A Short Course on Galois Cohomology - William Stein - University of ...

• The homomorphism cores: ˆ H 0 (H, A) → ˆ H 0 (G, A) is induced by the
homomorphism N G/H : A H /NH(A) → A G /NG(A) where N G/H(a) =
si.a.
Proof. See Cassels–Fröhlich.
Proposition 44. For all q ∈ Z,
cores G/H ◦ resH : ˆ H q (G, A) → ˆ H q (G, A)
and the composition cores ◦ res is multiplication by [G : H].
Proof. We check this for q = 0; the general result will then follow by dimension
shifting.
ˆH 0 (G, A) = AG /N(A)
res
ˆH 0 (H, A) = AH cores
/N(A)
where the restriction map is induced by AG ⊂ AH and the co-restriction
map is induced by NG/H(a) = si.a, where the si are coset representatives
of H in G. Observe that, for a ∈ AG ,
cores(res(a)) = si.a = n.a
where n = [G : H].
Corollary 45. (i) ˆ H q (G, A) is killed by |G|, for all q ∈ Z.
(ii) If A is finitely generated then ˆ H q (G, A) is finite for all q ∈ Z.
(iii) Suppose S ≤ G is a Sylow p-subgroup, that is, |S| = p n | |G| but
p n+1 ∤ |G|. Then ˆ H q (G, A)(p) → ˆ H q (S, A), for all q ∈ Z.
Proof. (i) Take H = {1} ≤ G above and note that ˆ H q (H, A) = 0 for all
q ∈ Z.
(ii) This is a calculation of ˆ H q (G, A) using the standard resolution. One
shows that ˆ H q (G, A) is finitely generated, but since every element is
killed by |G|, it follows that it is finite.
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