Test properties of some Cohen-Macaulay modules and criteria for local rings via finite vanishing of Ext or Tor
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917204375633920 |
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| author | Dey, Souvik Ghosh, Dipankar Saha, Aniruddha |
| author_facet | Dey, Souvik Ghosh, Dipankar Saha, Aniruddha |
| contents | In this article, we show test properties, in the sense of finitely many vanishing of Ext or Tor, of CM (Cohen-Macaulay) modules whose multiplicity and number of generators (resp., type) are related by certain inequalities. We apply these test behaviour, along with other results, to characterize various kinds of local rings, including hypersurface rings of multiplicity at most two, surprisingly requiring only finitely many vanishing of Ext or Tor involving such CM modules. As further applications, we verify the long-standing (Generalized) Auslander-Reiten Conjecture for every CM module of minimal multiplicity over a Noetherian local ring, thus vastly extending a result of Huneke-Şega-Vraciu. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_01636 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Test properties of some Cohen-Macaulay modules and criteria for local rings via finite vanishing of Ext or Tor Dey, Souvik Ghosh, Dipankar Saha, Aniruddha Commutative Algebra 13D07, 13C14, 13H15, 13D05, 13H10 In this article, we show test properties, in the sense of finitely many vanishing of Ext or Tor, of CM (Cohen-Macaulay) modules whose multiplicity and number of generators (resp., type) are related by certain inequalities. We apply these test behaviour, along with other results, to characterize various kinds of local rings, including hypersurface rings of multiplicity at most two, surprisingly requiring only finitely many vanishing of Ext or Tor involving such CM modules. As further applications, we verify the long-standing (Generalized) Auslander-Reiten Conjecture for every CM module of minimal multiplicity over a Noetherian local ring, thus vastly extending a result of Huneke-Şega-Vraciu. |
| title | Test properties of some Cohen-Macaulay modules and criteria for local rings via finite vanishing of Ext or Tor |
| topic | Commutative Algebra 13D07, 13C14, 13H15, 13D05, 13H10 |
| url | https://arxiv.org/abs/2412.01636 |