Abstract
In this paper, we compare annihilators of Tor and Ext modules of finitely generated modules over a commutative noetherian ring. For local Cohen–Macaulay rings, one of our results refines a theorem of Dao and Takahashi.
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Notes
These two equalities can also be deduced from the inclusions given in the first part of Proposition 4.2 (1). In fact, letting t = 0 yields \(\mathbb {{T}}_{n}(\mathcal {X},\mathcal {Y})\subseteq \mathbb {{E}}^{d+n}(\mathcal {X},\mathcal {Y}^{\dag })\) and \(\mathbb {{E}}^{d+n}(\mathcal {X},\mathcal {Y})\subseteq \mathbb {{T}}_{n}(\mathcal {X},\mathcal {Y}^{\dag })\). Then replace \(\mathcal {Y}\) with \(\mathcal {Y}^{\dag }\).
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Acknowledgements
The authors sincerely thank the anonymous referee for carefully reading the paper and giving numerous valuable suggestions.
Funding
Takahashi was partly supported by JSPS Grant-in-Aid for Scientific Research 19K03443.
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Dedicated to Professor Nguyen Tu Cuong on the occasion of his seventieth birthday
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Dey, S., Takahashi, R. Comparisons Between Annihilators of Tor and Ext. Acta Math Vietnam 47, 123–139 (2022). https://doi.org/10.1007/s40306-021-00443-0
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DOI: https://doi.org/10.1007/s40306-021-00443-0
Keywords
- Annihilator
- Canonical module
- Cohen–Macaulay ring
- Ext
- Maximal Cohen–Macaulay module
- Cosyzygy
- Non-Gorenstein locus
- Punctured spectrum
- Syzygy
- Tor
- Trace ideal