On the structure of Witt-Burnside rings attached to pro-\(p\) groups (Q2339191)

Let \(G\) be a pro-\(p\) group for some prime \(p\). Then \textit{A. W. M. Dress} and \textit{C. Siebeneicher} [Adv. Math. 70, No. 1, 87--132 (1988; Zbl 0691.13026)] introduced the ring-valued functor \(\mathbf{W}_{G}.\;\)In the case where \(G=\mathbb{Z}_{p}\) it recovers the the \(p\)-typical Witt vectors; and in the case \(G=\mathbb{\hat{Z}}\) it gives the ``big'' Witt vector construction. The rings in the image of \(\mathbf{W}_{G}\) are called Witt-Burnside rings, and while the structure of the Witt-Burnside rings in the \(G=\mathbb{Z}_{p}\) have been studied, little is known about these rings in general. Now let \(G\) be an infinite pro-\(p\) group and \(k\) a field of characteristic \(p\). In the work under review, the author studies the Witt-Burnside rings which arise in this case, highlighting their ``pathological'' nature. It is shown that \(\mathbf{W}_{G}\left( A\right) \) is local for any local ring \(A\) of characteristic \(p;\) in particular \(\mathbf{W}_{G}\left( k\right) \) is local. However, for \(d\geq2,\) the ring \(\mathbf{W}_{\mathbb{Z}_{p}^{d}}\left( k\right) \) is shown to be not Noetherian due to its maximal ideal not being finitely generated, a consequence of the fact that \(\mathbb{Z}_{p}^{d}\) has more than one maximal ideal; the non-Noetherian property is in stark contrast to the classical case \(G=\mathbb{Z}_{p}\) (i.e., \(d=1\)). Additionally, it is shown that \(\mathbf{W}_{\mathbb{Z}_{p}^{2}}\left( k\right) \) is reduced; the proof does not hold for \(d>2,\) though a definitive answer as to whether the result holds for \(d>2\) is not given.