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On the Ideal Case of a Conjecture of Huneke and Wiegand

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

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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