Lifting endo-trivial modules (Q2705004)

Let \(R\) be a complete discrete valuation ring of characteristic 0 with residue class field \(k\) of prime characteristic \(p\), and let \(P\) be a finite \(p\)-group. An \(RP\)-lattice \(L\) is called endo-trivial if the \(RP\)-lattice \(\text{End}_R(L)\) is the direct sum of the trivial \(RP\)-lattice \(R\) and a free \(RP\)-lattice. Endo-trivial \(kP\)-modules are defined in a similar way. The author shows that any endo-trivial \(kP\)-module \(M\) lifts to an endo-trivial \(RP\)-lattice \(L\) (i.e. \(k\otimes_RL\cong M\)). This solves an important special case of the problem whether any endo-permutation \(kP\)-module lifts to an endo-permutation \(RP\)-lattice.