On the Picard group of the integer group ring of the cyclic \(p\)-group and of rings close to it (Q2785951)

Let \(p\) be an odd prime number, \(C_n\) the cyclic group of order \(p^n\), \(\xi_n\) a primitive \(p^n\)-th root of unity. The main result of the paper is that \(C\ell (\mathbb{Z} [\xi_n])\) is a direct summand of \(\text{Pic} (\mathbb{Z}[C_n])\). More precisely, the Mayer-Vietoris exact sequence provides a natural epimorphism \(\text{Pic} (\mathbb{Z}[C_n]) \to C\ell (\mathbb{Z} [\xi_n]) \oplus\text{Pic} (\mathbb{Z} [C_{n-1}])\), and the author constructs a splitting \(C\ell (\mathbb{Z} [\xi_n])\to \text{Pic} (\mathbb{Z} [C_n])\).NEWLINENEWLINENEWLINEA natural question arises, whether there is a splitting \(C\ell (\mathbb{Z} [\xi_{n-1}]) \to\text{Pic} (\mathbb{Z}[C_n])\) or not. According to the author, the answer is no in general: if the prime \(p\) is properly irregular, the \(p\)-part of \(C\ell (\mathbb{Z} [\xi_{n-1}])\) is an obstruction for the splitting (see a forthcoming paper).NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].