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arXiv · 1705.06563

Totally Reflexive Modules and Poincar\'{e} Series

Mohsen Gheibi·Ryo Takahashi
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Abstract

We study Cohen-Macaulay non-Gorenstein local rings $(R,\mathfrak{m},k)$ admitting certain totally reflexive modules. More precisely, we give a description of the Poincar\'{e} series of $k$ by using the Poincar\'{e} series of a non-zero totally reflexive module with minimal multiplicity. Our results generalize a result of Yoshino to higher-dimensional Cohen-Macaulay local rings. Moreover, from a quasi-Gorenstein ideal satisfying some conditions, we construct a family of non-isomorphic indecomposable totally reflexive modules having an arbitrarily large minimal number of generators.

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Mohsen Gheibi, Ryo Takahashi. 2017-05-18. Totally Reflexive Modules and Poincar\'{e} Series. https://arxiv.org/abs/1705.06563

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