Title: 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 question is
affrmative, then R is a hypersurface. If moreover R has dimension two, then we
show that the converse also holds true.