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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


A converse of the Hilbert syzygy theorem

Authors: Charles Ching-an Cheng and Jay Shapiro
Journal: Proc. Amer. Math. Soc. 89 (1983), 11-15
MSC: Primary 13D05; Secondary 20M10
MathSciNet review: 706499
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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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PII: S 0002-9939(1983)0706499-5
Keywords: Monoid, cohomological dimension, global dimension, Hochschild dimension
Article copyright: © Copyright 1983 American Mathematical Society