Ranks of syzygies of perfect modules

Let R be a commutative ring with identity and M be a finitely generated perfect R-module of homological dimension n with a projective resolution F:0→Fn→fnfn−1→·→Fk→fkFk−1→·→F1→1F0→M→0 where rank fk is rk. In this paper, we show that for 1