 
Summary: Lifting Endotrivial Modules
J.L. Alperin
Endopermutation modules arise in a number of ways in the representation theory of
nite groups, in particular in connection with equivalences between blocks and as sources of
simple modules. Great progress has been made recently towards a complete classication of
such modules in characteristic p. The question of lifting such modules to valuation rings has
been open now for some time and is now quite relevant to the classication of such modules
over such rings. In this paper, we solve the problem in the key case of endotrivial modules.
Let R be a complete discrete valuation ring of characteristic zero with maximal ideal
J and residue class eld k of prime characteristic p. For a xed nite pgroup P , all the
kPmodules considered will be nite dimensional and all the RPmodules will be nitely
generated and free as Rmodules. An RPmodule M (and similarly for kPmodules) is an
endotrivial module if the tensor product M
M is the direct sum of the trivial RPmodule
R and a free RGmodule. The standard results of the theory of such modules may be found
in the references [4, 5, 8].
In order to state the result of this paper, recall that an RPmodule M lifts a kPmodule
U if M=JM is isomorphic with U .
Theorem. Any endotrivial kPmodule lifts to an endotrivial RPmodule.
