A converse of the Hilbert syzygy theorem

Cheng, Charles Ching-an; Shapiro, Jay

doi:10.1090/S0002-9939-1983-0706499-5

A converse of the Hilbert syzygy theorem
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by Charles Ching-an Cheng and Jay Shapiro PDF

Proc. Amer. Math. Soc. 89 (1983), 11-15 Request permission

Abstract:

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

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