On a question of Buchweitz about ranks of syzygies of modules of finite length

Abstract:

Let $R$ be a local ring of dimension $d$. Buchweitz asks if the rank of the $d$th syzygy of a module of finite lengh is greater than or equal to the rank of the $d$th syzygy of the residue field, unless the module has finite projective dimension. Assuming that $R$ is Gorenstein, we prove that if the answer is affirmative, then $R$ is a hypersurface. If moreover $R$ has dimension two, then we show that the converse also holds true.