HochschildSerre

Hochschild-Serre refers to the Hochschild-Serre spectral sequence, a fundamental tool in group cohomology that relates the cohomology of a group to the cohomology of a normal subgroup and its quotient. It is named after Ludwig Hochschild and Jean-Pierre Serre, who introduced it in their 1953 work on the cohomology of group extensions.

For a short exact sequence of groups 1 → N → G → Q → 1 and a G-module A, there is a first-quadrant spectral sequence with E2-term E2^{p,q} ≅ H^p(Q, H^q(N, A)) that converges to H^{p+q}(G, A). The Q-action on H^q(N, A) is induced by the G-action on N and A. This framework allows one to compute the cohomology of G from the cohomology of N and Q, and it yields connecting information in low degrees.

Consequences and use include practical computational leverage: it provides exact sequences in low degrees and clarifies how extensions and actions interact within cohomology. The Hochschild-Serre construction has variants for different contexts, such as continuous cohomology, and it has parallel versions for Lie algebras, commonly known as the Lyndon-Hochschild-Serre spectral sequence, stated for a short exact sequence of Lie algebras 0 → h → g → g/h → 0 with a g-module M: E2^{p,q} ≅ H^p(g/h, H^q(h, M)) ⇒ H^{p+q}(g, M).