Generalization of fine structural properties from Z[[Zp]] to broader completed group algebras

Determine the extent to which the fine structural properties established for the completed integral group algebra Z[[Zp]]—specifically, having weak Krull dimension 2 and, for almost all primes p, being a (2,2)-domain that is neither a (1,2)-domain nor a (2,1)-domain—generalize to completed integral group algebras Z[[G]] for groups G beyond Zp, such as compact p-adic analytic groups. Clarify for which classes of profinite groups these properties persist and identify any necessary conditions or obstructions to such generalizations.

Background

The paper establishes non-coherence of Z[[G]] under mild hypotheses and proves (d+3)-coherence for Z[[G]] when G is a compact p-adic analytic group of rank d. In Remark 3.7, the authors discuss possible strengthening of their main results and alternative approaches in small rank cases.

In particular, they refer to earlier work (Burns–Daoud) where, for G = Zp, additional fine structural properties of Z[[Zp]] were proved: weak Krull dimension 2 and, for most primes p, the (2,2)-domain property while failing to be (1,2) or (2,1) in Costa’s hierarchy.

The authors explicitly state that they do not know how far these finer structural properties extend beyond Z[[Zp]], motivating the open problem of determining their scope across more general completed group algebras Z[[G]].