Lifting to non-integral idempotents (Q5946461)

Let \(G\) be a finite group, \(p\) a prime divisor of the order of \(G\) and \(R\) a complete discrete valuation ring with valuation group \(\mathbb{Z}\). Let \((\pi)\) be the Jacobson radical of \(R\), \(A\) the centre of the group ring \(RG\) and \(J\subset A\) the inverse image of the Jacobson radical of \(A/\pi A\). Suppose that the field of fractions \(\mathbb{K}\) of \(R\) is a splitting field (of characteristic \(0\)) for \(G\) and all its subgroups. Assume moreover that the characteristic of \(F=R/(\pi)\) is \(p\). It is known how to lift the central idempotents of \(FG\) to idempotents of \(A\). In Section 1 the author discusses a way of obtaining all central idempotents of \(\mathbb{K} G\) from the knowledge of certain elements of \(A/\pi^{2t}J\), where \(t\) is a positive integer less than or equal to \(\nu_\pi(|G|)\). In Section 2 another approach to lifting elements is outlined which is related to idempotents in more general rings.