Lifting theorems for tilting complexes (Q1178924)

Green showed that a module \(M\) for a group algebra \(kG\) can be lifted to an \({\mathcal O}G\)-lattice, where \(\mathcal O\) is a complete discrete valuation ring with residue field \(k\), if \(\text{Ext}^ 2_{kG}(M,M)\) vanishes. When we consider equivalences between derived categories of module categories, we are led to consider tilting complexes, which are objects \(P^*\) of the derived category for which, among other conditions, \(\text{Hom}(P^*,P^*[i])\) vanishes for \(i \neq 0\). The case \(i = 2\) gives a condition analogous to Green's, and we may ask whether his theorem has an analogy for tilting complexes. The aim of this paper is to show that it does; in fact we shall consider more general situations where we have a quotient ring \(\overline{R}\) of a ring \(R\) and a tilting complex for \(\overline{R}\) that can be lifted to a tilting complex for \(R\).