A conductor formula for completed group algebras.
Let \(\mathfrak o\) be the ring of integers of a finite extension \(K\) of \(\mathbb Q_p\), let \(\mathfrak o[G]\) be the group ring of a finite group \(G\) over \(\mathfrak o\); then \(\mathcal F(\mathfrak o[G])=\{x\in\zeta(\mathfrak o[G]): x\Gamma\subseteq\mathfrak o[G]\}\) is called the central conductor, where \(\Gamma\subseteq K[G]\) is a maximal \(\mathfrak o\)-order containing \(\mathfrak o[G]\) and \(\zeta(\Lambda)\) denotes the center of a ring \(\Lambda\). \textit{H. Jacobinski} [Mich. Math. J. 13, 471-475 (1966; Zbl 0143.05702)] gave a complete description of \(\mathcal F(\mathfrak o[G])\) in terms of characters of \(G\) and he showed that \(\mathcal F(\mathfrak o[G])\) annihilates some Ext groups.NEWLINENEWLINE The aim of the paper under review is to extend Jacobinski's results to completed group algebras \(\mathfrak oG\) when \(G\) is a \(p\)-adic Lie group of dimension \(1\) (\(p\) is an odd prime number). In fact such a group \(G\) can be written as \(G=H\rtimes\Gamma\) with \(H\) finite and \(\Gamma\cong\mathbb Z_p\). The main result for \(\mathcal F(\mathfrak oG)\) takes the form of an inclusion, which is shown to be an equality under certain additional hypotheses or after localizations at prime ideals not containing \(p\). However it is shown by an example that equality does not hold in general. In the last section the author derives some consequences involving Ext groups, in analogy to the group ring case, though it is shown that in this setting it is not true that the central conductor annihilates the considered Ext groups. Some applications to non-commutative Fitting invariants are also described.
scientific article; zbMATH DE number 6358804