Ranks of syzygies of perfect modules

AbstractLet 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<k<n, the ideals Ir−1(fk),Ir−1(fk) and I(fk) have the same radical as the annihilator of M, provided either n is even or M is Gorenstein and n≡1mod4. Thus we show that the rank of the kth syzygy of M,rk≥3, for 1<k