On Hochschild-Serre spectral sequence of Lie algebras (Q713285)

The Hochschild-Serre spectral sequence associates to a Lie algebra \(L\), an ideal \(K\) of \(L\) and an \(L\)-module \(M\) a spectral sequence converging to the Lie algebra cohomology \(H^*(L,M)\) with \(E_2\)-term involving cohomology w.r.t. \(K\) and \(L/K\). To each spectral sequence, there is a 5-term exact sequence of low degree terms. It reads in our framework for trivial \(L\)-modules \(M\) \[ 0\to\text{Hom}(L/K,M)\to\text{Hom}(L,M)\to\text{Hom}_L(K,M)\to H^2(L/K,M) \to H^2(L,M). \] The author extends this exact sequence to a 6-term exact sequence, the last term being \(\text{Hom}(\text{Ker}(L\wedge K\to[L,K]),M)\), where the map \(L\wedge K\to[L,K])\) is given by the Lie bracket. Moreover, in case the Lie algebra \(L\) comes with a free presentation \(L=F/R\) such that \(K=S/R\) for a suitable ideal \(S\) of the free Lie algebra \(F\), the author shows that the last term of the above 6-term exact sequence takes the form \[ \text{Hom}((R\cap[F,S])/[F,R],M) \] and that the last map of the sequence is surjective.