A converse of the Hilbert syzygy theorem

Abstract

The following converse of the Hilbert Syzygy Theorem is proved. Suppose K K is a noetherian commutative ring with identity that has finite global dimension, and suppose that M M is a finitely generated abelian cancellative monoid. If gl dim ⁡ K M = n + gl dim ⁡ K {\text {gl}}\dim KM = n + {\text {gl}}\dim K then M M is of the form ( × i = 1 n M i ) × H ( \times _{i = 1}^n{M_i}) \times H where M i ≅ Z {M_i} \cong {\mathbf {Z}} or N {\mathbf {N}} and where H H is a finite group with no K K -torsion.