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On the second rigidity theorem of Huneke and Wiegand


Authors: Olgur Celikbas and Ryo Takahashi
Journal: Proc. Amer. Math. Soc. 147 (2019), 2733-2739
MSC (2010): Primary 13D07; Secondary 13C13, 13C14, 13H10
DOI: https://doi.org/10.1090/proc/14564
Published electronically: April 9, 2019
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Abstract: In 2007 Huneke and Wiegand announced in an erratum that one of the conclusions of their depth formula theorem is flawed due to an incorrect convention for the depth of the zero module. Since then, the deleted claim has remained unresolved. In this paper we give examples to prove that the deleted claim is false, in general. Moreover, we point out several places in the literature which relied upon this deleted claim or the initial argument from 2007.


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Additional Information

Olgur Celikbas
Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310
Email: olgur.celikbas@math.wvu.edu

Ryo Takahashi
Affiliation: Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan; Department of Mathematics, University of Kansas, Snow Hall, 1460 Jayhawk Blvd., Lawrence, Kansas 66045-7523
Email: takahashi@math.nagoya-u.ac.jp

DOI: https://doi.org/10.1090/proc/14564
Keywords: Serre's condition, second rigidity theorem, tensor products of modules, reflexivity, Tor-rigidity
Received by editor(s): May 26, 2018
Published electronically: April 9, 2019
Additional Notes: The second author was partially supported by JSPS Grant-in-Aid for Scientific Research 16K05098 and 16KK0099.
Dedicated: Dedicated to the memory of Ragnar-Olaf Buchweitz
Communicated by: Claudia Polini
Article copyright: © Copyright 2019 American Mathematical Society