By Otto F.G. Schilling, W.Stephen Piper

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Ip are acyclic G-modules. Then Hˆ n (G, A) ∼ = Hˆ n−p (G, B) for n ≥ p. . Exercise 6. Let C = (C n , dn )n∈ ZZ and C = (C n , dn )n∈ ZZ be two complexes in an abelian category and let f = (fn )n∈ ZZ and g = (gn )n∈ ZZ be two morphisms from C to C . A homotopy from f to g is a family h = (hn )n∈ ZZ of morphisms hn : C n+1 → C n such that . hn dn+1 + dn hn−1 = fn − gn . We say that f and g are homotopic and write f g if such a family exists. Show that in this case f and g induce the same homomorphisms H n (C ) → H n (C the homology.

Free for private, not for commercial use. 50 Chapter I. Cohomology of Profinite Groups Proof. (i), (ii) and (iii) are seen at once on the level of cochains. (iv) is equivalent to the commutativity of the diagram ❭❬❩❨❳ A) × H q (H, B) H p (H, ∪ H p+q (H, A ⊗ B) cor res cor H p (G, A) × H q (G, B) ∪ H p+q (G, A ⊗ B). 5) assume that G is finite: apply lim to the diagram with G, H −→ U replaced by G/U, H/U , where U runs through the open normal subgroups U contained in H. By dimension shifting, we may transform the above diagram into the diagram ❛❵❴❫❪ Ap ) × H ˆ 0 (H, Bq ) Hˆ 0 (H, ∪ Hˆ 0 (H, Ap ⊗ Bq ) res cor cor Hˆ 0 (G, Ap ) × Hˆ 0 (G, Bq ) ∪ Hˆ 0 (G, Ap ⊗ Bq ), which comes from the diagram A❜❢❞❡❝H × BqH p ⊗ (Ap ⊗ Bq )H NG/H NG/H G AG p × Bq ⊗ (Ap ⊗ Bq )G .

An inhomogeneous cochain x ∈ C n (G, A), n ≥ 1, is called normalized if x(σ1 , . . , σn ) = 0 whenever one of the σi is equal to 1. Show that every class in H n (G, A) is represented by a normalized cocycle. Hint: Construct inductively cochains x0 , x1 , . . , xn ∈ C n (G, A) and y1 , . . , yn C n−1 (G, A) such that x0 = x, xi = xi−1 − ∂yi , i = 1, . . , n, yi (σ1 , . . , σn−1 ) = (−1)i−1 xi−1 (σ1 , . . , σi−1 , 1, σi , . . , σn−1 ). Then xn is normalized and x − xn is a coboundary. de/∼schmidt/NSW/ ∈ Electronic Edition.

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