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The situation is different for restriction, however. In fact, this map can be extended to all dimensions q ≤ 0. 5) remains valid for q = 0. 8) Lemma. Let 0 → A → B → C → 0 be an exact sequence of G-modules, and let g be a subgroup of G. Then the following diagram commutes ↔➣→➔ C) H 0 (G, δ res0 H 0 (g, C) H 1 (G, A) res1 δ H 1 (g, A). Proof. Let c ∈ C G be a 0-cocycle of the G-module C and c = c + NG C its cohomology class. , c is also a 0-cocycle for the g-module C. If we choose b ∈ B with jb = c, then j∂b = ∂c = 0 implies that there exists a 1-cocycle a1 : G → A such that ia1 = ∂b.

If j i 0 −→ A −→ B −→ C −→ 0 is an exact sequence of G-modules and G-homomorphisms, then there exists a canonical homomorphism δq : H q (G, C) −→ H q+1 (G, A). The map δq is called the connecting homomorphism or also the δhomomorphism. For the construction of δq consider the following commutative diagram 0✶✵✴✳✲✰✱✯✬✭✮✫★✩✪✧✥✦ Aq−1 i ∂ 0 Aq Aq+1 j ∂ i ∂ 0 Bq−1 Bq Bq+1 0 ∂ j ∂ i Cq−1 Cq 0 ∂ j Cq+1 0. ) The rows in this diagram result from applying the functor HomG (Xi , ) (i = q−1, q, q+1) to the exact sequence 0 → A → B → C → 0; since the G-modules Xi are free (cf.

IG ⊗ A for q ≤ 0. Therefore one may define the cohomology groups of the G-module A from the beginning as the quotient group H q (G, A) = (Aq )G /NG Aq . 11) For cohomology theory developed along these lines, see C. Chevalley [12]. An abelian group A is said to be uniquely divisible if for every a ∈ A and every natural number n the equation nx = a has a unique solution x ∈ A. § 3. The Exact Cohomology Sequence 31 In particular, the G-module Q (on which the group G always acts trivially) has trivial cohomology.

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