Resolve the extension problem and determine the multiplicative structure of the THH computation

Resolve the extension problem for the collapsed descent spectral sequence computing the homotopy groups of the p-completed topological Hochschild homology spectrum THH(Z_p[x]/(px))^\wedge_p, and specify its multiplicative structure by constructing a suitable differential graded algebra model.

Background

The paper constructs a multiplicative second-quadrant descent spectral sequence converging to $\pi_*THH(\mathbb Z_p[x]/(px))^\wedge_p$. Its $E^2$-page is computed explicitly using a resolution of a graded Hopf algebroid, and the spectral sequence is shown to collapse at the $E^2$-page for degree reasons.

Although the associated graded information is therefore available, collapse does not determine the extensions needed to reconstruct the target homotopy groups, nor does it determine the multiplicative products among the resulting classes. The unresolved task is to solve these extension problems and identify the multiplicative structure, which the paper indicates can be achieved using a suitable DGA model.