊂ B be an inclusion of discrete valuation rings with
associated inclusion K ⊂ L of their fields of fractions (which need not be a finite extension
of fields). Let κ A and κ B be their respective residue fields. Let n be an integer invertible in
A, hence in B, and let e = e B/A be the ramification index of B over A, i.e. the valuation
in B of a uniformizing parameter of A. The following diagram commutes :
H i (K, µ ⊗j
n )
↓ Res K,L
H i (L, µ ⊗j
n )
∂ A
−−−−−→
∂ B
−−−−−→
H i−1 (κ A , µ ⊗j−1
n )
↓ ×e B/A .Res κA ,κ B
H i−1 (κ B , µ ⊗j−1
n ).