  Let $(R, \m)$ be a commutative Noetherian local ring with $\m^3 =(0)$. We
give a condition for $R$ to have a non-free module of G-dimension zero. We
shall also construct a family of non-isomorphic indecomposable modules of
G-dimension zero with parameters in an open subset of projective space. We
shall finally show that the subcategory consisting of modules of G-dimension
zero over $R$ is not necessarily a contravariantly finite subcategory in the
category of finitely generated $R$-modules.
