By the complete group algebra of $\mathbb{Z}^2$, I mean the inverse limit of normal group algebras: $$\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}^2]] := \varprojlim_{n,K}\mathbb{Z}/n[\mathbb{Z}^2/K]$$ as $n\rightarrow\infty$ and $K$ ranges over subgroups of $\mathbb{Z}^2$.
It's well known that $$\mathbb{Z}_p[[\mathbb{Z}_p^2]] := \varprojlim_{n,}\mathbb{Z}/p^n[(\mathbb{Z}/p^n)^2] = \varprojlim_n\mathbb{Z}/p^n[x,y]/(x^{p^n}-1,y^{p^n}-1)$$ is isomorphic to the ring of formal power series $\mathbb{Z}_p[[s,t]]$, the isomorphism given by sending $s\mapsto x-1$ and $t\mapsto y-1$.
The proof in Serre's Galois Cohomology uses the analogous result for the complete group algebra of a free group, which seems to use essential properties of pro-$p$ groups, and hence at first I thought that the analogous comparison: $$\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}^2]]\quad??\cong?? \quad\widehat{\mathbb{Z}}[[s,t]]$$ would not hold. On the other hand, any commutative profinite ring is a product of local rings, $\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}^2]]$ certainly admits $\mathbb{Z}_p$ as quotients, and so one might hope that $$\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}^2]] \cong \prod_p\mathbb{Z}_p[[s,t]]$$
My questions are:
- What are the local direct factors of the commutative profinite ring $\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}^2]]$?
- If $a,b$ are generators of $\widehat{\mathbb{Z}}^2$, then are $a-1,b-1$ prime elements of $\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}^2]]$?