Multiplicativity of the Stefan spectral sequence beyond faithfully flat extensions

Establish the multiplicative structure of the Stefan spectral sequence for arbitrary flat right Hopf–Galois extensions, including those that are not faithfully flat.

Background

For a flat right Hopf–Galois extension B/A, Stefan’s spectral sequence converges to the Hochschild cohomology of B and has second page given by the cohomology of the Hopf algebra H with coefficients in the Hochschild cohomology of the coinvariant subalgebra A. The paper proves multiplicativity when B/A is faithfully flat, using a functorial multiplicative spectral-sequence formalism and explicit coassociative diagonals on suitable resolutions.

The authors explicitly identify the extension from faithfully flat Hopf–Galois extensions to the full class of flat right Hopf–Galois extensions as unresolved. Thus, the remaining problem concerns whether the multiplicative structure persists without the faithful-flatness hypothesis.