High-Degree Restricted Cohomology for $K(3;1)^{(1)}$ in Characteristics Five and Seven

Abstract

We determine the high-degree restricted first cohomology of the positive subalgebra of the rank-one contact algebra in characteristics five and seven, with parabolically induced coefficients. In degrees from twice the characteristic through four times the characteristic minus three, each characteristic has six nonzero homogeneous components, of total dimension thirteen. The constituents include Frobenius twists of the natural horizontal module, represented by Cartier cocycles. PBW duality converts the calculation into bounded-degree homology, where the coefficient action couples the natural and cubic relation coordinates. The resulting modules determine the high-degree part of the coefficient cohomology used in the Lyndon-Hochschild-Serre term $E_2^{0,1}$ for extensions of induced contact modules. In particular, extensions from the Steinberg induced modules vanish in these degrees.