The structure of the cohomology of Morava stabilizer algebra \(S(3)\) (Q1802971)

The cohomology of the Morava stabilizer algebra \(S(3)\) is computed for \(p\geq 5\). It has 26 generators, and a long but explicit list of relations. Here \(S(n)\) is defined as \(K(n)_ * \otimes_{BP_ *} BP_ *BP \otimes_{BP_ *}F_ p\). The cohomology \(\text{Ext}_{S(n)}(F_ p,F_ p)\) gives the portion of \(\text{Ext}_{BP*BP}(BP_ *,BP_ *I_ n)\) which is free of \(v_ n\)-torsion. The result of this paper also yields the \(v_ 3\)-localization of the \(E_ 2\)-term of the Adams- Novikov spectral sequence converging to \(\pi_ *(V(2))\). The method involves defining a filtration on \(S(3)\) so that the dual of the associated graded Hopf algebra is isomorphic to the universal enveloping algebra of a restricted Lie algebra. This is utilized via several spectral sequences.