Bourbaki

Exercise 11, § 6

Algebra, X, p. 190

Let GG be a group, HH a normal subgroup of GG, MM a GG-module. According to Exercise 10, the group GG acts by conjugation on the Z\mathbf Z-modules Hq(H,M)H^q(H,M); the action of the subgroup HH being trivial, these are therefore endowed with a structure of Z(G/H)\mathbf Z^{(G/H)}-modules.

a) Now prove that there exists a spectral sequence EE (X, p. 175, exercise 14), converging to ⨁nHn(G,M)\displaystyle\bigoplus_n H^n(G,M), such that

E2pq=Hp(G/H,Hq(H,M)) E_2^{pq}=H^p(G/H,H^q(H,M))

(consider the spectral sequence of X, p. 188, exercise 6, d), taking into account the canonical isomorphism (12), p. 109).

b) Now prove that there exists an exact sequence

0⟶H1(G/H,H0(H,M))⟶H1(G,M)⟶H0(G/H,H1(H,M))⟶H2(G/H,H0(H,M))⟶H2(G,M). 0\longrightarrow H^1(G/H,H^0(H,M)) \longrightarrow H^1(G,M) \longrightarrow H^0(G/H,H^1(H,M)) \longrightarrow H^2(G/H,H^0(H,M)) \longrightarrow H^2(G,M).

Describe the homomorphisms of this sequence by using the description of Hq(G,M)H^q(G,M) in terms of Cq(G,M)C^q(G,M).

c) Define a spectral sequence analogous to that of a) for homology.

Solution

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Machine translation of the English, by gpt-5-6-mini. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.