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On the Ideal Case of a Conjecture of Huneke and Wiegand

Part of: Commutative algebra: Homological methods Theory of modules and ideals

Published online by Cambridge University Press:  11 February 2019

Olgur Celikbas ,
Shiro Goto ,
Ryo Takahashi  and
Naoki Taniguchi
Olgur Celikbas
Affiliation:
Department of Mathematics, West Virginia University, Morgantown, WV 26506USA (olgur.celikbas@math.wvu.edu)
Shiro Goto
Affiliation:
Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan (shirogoto@gmail.com)
Ryo Takahashi
Affiliation:
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan (takahashi@math.nagoya-u.ac.jp)
Naoki Taniguchi*
Affiliation:
Global Education Center, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, Japan (naoki.taniguchi@aoni.waseda.jp)
*
*Corresponding author
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Abstract

A conjecture of Huneke and Wiegand claims that, over one-dimensional commutative Noetherian local domains, the tensor product of a finitely generated, non-free, torsion-free module with its algebraic dual always has torsion. Building on a beautiful result of Corso, Huneke, Katz and Vasconcelos, we prove that the conjecture is affirmative for a large class of ideals over arbitrary one-dimensional local domains. Furthermore, we study a higher-dimensional analogue of the conjecture for integrally closed ideals over Noetherian rings that are not necessarily local. We also consider a related question on the conjecture and give an affirmative answer for first syzygies of maximal Cohen–Macaulay modules.

Keywords

integrally closed idealsweakly m-full idealstorsion in tensor products of modules

MSC classification

Primary: 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13C14: Cohen-Macaulay modules 13C15: Dimension theory, depth, related rings (catenary, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 847 - 859
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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References

1Auslander, M., Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647.Google Scholar
2Auslander, M. and Bridger, M., Stable module theory, Memoirs of the American Mathematical Society, Volume 94 (American Mathematical Society, Providence, RI, 1969).Google Scholar
3Bruns, W. and Herzog, J., Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, Volume 39 (Cambridge University Press, Cambridge, 1993).Google Scholar
4Celikbas, O. and Takahashi, R., On the second rigidity theorem of Huneke and Wiegand, preprint (arXiv:1805.02204, 2018).Google Scholar
5Celikbas, O. and Wiegand, R., Vanishing of Tor, and why we care about it, J. Pure Appl. Algebra 219(3) (2015), 429–448.Google Scholar
6Celikbas, O., Gheibi, M., Zargar, M. R. and Sadeghi, A., Homological dimensions of rigid modules, Kyoto J. Math. 58(2018), 639–669.Google Scholar
7Celikbas, O., Iima, K.-I., Sadeghi, A. and Takahashi, R., On the ideal case of a conjecture of Auslander and Reiten, Bull. Sci. Math. 142 (2018), 94–107.Google Scholar
8Constapel, P., Vanishing of Tor and torsion in tensor products, Commun. Algebra 24(3) (1996), 833–846.Google Scholar
9Corso, A. and Polini, C., Links of prime ideals and their Rees algebras, J. Algebra 178(1) (1995), 224–238.Google Scholar
10Corso, A., Huneke, C., Katz, D. and Vasconcelos, W. V., Integral closure of ideals and annihilators of homology, Commutative Algebra, Lecture Notes in Pure and Applied Mathematics, Volume 244 (Chapman & Hall/CRC, Boca Raton, FL, 2006), pp. 33–48.Google Scholar
11Dao, H., Eghbali, M. and Lyle, J., Hom and Ext, revisited, preprint (arxiv:1710.05123, 2017).Google Scholar
12Dao, H., Li, J. and Miller, C., On the (non)rigidity of the Frobenius endomorphism over Gorenstein rings, Algebra Number Theory 4(8) (2010), 1039–1053.Google Scholar
13Eisenbud, D., Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260(1) (1980), 35–64.Google Scholar
14Goto, S., Integral closedness of complete-intersection ideals, J. Algebra 108 (1987), 151–160.Google Scholar
15Goto, S. and Takahashi, R., On the Auslander–Reiten conjecture for Cohen–Macaulay local rings, Proc. Am. Math. Soc. 145(8) (2017), 3289–3296.Google Scholar
16Goto, S., Takahashi, R., Taniguchi, N. and Le Truong, H., Huneke–Wiegand conjecture and change of rings, J. Algebra 422 (2015), 33–52.Google Scholar
17Grayson, D. R. and Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, Available at https://faculty.math.illinois.edu/Macaulay2/.Google Scholar
18Herzinger, K. D., Minimal generating sets of ideals and torsion in I ⊗R I −1, PhD thesis, University of Nebraska at Lincoln (1996).Google Scholar
19Huneke, C. and Wiegand, R., Tensor products of modules and the rigidity of Tor, Math. Ann. 299(3) (1994), 449–476.Google Scholar
20Matsuoka, T., On the torsion submodule of a module of type (F 1), J. Sci. Hiroshima Univ. AI 31(2) (1967), 151–160.Google Scholar
21Sally, J. D., On the associated graded ring of a local Cohen–Macaulay ring, J. Math. Kyoto Univ. 17(1) (1977), 19–21.Google Scholar
22Watanabe, J., m-full ideals, Nagoya Math. J. 106 (1987), 101–111.Google Scholar