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Rigid and Test Modules

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Extended Abstracts Spring 2015

Part of the book series: Trends in Mathematics ((RPCRMB,volume 5))

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Abstract

We discuss a class of modules, called test modules, that are, in principle, those detecting finite homological dimensions. The main purpose of this mote is to report the following result: if a commutative Noetherian local ring \((R, {\mathfrak m}, k)\) admits a test module (e.g., the residue field k) of finite Gorenstein dimension, then R is Gorenstein. This extended abstract is based on joint works with Celikbas et al. (Kyoto J Math 54:295–310, 2014) [9], Celikbas et al. (Homological dimensions of rigid modules) [10], and Celikbas and Wagstaff (Testing for the Gorenstein property) [11].

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Acknowledgements

We would like to thank Ryo Takahashi for his valuable comments and for explaining us Example 17.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    Olgur Celikbas

Authors
  1. Olgur Celikbas
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Correspondence to Olgur Celikbas .

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Editors and Affiliations

  1. Universitat Autònoma de Barcelona, Departament de Matemàtiques Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain

    Dolors Herbera

  2. Universitat Autònoma de Barcelona, Departament de Matemàtiques Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain

    Wolfgang Pitsch

  3. Universitat de Barcelona, Departament de Matemàtiques i Informàtica, Barcelona, Barcelona, Spain

    Santiago Zarzuela

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Celikbas, O. (2016). Rigid and Test Modules. In: Herbera, D., Pitsch, W., Zarzuela, S. (eds) Extended Abstracts Spring 2015. Trends in Mathematics(), vol 5. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-45441-2_7

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