Derived equivalence for cyclic blocks over a \(p\)-adic ring (Q750603)

Let \(p\) be a prime number, \({\mathfrak O}\) a complete discrete valuation ring having an algebraically closed residue field \({\mathfrak k}\) of characteristic \(p\), \(G\) a finite group, \(b\) a block of \(G\) having a non trivial cyclic defect group \(P\) and \(E\) the cyclic \(p'\)-subgroup of \(\Aut(P)\) isomorphic to the inertial quotient of \(b\). Denote by \(P\rtimes E\) the semidirect product of P by E.Suppose that \({\mathfrak O}\) contains all \(|G|\)-th roots of unity. The aim of the paper is to show the following Theorem. The derived categories of \({\mathfrak O}Gb\) and \({\mathfrak O}(P\rtimes E)\) are equivalent. In the case \({\mathfrak O}={\mathfrak k}\) this follows from the work of J. Rickard, where he shows the existence of what he calls a tilting complex, which implies an equivalence of derived categories. The starting point for the proof of the above theorem is the observation that this tilting complex can be lifted canonically to a tilting complex for \({\mathfrak O}Gb\), having an endomorphism algebra isomorphic to \({\mathfrak O}(P\rtimes E)\). This amounts to showing that the basic algebra of \({\mathfrak O}Gb\) is up to isomorphism uniquely determined by the Brauer tree of the block \(b\).