  Let R be a commutative noetherian local ring and consider the set of
isomorphism classes of indecomposable totally reflexive R-modules. We prove
that if this set is finite, then either it has exactly one element, represented
by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R
is complete, then it is even a simple hypersurface singularity). The crux of
our proof is to argue that if the residue field has a totally reflexive cover,
then R is Gorenstein or every totally reflexive R-module is free.
